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Graduate Texts in Mathematics239
Editorial Board
S. Axler
K.A. Ribet
Cohen I FM.qxd 16/4/07 11:17 AM Page i

Graduate Texts in Mathematics
1TAKEUTI/ZARING. Introduction to
Axiomatic Set Theory. 2nd ed.
2O
XTOBY. Measure and Category. 2nd ed.
3S
CHAEFER. Topological Vector Spaces.
2nd ed.
4H
ILTON/STAMMBACH. A Course in
Homological Algebra. 2nd ed.
5M
ACLANE. Categories for the Working
Mathematician. 2nd ed.
6H
UGHES/PIPER. Projective Planes.
7 J.-P. S
ERRE. A Course in Arithmetic.
8T
AKEUTI/ZARING. Axiomatic Set Theory.
9H
UMPHREYS. Introduction to Lie
Algebras and Representation Theory.
10 C
OHEN. A Course in Simple Homotopy
Theory.
11 C
ONWAY. Functions of One Complex
Variable I. 2nd ed.
12 B
EALS. Advanced Mathematical Analysis.
13 A
NDERSON/FULLER. Rings and
Categories of Modules. 2nd ed.
14 G
OLUBITSKY/GUILLEMIN. Stable
Mappings and Their Singularities.
15 B
ERBERIAN. Lectures in Functional
Analysis and Operator Theory.
16 W
INTER. The Structure of Fields.
17 R
OSENBLATT. Random Processes. 2nd ed.
18 H
ALMOS. Measure Theory.
19 H
ALMOS. A Hilbert Space Problem
Book. 2nd ed.
20 H
USEMOLLER. Fibre Bundles. 3rd ed.
21 H
UMPHREYS. Linear Algebraic Groups.
22 B
ARNES/MACK. An Algebraic
Introduction to Mathematical Logic.
23 G
REUB. Linear Algebra. 4th ed.
24 H
OLMES. Geometric Functional
Analysis and Its Applications.
25 H
EWITT/STROMBERG. Real and Abstract
Analysis.
26 M
ANES. Algebraic Theories.
27 K
ELLEY. General Topology.
28 Z
ARISKI/SAMUEL. Commutative
Algebra. Vol. I.
29 Z
ARISKI/SAMUEL. Commutative
Algebra. Vol. II.
30 J
ACOBSON. Lectures in Abstract Algebra
I. Basic Concepts.
31 J
ACOBSON. Lectures in Abstract Algebra
II. Linear Algebra.
32 J
ACOBSON. Lectures in Abstract Algebra
III. Theory of Fields and Galois
Theory.
33 H
IRSCH. Differential Topology.
34 S
PITZER. Principles of Random Walk.
2nd ed.
35 A
LEXANDER/WERMER. Several Complex
Variables and Banach Algebras. 3rd ed.
36 K
ELLEY/NAMIOKAet al. Linear
Topological Spaces.
37 M
ONK. Mathematical Logic.
38 G
RAUERT/FRITZSCHE. Several Complex
Variables.
39 A
RVESON. An Invitation to C
*
-Algebras.
40 K
EMENY/SNELL/KNAPP. Denumerable
Markov Chains. 2nd ed.
41 A
POSTOL. Modular Functions and
Dirichlet Series in Number Theory.
2nd ed.
42 J.-P. S
ERRE. Linear Representations of
Finite Groups.
43 G
ILLMAN/JERISON. Rings of
Continuous Functions.
44 K
ENDIG. Elementary Algebraic
Geometry.
45 L
OÈVE. Probability Theory I. 4th ed.
46 L
OÈVE. Probability Theory II. 4th ed.
47 M
OISE. Geometric Topology in
Dimensions 2 and 3.
48 S
ACHS/WU. General Relativity for
Mathematicians.
49 G
RUENBERG/WEIR. Linear Geometry.
2nd ed.
50 E
DWARDS. Fermat's Last Theorem.
51 K
LINGENBERG. A Course in Differential
Geometry.
52 H
ARTSHORNE. Algebraic Geometry.
53 M
ANIN. A Course in Mathematical Logic.
54 G
RAVER/WATKINS. Combinatorics with
Emphasis on the Theory of Graphs.
55 B
ROW N/PEARCY. Introduction to
Operator Theory I: Elements of
Functional Analysis.
56 M
ASSEY. Algebraic Topology: An
Introduction.
57 C
ROWELL/FOX. Introduction to Knot
Theory.
58 K
OBLITZ.p-adic Numbers,p-adic
Analysis, and Zeta-Functions. 2nd ed.
59 L
ANG. Cyclotomic Fields.
60 A
RNOLD. Mathematical Methods in
Classical Mechanics. 2nd ed.
61 W
HITEHEAD. Elements of Homotopy
Theory.
62 K
ARGAPOLOV/MERIZJAKOV.
Fundamentals of the Theory of Groups.
63 B
OLLOBAS. Graph Theory.
(continued after index)
Cohen I FM.qxd 16/4/07 11:17 AM Page ii

Henri Cohen
Number Theory
Volume I:
Tools and Diophantine
Equations
Cohen I FM.qxd 16/4/07 11:17 AM Page iii

Henri Cohen
Université Bordeaux I
Institut de Mathématiques de Bordeaux
351, cours de la Libération
33405, Talence cedex
France
[email protected]
Editorial Board
S. Axler K.A. Ribet
Mathematics Department Mathematics Department
San Francisco State University University of California at Berkeley
San Francisco, CA 94132 Berkeley, CA 94720-3840
USA USA
[email protected] [email protected]
Mathematics Subject Classification (2000): 11-xx 11-01 11Dxx 11Rxx 11Sxx
Library of Congress Control Number: 2007925736
ISBN-13: 978-0-387-49922-2 eISBN-13: 978-0-387-49923-9
Printed on acid-free paper.
© 2007 Springer Science+Business Media, LLC
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street,
New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly
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Cohen I FM.qxd 16/4/07 11:17 AM Page iv

Preface
This book deals with several aspects of what is now called “explicit number
theory,” not including the essential algorithmic aspects, which are for the
most part covered by two other books of the author [Coh0] and [Coh1]. The
central (although not unique) theme is the solution of Diophantine equa-
tions, i.e., equations or systems of polynomial equations that must be solved
in integers, rational numbers, or more generally in algebraic numbers. This
theme is in particular the central motivation for the modern theory of arith-
metic algebraic geometry. We will consider it through three of its most basic
aspects.
The ﬁrst is thelocalaspect: the invention ofp-adic numbers and their
generalizations by K. Hensel was a major breakthrough, enabling in particular
the simultaneous treatment of congruences modulo prime powers. But more
importantly, one can doanalysisinp-adic ﬁelds, and this goes much further
than the simple deﬁnition ofp-adic numbers. The local study of equations
is usually not very diﬃcult. We start by looking at solutions inﬁnite ﬁelds,
where important theorems such as the Weil bounds and Deligne’s theorem
on the Weil conjectures come into play. We thenliftthese solutions to local
solutions usingHensel lifting.
The second aspect is theglobalaspect: the use of number ﬁelds, and
in particular of class groups and unit groups. Although local considerations
can give a considerable amount of information on Diophantine problems,
the “local-to-global” principles are unfortunately rather rare, and we will
see many examples of failure. Concerning the global aspect, we will ﬁrst
require as a prerequisite of the reader that he or she be familiar with the
standard basic theory of number ﬁelds, up to and including the ﬁniteness of
the class group and Dirichlet’s structure theorem for the unit group. This can
be found in many textbooks such as [Sam] and [Marc]. Second, and this is
less standard, we will always assume that we have at our disposal a computer
algebra system (CAS) that is able to compute rings of integers, class and unit
groups, generators of principal ideals, and related objects. Such CAS are now
very common, for instanceKash, magma,andPari/GP, to cite the most useful
in algebraic number theory.

vi Preface
The third aspect is the theory of zeta andL-functions. This can be consid-
ered aunifying theme
1
for the whole subject, and it embodies in a beautiful
way the local and global aspects of Diophantine problems. Indeed, these func-
tions are deﬁned through the local aspects of the problems, but their analytic
behavior is intimately linked to the global aspects. A ﬁrst example is given by
the Dedekind zeta function of a number ﬁeld, which is deﬁned only through
the splitting behavior of the primes, but whose leading term ats= 0 contains
at the same time explicit information on the unit rank, the class number, the
regulator, and the number of roots of unity of the number ﬁeld. A second
very important example, which is one of the most beautiful and important
conjectures in the whole of number theory (and perhaps of the whole of math-
ematics), the Birch and Swinnerton-Dyer conjecture, says that the behavior
ats= 1 of theL-function of an elliptic curve deﬁned overQcontains at the
same time explicit information on the rank of the group of rational points
on the curve, on the regulator, and on the order of the torsion group of the
group of rational points, in complete analogy with the case of the Dedekind
zeta function. In addition to the purelyanalyticalproblems, the theory of
L-functions contains beautiful results (and conjectures) onspecial values,of
which Euler’s formula
α
nα1
1/n
2
=π
2
/6 is a special case.
This book can be considered as having four main parts. The ﬁrst part gives
the tools necessary for Diophantine problems: equations over ﬁnite ﬁelds,
number ﬁelds, and ﬁnally local ﬁelds such asp-adic ﬁelds (Chapters 1, 2, 3,
4, and part of Chapter 5). The emphasis will be mainly on the theory of
p-adic ﬁelds (Chapter 4), since the reader probably has less familiarity with
these. Note that we will consider function ﬁelds only in Chapter 7, as a tool
for proving Hasse’s theorem on elliptic curves. An important tool that we will
introduce at the end of Chapter 3 is the theory of the Stickelberger ideal over
cyclotomic ﬁelds, together with the important applications to the Eisenstein
reciprocity law, and the Davenport–Hasse relations. Through Eisenstein reci-
procity this theory will enable us to prove Wieferich’s criterion for the ﬁrst
case of Fermat’s last theorem (FLT), and it will also be an essential tool in
the proof of Catalan’s conjecture given in Chapter 16.
The second part is a study of certain basic Diophantine equations or
systems of equations (Chapters 5, 6, 7, and 8). It should be stressed that
even though a number of general techniques are available, each Diophantine
equation poses a new problem, and it is diﬃcult to know in advance whether
it will be easy to solve. Even without mentioningfamiliesof Diophantine
equations such as FLT, the congruent number problem, or Catalan’s equation,
all of which will be stated below, proving for instance that a speciﬁc equation
such asx
3
+y
5
=z
7
withx,ycoprime integers has no solution withxyzα=0
seems presently out of reach, although it has been proved (based on a deep
theorem of Faltings) that there are only ﬁnitely many solutions; see [Dar-Gra]
1
Expression due to Don Zagier.

Preface vii
and Chapter 14. Note also that it has been shown by Yu. Matiyasevich (after
a considerable amount of work by other authors) in answer to Hilbert’s tenth
problem that there cannot exist a general algorithm for solving Diophantine
equations.
The third part (Chapters 9, 10, and 11) deals with the detailed study
of analytic objects linked to algebraic number theory: Bernoulli polynomi-
als and numbers, the gamma function, and zeta andL-functions of Dirichlet
characters, which are the simplest types ofL-functions. In Chapter 11 we
also studyp-adic analogues of the gamma, zeta, andL-functions, which have
come to play an important role in number theory, and in particular the Gross–
Koblitz formula for Morita’sp-adic gamma function. In particular, we will
see that this formula leads to remarkably simple proofs of Stickelberger’s con-
gruence and the Hasse–Davenport product relation. More generalL-functions
such as HeckeL-functions for Gr¨ossencharacters, ArtinL-functions for Galois
representations, orL-functions attached to modular forms, elliptic curves, or
higher-dimensional objects are mentioned in several places, but a systematic
exposition of their properties would be beyond the scope of this book.
Much more sophisticated techniques have been brought to bear on the
subject of Diophantine equations, and it is impossible to be exhaustive. Be-
cause the author is not an expert in most of these techniques, they are not
studied in the ﬁrst three parts of the book. However, considering their impor-
tance, I have asked a number of much more knowledgeable people to write
a few chapters on these techniques, and I have written two myself, and this
forms the fourth and last part of the book (Chapters 12 to 16). These chap-
ters have a diﬀerent ﬂavor from the rest of the book: they are in general not
self-contained, are of a higher mathematical sophistication than the rest, and
usually have no exercises. Chapter 12, written by Yann Bugeaud, Guillaume
Hanrot, and Maurice Mignotte, deals with the applications of Baker’s explicit
results on linear forms in logarithms of algebraic numbers, which permit the
solution of a large class of Diophantine equations such as Thue equations
and norm form equations, and includes some recent spectacular successes.
Paradoxically, the similar problems on elliptic curves are considerably less
technical, and are studied in detail in Section 8.7. Chapter 13, written by
Sylvain Duquesne, deals with the search for rational points on curves of genus
greater than or equal to 2, restricting for simplicity to the case of hyperelliptic
curves of genus 2 (the case of genus 0—in other words, of quadratic forms—is
treated in Chapters 5 and 6, and the case of genus 1, essentially of elliptic
curves, is treated in Chapters 7 and 8). Chapter 14, written by the author,
deals with the so-called super-Fermat equationx
p
+y
q
=z
r
, on which several
methods have been used, including ordinary algebraic number theory, classi-
cal invariant theory, rational points on higher genus curves, and Ribet–Wiles
type methods. The only proofs that are included are those coming from alge-
braic number theory. Chapter 15, written by Samir Siksek, deals with the use
of Galois representations, and in particular of Ribet’s level-lowering theorem

viii Preface
and Wiles’s and Taylor–Wiles’s theorem proving the modularity conjecture.
The main application is to equations of “abc” type, in other words, equations
of the forma+b+c= 0 witha,b,andchighly composite, the “easiest”
application of this method being the proof of FLT. The author of this chapter
has tried to hide all the sophisticated mathematics and to present the method
as a black box that can be used without completely understanding the un-
derlying theory. Finally, Chapter 16, also written by the author, gives the
complete proof of Catalan’s conjecture by P. Mih˘ailescu. It is entirely based
on notes of Yu. Bilu, R. Schoof, and especially of J. Bo´echat and M. Mischler,
and the only reason that it is not self-contained is that it will be necessary to
assume the validity of an important theorem of F. Thaine on the annihilator
of the plus part of the class group of cyclotomic ﬁelds.
Warnings
Since mathematical conventions and notation are not the same from one
mathematical culture to the next, I have decided to use systematically un-
ambiguous terminology, and when the notations clash, the French notation.
Here are the most important:
–We will systematically say thatais strictly greater thanb, or greater than
or equal tob(orbis strictly less thana, or less than or equal toa), although
the English terminologyais greater thanbmeans in fact one of the two
(I don’t remember which one, and that is one of the main reasons I refuse
to use it) and the French terminology means the other. Similarly, positive
and negative are ambiguous (does it include the number 0)? Even though
the expression “x is nonnegative” is slightly ambiguous, it is useful, and I
willallow myself to use it, with the meaningxﬀ0.
–Although we will almost never deal with noncommutative ﬁelds (which is
a contradiction in terms since in principle the word ﬁeld implies commu-
tativity), we will usually not use the word ﬁeld alone. Either we will write
explicitly commutative (or noncommutative) ﬁeld, or we will deal with spe-
ciﬁc classes of ﬁelds, such as ﬁnite ﬁelds,p-adic ﬁelds, local ﬁelds, number
ﬁelds, etc., for which commutativity is clear. Note that the “proper” way
in English-language texts to talk about noncommutative ﬁelds is to call
them either skew ﬁelds or division algebras. In any case this will not be an
issue since the only appearances of skew ﬁelds will be in Chapter 2, where
we will prove that ﬁnite division algebras are commutative, and in Chapter
7 about endomorphism rings of elliptic curves over ﬁnite ﬁelds.
–The GCD (respectively the LCM) of two integers can be denoted by (a, b)
(respectively by [a, b]), but to avoid ambiguities, I will systematically use
the explicit notation gcd(a, b) (respectively lcm(a, b)), and similarly when
more than two integers are involved.

Preface ix
–An open interval with endpointsaandbis denoted by (a, b) in the En-
glish literature, and by ]a, b[ in the French literature. I will use the French
notation, and similarly for half-open intervals (a, b] and [a, b), which I will
denote by ]a, b] and [a, b[. Although it is impossible to change such a well-
entrenched notation, I urge my English-speaking readers to realize the
dreadful ambiguity of the notation (a, b), which can mean either the or-
dered pair (a, b), the GCD ofaandb, the inner product ofaandb,orthe
open interval.
–The trigonometric functions sec(x) and csc(x) do not exist in France, so
I will not use them. The functions tan(x), cot(x), cosh(x), sinh( x), and
tanh(x) are denoted respectively by tg(x), cotg(x), ch( x), sh(x), and th( x)
in France, but for once to bow to the majority I will use the English names.
–Θ(s)andΩ( s) denote the real and imaginary parts of the complex number
s, the typography coming from the standard TEX macros.
Notation
In addition to the standard notation of number theory we will use the fol-
lowing notation.
–We will often use the practical self-explanatory notationZ
>0,ZΛ0,Z<0,
Z
Θ0, and generalizations thereof, which avoid using excessive verbiage. On
the other hand, I prefer not to use the notationN(forZ
Λ0,orisitZ >0?).
–Ifaandbare nonzero integers, we write gcd(a, b
∞
) for the limit of the
ultimately constant sequence gcd(a, b
n
)asn→∞.Wehaveofcourse
gcd(a, b
∞
)=
Θ
p|gcd(a,b)
p
vp(a)
,anda/gcd(a, b
∞
) is the largest divisor ofa
coprime tob.
–Ifnis a nonzero integer andd|n, we writedψnif gcd(d, n/d) = 1. Note
that this isnotthe same thing as the conditiond
2
Λn, except ifdis prime.
–Ifx∈R, we denote byλx the largest integer less than or equal tox(the
ﬂoorofx), byxthe smallest integer greater than or equal tox(theceiling
ofx, which is equal toλx + 1 if and only ifx/∈Z), and byλxthe nearest
integer tox(or one of the two ifx∈1/2+Z), so that λx=λx+1/2 .
We also set{x}=x?x ,thefractional partofx. Note that for instance
?1.4 =−2, and not−1 as almost all computer languages would lead us
to believe.
–For anyαbelonging to a ﬁeldKof characteristic zero and anyk∈Z
Λ0
we set Ω
α
k
∆
=
α(α−1)···(α−k+1)
k!
.
In particular, ifα∈Z
Λ0we have
ε
α
k
ψ
=0ifk>α, and in this case we will
set
ε
α
k
ψ
= 0 also whenk<0. On the other hand,
ε
α
k
ψ
isundeterminedfor
k<0ifα/∈Z
Λ0.

xPreface
–Capital italic letters such asKandLwill usually denote number ﬁelds.
–Capital calligraphic letters such asKandLwill denote generalp-adic ﬁelds
(for speciﬁc ones, we write for instanceK
p).
–Letters such asEandFwill always denote ﬁnite ﬁelds.
–The letterZindexed by a capital italic or calligraphic letter such asZ
K,
Z
L,ZK, etc., will always denote the ring of integers of the corresponding
ﬁeld.
–Capital italic letters such asA,B,C,G,H,S,T,U,V,W, or lowercase
italic letters such asf,g,h, will usually denote polynomials or formal power
series with coeﬃcients in some base ring or ﬁeld. The coeﬃcient of degreem
of these polynomials or power series will be denoted by the corresponding
letter indexed bym, such asA
m,Bm, etc. Thus we will always write (for
instance)A(X)=A
dX
d
+Ad−1X
d−1
+···+A 0,sothatthei th elementary
symmetric function of the roots is equal to (−1)
i
Ad−i/Ad.
Acknowledgments
A large part of the material on local ﬁelds has been taken with little change
from the remarkable book by Cassels [Cas1], and also from unpublished notes
of Jaulent written in 1994. Forp-adic analysis, I have also liberally borrowed
from work of Robert, in particular his superb GTM volume [Rob1]. For part of
the material on elliptic curves I have borrowed from another excellent book by
Cassels [Cas2], as well as the treatises of Cremona and Silverman [Cre2], [Sil1],
[Sil2], and the introductory book by Silverman–Tate [Sil-Tat]. I have also
borrowed from the classical books by Borevich–Shafarevich [Bor-Sha], Serre
[Ser1], Ireland–Rosen [Ire-Ros], and Washington [Was]. I would like to thank
my former students K. Belabas, C. Delaunay, S. Duquesne, and D. Simon,
who have helped me to write speciﬁc sections, and my colleagues J.-F. Jaulent
and J. Martinet for answering many questions in algebraic number theory. I
would also like to thank M. Bennett, J. Cremona, A. Kraus, and F. Rodriguez-
Villegas for valuable comments on parts of this book. I would especially like
to thank D. Bernardi for his thorough rereading of the ﬁrst ten chapters
of the manuscript, which enabled me to remove a large number of errors,
mathematical or otherwise. Finally, I would like to thank my copyeditor,
who was very helpful and who did an absolutely remarkable job.
It is unavoidable that there still remain errors, typographical or otherwise,
and the author would like to hear about them. Please send e-mail to
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be obtained on the author’s home page at the URL
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Table of Contents
Volume I
Preface...................................................v
1. Introduction to Diophantine Equations ...................1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Examples of Diophantine Problems . . . . . . . . . . . . . . . . . 1
1.1.2 Local Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.3 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 ExercisesforChapter1 ................................. 8
Part I. Tools
2. Abelian Groups, Lattices, and Finite Fields...............11
2.1 Finitely Generated Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 Description of Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.3 Characters of Finite Abelian Groups . . . . . . . . . . . . . . . . 17
2.1.4 The Groups (Z/mZ)
∗
............................. 20
2.1.5 Dirichlet Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.6 Gauss Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 The Quadratic Reciprocity Law . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.1 The Basic Quadratic Reciprocity Law . . . . . . . . . . . . . . . 33
2.2.2 Consequences of the Basic Quadratic Reciprocity Law 36
2.2.3 Gauss’s Lemma and Quadratic Reciprocity . . . . . . . . . . 39
2.2.4 Real Primitive Characters . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2.5 The Sign of the Quadratic Gauss Sum . . . . . . . . . . . . . . 45
2.3 Lattices and the Geometry of Numbers . . . . . . . . . . . . . . . . . . . . 50
2.3.1 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.3.2 Hermite’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.3.3 LLL-Reduced Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3.4 The LLL Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.3.5 Approximation of Linear Forms . . . . . . . . . . . . . . . . . . . . 60
2.3.6 Minkowski’s Convex Body Theorem . . . . . . . . . . . . . . . . 63

xii Table of Contents
2.4 BasicPropertiesofFiniteFields.......................... 65
2.4.1 General Properties of Finite Fields . . . . . . . . . . . . . . . . . 65
2.4.2 Galois Theory of Finite Fields . . . . . . . . . . . . . . . . . . . . . 69
2.4.3 Polynomials over Finite Fields . . . . . . . . . . . . . . . . . . . . . 71
2.5 Bounds for the Number of Solutions in Finite Fields . . . . . . . . 72
2.5.1 The Chevalley–Warning Theorem . . . . . . . . . . . . . . . . . . 72
2.5.2 Gauss Sums for Finite Fields. . . . . . . . . . . . . . . . . . . . . . . 73
2.5.3 Jacobi Sums for Finite Fields . . . . . . . . . . . . . . . . . . . . . . 79
2.5.4 The Jacobi SumsJ(χ
1,χ2) ........................ 82
2.5.5 The Number of Solutions of Diagonal Equations . . . . . . 87
2.5.6 The Weil Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.5.7 The Weil Conjectures (Deligne’s Theorem) . . . . . . . . . . 92
2.6 ExercisesforChapter2 ................................. 93
3. Basic Algebraic Number Theory ..........................101
3.1 Field-Theoretic Algebraic Number Theory . . . . . . . . . . . . . . . . . 101
3.1.1 Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.1.2 Number Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.1.4 Characteristic Polynomial, Norm, Trace . . . . . . . . . . . . . 109
3.1.5 Noether’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.1.6 The Basic Theorem of Kummer Theory . . . . . . . . . . . . . 111
3.1.7 Examples of the Use of Kummer Theory . . . . . . . . . . . . 114
3.1.8 Artin–Schreier Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.2 TheNormalBasisTheorem.............................. 117
3.2.1 Linear Independence and Hilbert’s Theorem 90. . . . . . . 117
3.2.2 The Normal Basis Theorem in the Cyclic Case . . . . . . . 119
3.2.3 Additive Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.2.4 Algebraic Independence of Homomorphisms . . . . . . . . . 121
3.2.5 The Normal Basis Theorem . . . . . . . . . . . . . . . . . . . . . . . . 123
3.3 Ring-TheoreticAlgebraicNumberTheory ................. 124
3.3.1 Gauss’s Lemma on Polynomials . . . . . . . . . . . . . . . . . . . . 124
3.3.2 Algebraic Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.3.3 Ring of Integers and Discriminant . . . . . . . . . . . . . . . . . . 128
3.3.4 Ideals and Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.3.5 Decomposition of Primes and Ramiﬁcation . . . . . . . . . . 132
3.3.6 Galois Properties of Prime Decomposition . . . . . . . . . . . 134
3.4 Quadratic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.4.1 Field-Theoretic and Basic Ring-Theoretic Properties . . 136
3.4.2 Results and Conjectures on Class and Unit Groups . . . 138
3.5 Cyclotomic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3.5.1 Cyclotomic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3.5.2 Field-Theoretic Properties ofQ(ζ
n)................. 144
3.5.3 Ring-Theoretic Properties . . . . . . . . . . . . . . . . . . . . . . . . . 146
3.5.4 The Totally Real Subﬁeld ofQ(ζ
p
k)................. 148

Table of Contents xiii
3.6 Stickelberger’sTheorem ................................. 150
3.6.1 Introduction and Algebraic Setting . . . . . . . . . . . . . . . . . 150
3.6.2 Instantiation of Gauss Sums . . . . . . . . . . . . . . . . . . . . . . . 151
3.6.3 Prime Ideal Decomposition of Gauss Sums . . . . . . . . . . . 154
3.6.4 The Stickelberger Ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
3.6.5 Diagonalization of the Stickelberger Element . . . . . . . . . 163
3.6.6 The Eisenstein Reciprocity Law . . . . . . . . . . . . . . . . . . . . 165
3.7 The Hasse–Davenport Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 170
3.7.1 Distribution Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
3.7.2 The Hasse–Davenport Relations . . . . . . . . . . . . . . . . . . . . 173
3.7.3 The Zeta Function of a Diagonal Hypersurface . . . . . . . 177
3.8 ExercisesforChapter3 ................................. 179
4.p-adic Fields..............................................183
4.1 Absolute Values and Completions . . . . . . . . . . . . . . . . . . . . . . . . 183
4.1.1 Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
4.1.2 Archimedean Absolute Values . . . . . . . . . . . . . . . . . . . . . . 184
4.1.3 Non-Archimedean and Ultrametric Absolute Values . . . 188
4.1.4 Ostrowski’s Theorem and the Product Formula . . . . . . 190
4.1.5 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
4.1.6 Completions of a Number Field . . . . . . . . . . . . . . . . . . . . 195
4.1.7 Hensel’s Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
4.2 Analytic Functions inp-adicFields ....................... 205
4.2.1 Elementary Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
4.2.2 Examples of Analytic Functions . . . . . . . . . . . . . . . . . . . . 208
4.2.3 Application of the Artin–Hasse Exponential . . . . . . . . . 217
4.2.4 Mahler Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
4.3 Additive and Multiplicative Structures . . . . . . . . . . . . . . . . . . . . 224
4.3.1 Concrete Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
4.3.2 Basic Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
4.3.3 Study of the GroupsU
i........................... 229
4.3.4 Study of the GroupU
1............................ 231
4.3.5 The GroupK
∗
p
/K
∗
p
2
.............................. 234
4.4 Extensions ofp-adicFields............................... 235
4.4.1 Preliminaries on Local Field Norms . . . . . . . . . . . . . . . . . 235
4.4.2 Krasner’s Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
4.4.3 General Results on Extensions . . . . . . . . . . . . . . . . . . . . . 239
4.4.4 Applications of the Cohomology of Cyclic Groups . . . . 242
4.4.5 Characterization of Unramiﬁed Extensions . . . . . . . . . . . 249
4.4.6 Properties of Unramiﬁed Extensions . . . . . . . . . . . . . . . . 251
4.4.7 Totally Ramiﬁed Extensions . . . . . . . . . . . . . . . . . . . . . . . 253
4.4.8 Analytic Representations ofpth Roots of Unity . . . . . . 254
4.4.9 Factorizations in Number Fields . . . . . . . . . . . . . . . . . . . . 258
4.4.10 Existence of the FieldC
p.......................... 260
4.4.11 Some Analysis inC
p.............................. 263

xiv Table of Contents
4.5 The Theorems of Strassmann and Weierstrass . . . . . . . . . . . . . . 266
4.5.1 Strassmann’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
4.5.2 Krasner Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . 267
4.5.3 The Weierstrass Preparation Theorem . . . . . . . . . . . . . . 270
4.5.4 Applications of Strassmann’s Theorem . . . . . . . . . . . . . . 272
4.6 ExercisesforChapter4 ................................. 275
5. Quadratic Forms and Local–Global Principles ............285
5.1 Basic Results on Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . 285
5.1.1 Basic Properties of Quadratic Modules . . . . . . . . . . . . . . 286
5.1.2 Contiguous Bases and Witt’s Theorem . . . . . . . . . . . . . . 288
5.1.3 Translations into Results on Quadratic Forms . . . . . . . . 291
5.2 Quadratic Forms over Finite and Local Fields . . . . . . . . . . . . . . 294
5.2.1 Quadratic Forms over Finite Fields . . . . . . . . . . . . . . . . . 294
5.2.2 Deﬁnition of the Local Hilbert Symbol . . . . . . . . . . . . . . 295
5.2.3 Main Properties of the Local Hilbert Symbol . . . . . . . . . 296
5.2.4 Quadratic Forms overQ
p.......................... 300
5.3 Quadratic Forms overQ................................. 303
5.3.1 Global Properties of the Hilbert Symbol . . . . . . . . . . . . . 303
5.3.2 Statement of the Hasse–Minkowski Theorem . . . . . . . . . 305
5.3.3 The Hasse–Minkowski Theorem fornφ2 ........... 306
5.3.4 The Hasse–Minkowski Theorem forn=3 ........... 307
5.3.5 The Hasse–Minkowski Theorem forn=4 ........... 308
5.3.6 The Hasse–Minkowski Theorem fornα5 ........... 309
5.4 Consequences of the Hasse–Minkowski Theorem . . . . . . . . . . . . 310
5.4.1 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
5.4.2 A Result of Davenport and Cassels . . . . . . . . . . . . . . . . . 311
5.4.3 Universal Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . 312
5.4.4 Sums of Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
5.5 TheHasseNormPrinciple............................... 318
5.6 TheHassePrincipleforPowers........................... 321
5.6.1 A General Theorem on Powers . . . . . . . . . . . . . . . . . . . . . 321
5.6.2 The Hasse Principle for Powers. . . . . . . . . . . . . . . . . . . . . 324
5.7 Some Counterexamples to the Hasse Principle . . . . . . . . . . . . . . 326
5.8 ExercisesforChapter5 ................................. 329
Part II. Diophantine Equations
6. Some Diophantine Equations .............................335
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
6.1.1 The Use of Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 6.1.2 Local Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
6.1.3 Global Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
6.2 Diophantine Equations of Degree 1 . . . . . . . . . . . . . . . . . . . . . . . 339

Table of Contents xv
6.3 Diophantine Equations of Degree 2 . . . . . . . . . . . . . . . . . . . . . . . 341
6.3.1 The General Homogeneous Equation . . . . . . . . . . . . . . . . 341
6.3.2 The Homogeneous Ternary Quadratic Equation . . . . . . 343
6.3.3 Computing a Particular Solution . . . . . . . . . . . . . . . . . . . 347
6.3.4 Examples of Homogeneous Ternary Equations . . . . . . . . 352
6.3.5 The Pell–Fermat Equationx
2
−Dy
2
=N........... 354
6.4 Diophantine Equations of Degree 3 . . . . . . . . . . . . . . . . . . . . . . . 357
6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
6.4.2 The Equationax
p
+by
p
+cz
p
= 0: Local Solubility . . . 359
6.4.3 The Equationax
p
+by
p
+cz
p
= 0: Number Fields . . . . 362
6.4.4 The Equationax
p
+by
p
+cz
p
=0:
Hyperelliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
6.4.5 The Equationx
3
+y
3
+cz
3
=0.................... 373
6.4.6 Sums of Two or More Cubes . . . . . . . . . . . . . . . . . . . . . . . 376
6.4.7 Skolem’s Equationsx
3
+dy
3
=1 ................... 385
6.4.8 Special Cases of Skolem’s Equations . . . . . . . . . . . . . . . . 386
6.4.9 The Equationsy
2
=x
3
±1 in Rational Numbers . . . . . 387
6.5 The Equationsax
4
+by
4
+cz
2
=0andax
6
+by
3
+cz
2
= 0 . 389
6.5.1 The Equationax
4
+by
4
+cz
2
= 0: Local Solubility . . . 389
6.5.2 The Equationsx
4
±y
4
=z
2
andx
4
+2y
4
=z
2
....... 391
6.5.3 The Equationax
4
+by
4
+cz
2
= 0: Elliptic Curves . . . . 392
6.5.4 The Equationax
4
+by
4
+cz
2
= 0: Special Cases . . . . . 393
6.5.5 The Equationax
6
+by
3
+cz
2
=0.................. 396
6.6 The Fermat Quarticsx
4
+y
4
=cz
4
....................... 397
6.6.1 Local Solubility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
6.6.2 Global Solubility: Factoring over Number Fields . . . . . . 400
6.6.3 Global Solubility: Coverings of Elliptic Curves . . . . . . . 407
6.6.4 Conclusion, and a Small Table . . . . . . . . . . . . . . . . . . . . . 409
6.7 The Equationy
2
=x
n
+t............................... 410
6.7.1 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
6.7.2 The Casep=3 .................................. 414
6.7.3 The Casep=5 .................................. 416
6.7.4 Application of the Bilu–Hanrot–Voutier Theorem . . . . . 417
6.7.5 Special Cases with Fixedt......................... 418
6.7.6 The Equationsty
2
+1=4x
p
andy
2
+y+1=3x
p
. . . 420
6.8 Linear Recurring Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
6.8.1 Squares in the Fibonacci and Lucas Sequences . . . . . . . 421
6.8.2 The Square Pyramid Problem . . . . . . . . . . . . . . . . . . . . . . 424
6.9 Fermat’s “Last Theorem”x
n
+y
n
=z
n
................... 427
6.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
6.9.2 General Primen:TheFirstCase ................... 428
6.9.3 Congruence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
6.9.4 The Criteria of Wendt and Germain . . . . . . . . . . . . . . . . 430
6.9.5 Kummer’s Criterion: Regular Primes . . . . . . . . . . . . . . . . 431

xvi Table of Contents
6.9.6 The Criteria of Furtw¨anglerandWieferich........... 434
6.9.7 General Primen: The Second Case . . . . . . . . . . . . . . . . . 435
6.10 An Example of Runge’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . 439
6.11 First Results on Catalan’s Equation . . . . . . . . . . . . . . . . . . . . . . 442
6.11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
6.11.2 The Theorems of Nagell and Ko Chao . . . . . . . . . . . . . . 444
6.11.3 Some Lemmas on Binomial Series . . . . . . . . . . . . . . . . . . 446
6.11.4 Proof of Cassels’s Theorem 6.11.5 . . . . . . . . . . . . . . . . . . 447
6.12 Congruent Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
6.12.1 Reduction to an Elliptic Curve . . . . . . . . . . . . . . . . . . . . . 451
6.12.2 The Use of the Birch and Swinnerton-Dyer Conjecture 452
6.12.3 Tunnell’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
6.13 Some Unsolved Diophantine Problems . . . . . . . . . . . . . . . . . . . . . 455
6.14 Exercises for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
7. Elliptic Curves...........................................465
7.1 Introduction and Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
7.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
7.1.2 Weierstrass Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
7.1.3 Degenerate Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . 467
7.1.4 The Group Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
7.1.5 Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
7.2 Transformations into Weierstrass Form . . . . . . . . . . . . . . . . . . . . 474
7.2.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 474
7.2.2 Transformation of the Intersection of Two Quadrics . . . 475
7.2.3 Transformation of a Hyperelliptic Quartic . . . . . . . . . . . 476
7.2.4 Transformation of a General Nonsingular Cubic . . . . . . 477
7.2.5 Example: The Diophantine Equationx
2
+y
4
=2z
4
. . . 480
7.3 Elliptic Curves overC,R,k(T),F
q,andK p............... 482
7.3.1 Elliptic Curves overC............................ 482
7.3.2 Elliptic Curves overR............................ 484
7.3.3 Elliptic Curves overk(T).......................... 486
7.3.4 Elliptic Curves overF
q............................ 494
7.3.5 Constant Elliptic Curves overR[[T]]: Formal Groups . . 500
7.3.6 Reduction of Elliptic Curves overK
p............... 505
7.3.7 Thep-adic Filtration for Elliptic Curves overK
p..... 507
7.4 ExercisesforChapter7 ................................. 512
8. Diophantine Aspects of Elliptic Curves...................517
8.1 Elliptic Curves overQ.................................. 517
8.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
8.1.2 Basic Results and Conjectures . . . . . . . . . . . . . . . . . . . . . 518
8.1.3 Computing the Torsion Subgroup . . . . . . . . . . . . . . . . . . 524
8.1.4 Computing the Mordell–Weil Group . . . . . . . . . . . . . . . . 528
8.1.5 The Na¨ıve and Canonical Heights . . . . . . . . . . . . . . . . . . 529

Table of Contents xvii
8.2 Description of 2-Descent with Rational 2-Torsion . . . . . . . . . . . 532
8.2.1 The Fundamental 2-Isogeny . . . . . . . . . . . . . . . . . . . . . . . . 532
8.2.2 Description of the Image ofφ...................... 534
8.2.3 The Fundamental 2-Descent Map . . . . . . . . . . . . . . . . . . . 535
8.2.4 Practical Use of 2-Descent with 2-Isogenies . . . . . . . . . . 538
8.2.5 Examples of 2-Descent using 2-Isogenies . . . . . . . . . . . . . 542
8.2.6 An Example of Second Descent . . . . . . . . . . . . . . . . . . . . 546
8.3 Description of General 2-Descent . . . . . . . . . . . . . . . . . . . . . . . . . 548
8.3.1 The Fundamental 2-Descent Map . . . . . . . . . . . . . . . . . . . 548
8.3.2 TheT-SelmerGroupofaNumberField............. 550
8.3.3 Description of the Image ofα...................... 552
8.3.4 Practical Use of 2-Descent in the General Case . . . . . . . 554
8.3.5 Examples of General 2-Descent . . . . . . . . . . . . . . . . . . . . . 555
8.4 Description of 3-Descent with Rational 3-Torsion Subgroup . . 557
8.4.1 Rational 3-Torsion Subgroups . . . . . . . . . . . . . . . . . . . . . . 557
8.4.2 The Fundamental 3-Isogeny . . . . . . . . . . . . . . . . . . . . . . . . 558
8.4.3 Description of the Image ofφ...................... 560
8.4.4 The Fundamental 3-Descent Map . . . . . . . . . . . . . . . . . . . 563
8.5 The Use ofL(E,s) ..................................... 564
8.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564
8.5.2 The Case of Complex Multiplication . . . . . . . . . . . . . . . . 565
8.5.3 Numerical Computation ofL
(r)
(E,1) ............... 572
8.5.4 Computation of Γ
r(1,x) for Smallx................ 575
8.5.5 Computation of Γ
r(1,x)forLargex................ 580
8.5.6 The Famous Curvey
2
+y=x
3
−7x+6 ............ 582
8.6 TheHeegnerPointMethod .............................. 584
8.6.1 Introduction and the Modular Parametrization . . . . . . . 584
8.6.2 Heegner Points and Complex Multiplication . . . . . . . . . 586
8.6.3 The Use of the Theorem of Gross–Zagier . . . . . . . . . . . . 589
8.6.4 Practical Use of the Heegner Point Method . . . . . . . . . . 591
8.6.5 Improvements to the Basic Algorithm, in Brief . . . . . . . 596
8.6.6 A Complete Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
8.7 Computation of Integral Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 600
8.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600
8.7.2 An Upper Bound for the Elliptic Logarithm onE(Z) . 601
8.7.3 Lower Bounds for Linear Forms in Elliptic Logarithms 603
8.7.4 A Complete Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
8.8 ExercisesforChapter8 ................................. 607
Bibliography..............................................615
Index of Notation........................................625
Index of Names...........................................633
General Index............................................639

xviii Table of Contents
Volume I I
Preface...................................................v
Part III. Analytic Tools
9. Bernoulli Polynomials and the Gamma Function ..........3
9.1 Bernoulli Numbers and Polynomials . . . . . . . . . . . . . . . . . . . . . . 3
9.1.1 Generating Functions for Bernoulli Polynomials . . . . . . 3
9.1.2 Further Recurrences for Bernoulli Polynomials . . . . . . . 10
9.1.3 Computing a Single Bernoulli Number . . . . . . . . . . . . . . 14
9.1.4 Bernoulli Polynomials and Fourier Series . . . . . . . . . . . . 16
9.2 Analytic Applications of Bernoulli Polynomials . . . . . . . . . . . . . 19
9.2.1 Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
9.2.2 The Euler–MacLaurin Summation Formula . . . . . . . . . . 21
9.2.3 The Remainder Term and the Constant Term . . . . . . . . 25
9.2.4 Euler–MacLaurin and the Laplace Transform . . . . . . . . 27
9.2.5 Basic Applications of the Euler–MacLaurin Formula . . 31
9.3 Applications to Numerical Integration . . . . . . . . . . . . . . . . . . . . . 35
9.3.1 Standard Euler–MacLaurin Numerical Integration . . . . 36
9.3.2 The Basic Tanh-Sinh Numerical Integration Method . . 37
9.3.3 General Doubly Exponential Numerical Integration . . . 39
9.4χ-Bernoulli Numbers, Polynomials, and Functions . . . . . . . . . . 43
9.4.1χ-Bernoulli Numbers and Polynomials . . . . . . . . . . . . . . 43
9.4.2χ-Bernoulli Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
9.4.3 Theχ-Euler–MacLaurin Summation Formula . . . . . . . . 50
9.5 Arithmetic Properties of Bernoulli Numbers . . . . . . . . . . . . . . . 52
9.5.1χ-PowerSums ................................... 52
9.5.2 The Generalized Clausen–von Staudt Congruence . . . . 61
9.5.3 The Voronoi Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . 64
9.5.4 The Kummer Congruences . . . . . . . . . . . . . . . . . . . . . . . . 67
9.5.5 The Almkvist–Meurman Theorem . . . . . . . . . . . . . . . . . . 70
9.6 The Real and Complex Gamma Functions . . . . . . . . . . . . . . . . . 71
9.6.1 The Hurwitz Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . 71
9.6.2 Deﬁnition of the Gamma Function . . . . . . . . . . . . . . . . . . 77
9.6.3 Preliminary Results for the Study of Γ(s)............ 81
9.6.4 Properties of the Gamma Function . . . . . . . . . . . . . . . . . 84
9.6.5 Speciﬁc Properties of the Functionψ(s) ............. 95
9.6.6 Fourier Expansions ofζ(s, x) and log(Γ(x)).......... 100
9.7 IntegralTransforms..................................... 103
9.7.1 Generalities on Integral Transforms . . . . . . . . . . . . . . . . . 104
9.7.2 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
9.7.3 The Mellin Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Table of Contents xix
9.7.4 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
9.8 Bessel Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
9.8.1 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
9.8.2 Integral Representations and Applications . . . . . . . . . . . 113
9.9 ExercisesforChapter9 ................................. 118
10. Dirichlet Series andL-Functions..........................151
10.1 Arithmetic Functions and Dirichlet Series. . . . . . . . . . . . . . . . . . 151
10.1.1 Operations on Arithmetic Functions . . . . . . . . . . . . . . . . 152
10.1.2 Multiplicative Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
10.1.3 Some Classical Arithmetical Functions . . . . . . . . . . . . . . 155
10.1.4 Numerical Dirichlet Series . . . . . . . . . . . . . . . . . . . . . . . . . 160
10.2 The Analytic Theory ofL-Series. . . . . . . . . . . . . . . . . . . . . . . . . . 162
10.2.1 Simple Approaches to Analytic Continuation . . . . . . . . . 163
10.2.2 The Use of the Hurwitz Zeta Functionζ(s, x) ........ 168
10.2.3 The Functional Equation for the Theta Function . . . . . 169
10.2.4 The Functional Equation for DirichletL-Functions . . . 172
10.2.5 Generalized Poisson Summation Formulas . . . . . . . . . . . 177
10.2.6 Voronoi’s Error Term in the Circle Problem. . . . . . . . . . 182
10.3 Special Values of DirichletL-Functions . . . . . . . . . . . . . . . . . . . . 186
10.3.1 Basic Results on Special Values . . . . . . . . . . . . . . . . . . . . 186
10.3.2 Special Values ofL-Functions and Modular Forms . . . . 193
10.3.3 The P´olya–Vinogradov Inequality . . . . . . . . . . . . . . . . . . 198
10.3.4 Bounds and Averages forL(χ,1) ................... 200
10.3.5 Expansions ofζ(s) Arounds=k∈Z
φ1............. 205
10.3.6 Numerical Computation of Euler Products and Sums . 208
10.4 Epstein Zeta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
10.4.1 The Nonholomorphic Eisenstein SeriesG(τ,s)........ 211
10.4.2 The Kronecker Limit Formula . . . . . . . . . . . . . . . . . . . . . . 213
10.5 Dirichlet Series Linked to Number Fields . . . . . . . . . . . . . . . . . . 216
10.5.1 The Dedekind Zeta Functionζ
K(s) ................. 216
10.5.2 The Dedekind Zeta Function of Quadratic Fields . . . . . 219
10.5.3 Applications of the Kronecker Limit Formula . . . . . . . . 223
10.5.4 The Dedekind Zeta Function of Cyclotomic Fields . . . . 230
10.5.5 The Nonvanishing ofL(χ,1) ....................... 235
10.5.6 Application to Primes in Arithmetic Progression . . . . . 237
10.5.7 Conjectures on DirichletL-Functions . . . . . . . . . . . . . . . 238
10.6 Science Fiction onL-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
10.6.1 LocalL-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
10.6.2 GlobalL-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
10.7 The Prime Number Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
10.7.1 Estimates forζ(s) ................................ 246
10.7.2 Newman’s Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
10.7.3 Iwaniec’s Proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
10.8 Exercises for Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

xx Table of Contents
11.p-adic Gamma and L-Functions...........................275
11.1 Generalities onp-adic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 275
11.1.1 Methods for Constructingp-adic Functions . . . . . . . . . . 275
11.1.2 A Brief Study of Volkenborn Integrals . . . . . . . . . . . . . . . 276
11.2 Thep-adic Hurwitz Zeta Functions . . . . . . . . . . . . . . . . . . . . . . . 280
11.2.1 Teichm¨uller Extensions and Characters onZ
p........ 280
11.2.2 Thep-adic Hurwitz Zeta Function forx∈CZ
p....... 281
11.2.3 The Functionζ
p(s, x) Arounds=1................. 288
11.2.4 Thep-adic Hurwitz Zeta Function forx∈Z
p........ 290
11.3p-adicL-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
11.3.1 Dirichlet Characters in thep-adicContext........... 300
11.3.2 Deﬁnition and Basic Properties ofp-adicL-Functions . 301
11.3.3p-adicL-Functions at Positive Integers . . . . . . . . . . . . . . 305
11.3.4χ-Power Sums Involvingp-adic Logarithms . . . . . . . . . . 310
11.3.5 The FunctionL
p(χ, s) Arounds=1................ 317
11.4 Applications ofp-adicL-Functions . . . . . . . . . . . . . . . . . . . . . . . . 319
11.4.1 Integrality and Parity ofL-Function Values . . . . . . . . . . 319
11.4.2 Bernoulli Numbers and Regular Primes . . . . . . . . . . . . . 324
11.4.3 Strengthening of the Almkvist–Meurman Theorem . . . 326
11.5p-adic Log Gamma Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
11.5.1 Diamond’sp-adic Log Gamma Function . . . . . . . . . . . . . 330
11.5.2 Morita’sp-adic Log Gamma Function . . . . . . . . . . . . . . . 336
11.5.3 Computation of somep-adic Logarithms . . . . . . . . . . . . . 346
11.5.4 Computation of Limits of some Logarithmic Sums . . . . 356
11.5.5 Explicit Formulas forψ
p(r/m)andψ p(χ, r/m)....... 359
11.5.6 Application to the Value ofL
p(χ,1) ................ 361
11.6 Morita’sp-adic Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . 364
11.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
11.6.2 Deﬁnitions and Basic Results . . . . . . . . . . . . . . . . . . . . . . 365
11.6.3 Main Properties of thep-adic Gamma Function . . . . . . 369
11.6.4 Mahler–Dwork Expansions Linked to Γ
p(x).......... 375
11.6.5 Power Series Expansions Linked to Γ
p(x)............ 378
11.6.6 The Jacobstahl–Kazandzidis Congruence . . . . . . . . . . . . 380
11.7 The Gross–Koblitz Formula and Applications . . . . . . . . . . . . . . 383
11.7.1 Statement and Proof of the Gross–Koblitz Formula . . . 383
11.7.2 Application toL
χ
p
(χ,0)............................ 389
11.7.3 Application to the Stickelberger Congruence . . . . . . . . . 390
11.7.4 Application to the Hasse–Davenport Product Relation 392
11.8 Exercises for Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

Table of Contents xxi
Part IV. Modern Tools
12. Applications of Linear Forms in Logarithms..............411
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
12.1.1 Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
12.1.2 Applications to Diophantine Equations and Problems . 413
12.1.3 A List of Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
12.2 A Lower Bound for|2
m
−3
n
|............................ 415
12.3 Lower Bounds for the Trace ofα
n
........................ 418
12.4 Pure Powers in Binary Recurrent Sequences . . . . . . . . . . . . . . . 420
12.5 Greatest Prime Factors of Terms of Some Recurrent Se-
quences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
12.6 Greatest Prime Factors of Values of Integer Polynomials . . . . . 422
12.7 The Diophantine Equationax
n
−by
n
=c.................. 423
12.8 Simultaneous Pell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
12.8.1 General Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
12.8.2 An Example in Detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
12.8.3 A General Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
12.9 Catalan’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
12.10 Thue Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
12.10.1 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
12.10.2 Algorithmic Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
12.11 Other Classical Diophantine Equations . . . . . . . . . . . . . . . . . . . 436
12.12 A Few Words on the Non-Archimedean Case . . . . . . . . . . . . . . 439
13. Rational Points on Higher-Genus Curves .................441
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
13.2 The Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
13.2.1 Functions on Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
13.2.2 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
13.2.3 Rational Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
13.2.4 The Group Law: Cantor’s Algorithm . . . . . . . . . . . . . . . . 446
13.2.5 The Group Law: The Geometric Point of View . . . . . . . 448
13.3 Rational Points on Hyperelliptic Curves . . . . . . . . . . . . . . . . . . . 449
13.3.1 The Method of Dem
χ
yanenko–Manin . . . . . . . . . . . . . . . . 449
13.3.2 The Method of Chabauty–Coleman . . . . . . . . . . . . . . . . . 452
13.3.3 Explicit Chabauty According to Flynn . . . . . . . . . . . . . . 453
13.3.4 When Chabauty Fails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
13.3.5 Elliptic Curve Chabauty . . . . . . . . . . . . . . . . . . . . . . . . . . 456
13.3.6 A Complete Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

xxii Table of Contents
14. The Super-Fermat Equation ..............................463
14.1 Preliminary Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
14.2 The Dihedral Cases (2, 2,r).............................. 465
14.2.1 The Equationx
2
−y
2
=z
r
........................ 465
14.2.2 The Equationx
2
+y
2
=z
r
........................ 466
14.2.3 The Equationsx
2
+3y
2
=z
3
andx
2
+3y
2
=4z
3
..... 466
14.3 The Tetrahedral Case (2, 3,3)............................ 467
14.3.1 The Equationx
3
+y
3
=z
2
........................ 467
14.3.2 The Equationx
3
+y
3
=2z
2
....................... 470
14.3.3 The Equationx
3
−2y
3
=z
2
....................... 472
14.4 The Octahedral Case (2, 3,4) ............................ 473
14.4.1 The Equationx
2
−y
4
=z
3
........................ 473
14.4.2 The Equationx
2
+y
4
=z
3
........................ 475
14.5 Invariants, Covariants, and Dessins d’Enfants . . . . . . . . . . . . . . 477
14.5.1 Dessins d’Enfants, Klein Forms, and Covariants . . . . . . 478
14.5.2 The Icosahedral Case (2,3,5) ...................... 479
14.6 The Parabolic and Hyperbolic Cases . . . . . . . . . . . . . . . . . . . . . . 481
14.6.1 The Parabolic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
14.6.2 General Results in the Hyperbolic Case . . . . . . . . . . . . . 482
14.6.3 The Equationsx
4
±y
4
=z
3
....................... 484
14.6.4 The Equationx
4
+y
4
=z
5
........................ 485
14.6.5 The Equationx
6
−y
4
=z
2
........................ 486
14.6.6 The Equationx
4
−y
6
=z
2
........................ 487
14.6.7 The Equationx
6
+y
4
=z
2
........................ 488
14.6.8 Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
14.7 Applications of Mason’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 490
14.7.1 Mason’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
14.7.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
14.8 Exercises for Chapter 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
15. The Modular Approach to Diophantine Equations ........495
15.1 Newforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
15.1.1 Introduction and Necessary Software Tools . . . . . . . . . . 495
15.1.2 Newforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
15.1.3 Rational Newforms and Elliptic Curves . . . . . . . . . . . . . 497
15.2 Ribet’s Level-Lowering Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 498
15.2.1 Deﬁnition of “Arises From” . . . . . . . . . . . . . . . . . . . . . . . . 498
15.2.2 Ribet’s Level-Lowering Theorem . . . . . . . . . . . . . . . . . . . 499
15.2.3 Absence of Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
15.2.4 How to use Ribet’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 503
15.3 Fermat’s Last Theorem and Similar Equations . . . . . . . . . . . . . 503
15.3.1 A Generalization of FLT . . . . . . . . . . . . . . . . . . . . . . . . . . 504
15.3.2EArises from a Curve with Complex Multiplication . . 505
15.3.3 End of the Proof of Theorem 15.3.1 . . . . . . . . . . . . . . . . . 506
15.3.4 The Equationx
2
=y
p
+2
r
z
p
forpα7andr α2..... 507

Table of Contents xxiii
15.3.5 The Equationx
2
=y
p
+z
p
forp∞7................ 509
15.4 An Occasional Bound for the Exponent . . . . . . . . . . . . . . . . . . . 509
15.5 An Example of Serre–Mazur–Kraus . . . . . . . . . . . . . . . . . . . . . . . 511
15.6 The Method of Kraus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514
15.7 “Predicting Exponents of Constants” . . . . . . . . . . . . . . . . . . . . . 517
15.7.1 The Diophantine Equationx
2
−2=y
p
.............. 517
15.7.2 Application to the SMK Equation . . . . . . . . . . . . . . . . . . 521
15.8 Recipes for Some Ternary Diophantine Equations . . . . . . . . . . . 522
15.8.1 Recipes for Signature (p, p, p) ...................... 523
15.8.2 Recipes for Signature (p, p,2) ...................... 524
15.8.3 Recipes for Signature (p, p,3) ...................... 526
16. Catalan’s Equation.......................................529
16.1 Mih˘ailescu’s First Two Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 529
16.1.1 The First Theorem: Double Wieferich Pairs . . . . . . . . . . 530
16.1.2 The Equation (x
p
−1)/(x−1) =py
q
............... 532
16.1.3 Mih˘ailescu’s Second Theorem:p|h
−
q
andq|h
−
p
...... 536
16.2 The + and−Subspaces and the GroupS................. 537
16.2.1 The + and−Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
16.2.2 The GroupS.................................... 540
16.3 Mih˘ailescu’s Third Theorem:p<4q
2
andq<4p
2
.......... 542
16.4 Mih˘ailescu’s Fourth Theorem:p≡1 (modq)orq ≡1 (modp) 547
16.4.1 Preliminaries on Commutative Algebra . . . . . . . . . . . . . . 547
16.4.2 Preliminaries on the Plus Part . . . . . . . . . . . . . . . . . . . . . 549
16.4.3 Cyclotomic Units and Thaine’s Theorem . . . . . . . . . . . . 552
16.4.4 Preliminaries on Power Series . . . . . . . . . . . . . . . . . . . . . . 554
16.4.5 Proof of Mih˘ailescu’s Fourth Theorem . . . . . . . . . . . . . . 557
16.4.6 Conclusion: Proof of Catalan’s Conjecture . . . . . . . . . . . 560
Bibliography..............................................561
Index of Notation........................................571
Index of Names...........................................579
General Index............................................585

1. Introduction to Diophantine Equations
1.1 Introduction
The study ofDiophantine equationsis the study of solutions of polynomial
equations or systems of equations in integers, rational numbers, or sometimes
more general number rings. It is one of the oldest branches of number theory,
in fact of mathematics itself, since its origins can be found in texts of the
ancient Babylonians, Chinese, Egyptians, and Greeks. One of the fascinations
of the subject is that the problems are usually easy to state, but more often
than not very diﬃcult to solve, and when they can be solved, they sometimes
involve extremely sophisticated mathematical tools.
Perhaps even more importantly, mathematicians must often invent or
extensively develop entirely new tools to solve the number-theoretical prob-
lems, and these become in turn important branches of mathematics per se,
which often have applications in completely diﬀerent problems from those
from which they originate.
1.1.1 Examples of Diophantine Problems
Let me give ﬁve examples. The ﬁrst and most famous is “Fermat’s last the-
orem” (FLT), stating that fornﬀ3, the curvex
n
+y
n
= 1 has no rational
points other than the ones withxoryequal to 0 (this is of course equivalent
to the usual statement).
1
In the nineteenth century, thanks in particular to the work of E. Kummer
and P.-G. Lejeune-Dirichlet, the theorem was proved for quite a large number
1
Incidentally, this is the place to destroy the legend concerning this statement,
which has produced an enormous number of “Fermatists” claiming to have found
an “elementary” proof that Fermat may have found himself: Fermat made this
statement in the margin of his copy of the book by Diophantus on number theory
(at the place where Diophantus discusses Pythagorean triples, see below), and
claimed to have found a marvelous proof and so on. However, he wrote this
statement when he was young, never claimed it publicly, and certainly never
imagined that it would be made public, so he forgot about it. Itmaybe possible
that there does exist an elementary proof (although this is unlikely), but we can
be positively sure that Fermat did not have it, for otherwise he would at least
have challenged his English colleagues, as was the custom at that time.

2 1. Introduction to Diophantine Equations
of values ofn, including alln∗100. Together with the theory of quadratic
forms initiated by A.-M. Legendre and especially by C. F. Gauss, one can
without exaggeration say that this single problem gave rise to algebraic num-
ber theory (rings, ideals, prime ideals, principal ideals, class numbers, units,
Dirichlet series,L-functions, etc.). As is well known, although these methods
were pushed to the extreme in the twentieth century, they did not succeed in
solving the problem completely. The next progress on FLT came from alge-
braic geometry thanks to the work of G. Faltings, who proved the so-called
Mordell conjecture, which in particular implies that for aﬁxedn∞3thenum-
ber of solutions to the Fermat equation is ﬁnite. However, it was only thanks
to the work of several mathematicians starting with Y. Hellegouarch and
G. Frey, and culminating with the work of K. Ribet, then ﬁnally of A. Wiles
(helped for a crucial part by R. Taylor), that the problem was ﬁnally com-
pletely solved using completely diﬀerent tools from those of Kummer (and
even Faltings): elliptic curves, Galois representations, and modular forms.
Although these subjects were not initiated by FLT, their development was
certainly accelerated by the impetus given by FLT. In particular, thanks to
the work of Wiles, the complete proof of the Taniyama–Shimura–Weil con-
jecture was obtained a few years later by C. Breuil, B. Conrad, F. Diamond,
and R. Taylor. This latter result can be considered in itself a more important
(and certainly a more useful) theorem than FLT.
A second rather similar problem whose history is slightly diﬀerent isCata-
lan’s conjecture. This states that when nandmare greater than or equal to
2, the only solutions in nonzero integersxandyof the equationx
m
−y
n
=1
come from the equality 3
2
−2
3
= 1. This problem can be naturally attacked
by the standard methods of algebraic number theory originating in the work
of Kummer. However, it came as a surprise that an elementary argument due
to Cassels (see Theorem 6.11.5) shows that the “ﬁrst case” is impossible, in
other words that ifx
p
−y
q
= 1 withpandqprimes thenp|yandq|x.
The next important result, due to R. Tijdeman using Baker’s theory of lin-
ear forms in logarithms of algebraic numbers, was that the total number of
quadruplets (m, n, x, y) satisfying the required conditions is ﬁnite. Note that
the proof of this ﬁniteness result is completely diﬀerent from Faltings’s proof
of the corresponding one for FLT, and in fact in the latter his result did not
imply the ﬁniteness of the number of triples (x, y, n) with n∞3andxy ∞=0
such thatx
n
+y
n
=1.
Until the end of the 1990s the situation was quite similar to that of FLT
before Wiles: under suitable conditions on the nondivisibility of the class
number of cyclotomic ﬁelds, the Catalan equation was known to have no
nontrivial solutions. It thus came as a total surprise that in 1999 P. Mih˘ailescu
proved that if Catalan’s equationx
p
−y
q
= 1 withpandqodd primes has a
solution thenpandqmust satisfy the so-called double Wieferich condition
p
q−1
≡1 (modq
2
)andq
p−1
≡1 (modp
2
). These conditions were known
before him, but he completely removed the conditions on class numbers. The

1.1 Introduction 3
last step was again taken by Mih˘ailescu in 2001, who ﬁnished the proof of
Catalan’s conjecture. His proof was improved and simpliﬁed by several people,
including in particular Yu. Bilu and H. W. Lenstra. The remarkable thing
about the ﬁnal proof is that it usesonlyalgebraic number theory techniques
on cyclotomic ﬁelds. However, it uses a large part of the theory, including the
relatively recent theorem of F. Thaine, that has had some very important
applications elsewhere. It does not use any computer calculations, while the
initial proof did.
A third example is thecongruent number problem, stated by Diophantus
in the fourth century A.D. The problem is to ﬁnd all integersn(called con-
gruent numbers) that are equal to the area of aPythagorean triangle, i.e.,
a right-angled triangle with all three sides rational. Very simple algebraic
transformations show thatnis congruent if and only if the Diophantine
equationy
2
=x
3
−n
2
xhas rational solutions other than those withy=0.
The problem was in an “experimental” state until the 1970s; more precisely,
one knew the congruent or noncongruent nature of numbersnup to a few
hundred (and of course of many other larger numbers). Remarkable progress
was made on this problem by J. Tunnell in 1980 using the theory of modular
forms, and especially of modular forms of half-integral weight. In eﬀect, he
completely solved the problem, by giving an easily checked criterion fornto
be a congruent number, assuming a weak form of the Birch–Swinnerton-Dyer
conjecture, see Theorem 6.12.4. This conjecture (for which a prize of 1 mil-
lion U.S. dollars has been oﬀered by the Clay foundation) is probably one of
the most important, and also one of the most beautiful, conjectures in all of
mathematics in the twenty-ﬁrst century.
A fourth important example is the Weil conjectures. These have to do
with the number of solutions of Diophantine equationsin ﬁnite ﬁelds.In-
deed, one of the main themes of this book is that to study a Diophantine
equation it is essential to start by studying it in ﬁnite ﬁelds. Let us give a
simple example. LetN(p) be the number of solutions modulopof the equa-
tiony
2
=x
5
−x. Then|N(p)−p|can never be very large compared top,
more precisely|N(p)−p|<4
√
p, and the constant 4 is best possible. This
result is already quite nontrivial, and the general study of the number of
points oncurvesculminated with work of A. Weil in 1949 proving that this
phenomenon occurs for all (nonsingular) curves and many other results be-
sides. It was then natural to ask the question for surfaces, and more generally
varieties of any dimension. This problem (in a very precise form, which in par-
ticular implied excellent bounds on the number of solutions) became known
as the Weil conjectures. A general strategy for solving these conjectures was
put forth by Weil himself, but the achievement of this goal was made possible
only by an amazing amount of work by numerous people. It included the cre-
ation of modern algebraic geometry by A. Grothendieck and his students (the
famous EGA and SGA treatises). The Weil conjectures were ﬁnally solved

4 1. Introduction to Diophantine Equations
by P. Deligne in the early 1970s, exactly following Weil’s strategy, but using
all the tools developed since.
As a last example we mention Waring’s problem. One of its forms (by far
not the only one) is the following: given an integerkα2, ﬁnd the smallest
integerg(k) such that any nonnegative integer can be represented as a sum
ofg(k) nonnegativekth powers. It has been known since J.-L. Lagrange that
any integer is a sum of 4 squares, and that integers congruent to 7 modulo 8
are not the sum of 3 squares, so thatg(2) = 4. It was proved by D. Hilbert
thatg(k) is ﬁnite (this can be proved with not too much diﬃculty from
Lagrange’s result, but is still not completely trivial: try it as an exercise).
However, the major advances on this problem were made by G. H. Hardy and
J. Littlewood, who invented thecircle methodin order to treat the problem.
One of the important aspects of the circle method is the so-calledsingular
series, which regroups all the arithmetic information obtained by studying
the problem modulopfor each primep. The other major advances were
made by I.-M. Vinogradov using the theory oftrigonometric sums. Both the
circle method and trigonometric sums have found universal application in the
branch of number theory called “additive number theory,” and also in other
branches of number theory. To ﬁnish this example, we note that Waring’s
problem as given above (as already mentioned, there are other versions) is
completely solved. Perhaps surprisingly, when one compares it with FLT for
example, the hardest cases are not for largekbut for smallk: the most diﬃcult
isk= 4, solved only in the 1980s by R. Balasubramanian, J.-M. Deshouillers,
and F. Dress, see [BDD]. For the record, we haveg(2) = 4,g(3) = 9, and
g(4) = 19.
Additive number theory forms a large part of what is usually called “an-
alytic number theory” because many sophisticated analytic techniques come
into play. Analytic number theory willnotbe studied in this book, with
the exception of a few basic results such as the prime number theorem and
Dirichlet’s theorem on primes in arithmetic progression. The expression “an-
alytic methods” used in the third part of this book (Chapters 9 to 11) refer to
the study of Bernoulli polynomials, gamma andL-functions, integral trans-
forms, summation formulas, and the like. We refer for instance to [Ell] and
[Iwa-Kow] among many others for excellent expositions of analytic number
theory.
1.1.2 Local Methods
As is explicit or implicit in all of the examples given above (and in fact in
all Diophantine problems), it is essential to start by studying a Diophantine
equationlocally, in other words prime by prime (we will see later precisely
what this means). Letpbe a prime number, and letF
pγZ/pZ be the prime
ﬁnite ﬁeld withpelements. We can begin by studying our problem inF
p(i.e.,
modulop), and this can already be considered as the start of a local study.

1.1 Introduction 5
This is sometimes suﬃcient, but usually not. In that case, keeping the same
primep, we will see that there are twototally diﬀerentways to reﬁne the
study of the equation.
The ﬁrst is to consider it modulop
2
,p
3
, and so on, i.e., to work inZ/p
2
Z,
Z/p
3
Z,...Animportantdiscovery,madebyK.Henselinthebeginningofthe
twentieth century, is that it is possible to regroup all these rings with zero di-
visors into a single object, called thep-adic integers, and denoted byZ
p,which
is an integral domain. Not only do we have the beneﬁt of being able to work
conveniently with all the congruences modulop,p
2
,p
3
,. . . simultaneously,
but we have the added beneﬁt of havingtopological properties,which add a
considerable number of tools that we may use, in particularanalytic methods
(note that this type of limiting construction is very frequent in mathematics,
with the same type of beneﬁts). When we say that we study our Diophantine
problem locally atp, this means that we study it inZ
p, or in the ﬁeld of
fractionsQ
pofZp. We will devote the entirety of Chapter 4 to the study
ofp-adic numbers and their generalizations. The reason for the word “local”
will become clear when we studyp-adic numbers.
A second way to reﬁne the study of our equation, which is explicit for
example in Weil’s estimates and conjectures, is to study our equations in
the ﬁnite ﬁeldsF
p
2,F
p
3, etc. (Note that usually this does not bring any
information for equations overQ, since in that case only local methods are
useful.) At this point, recall that the main theorem on ﬁnite ﬁelds (which
we will recall, with proof, in Chapter 2) is that for any prime powerq=p
n
there exists up to isomorphism exactly one ﬁnite ﬁeldF p
nof that cardinality,
and all ﬁnite ﬁelds have this form. They are of coursenotisomorphic to
Z/p
2
Z,Z/p
3
Z,...sincethelatter are not even ﬁelds. We will come back to the
structure of ﬁnite ﬁelds in the text. Once again, we can use a limiting process
of a slightly diﬀerent kind so as to put all these ﬁnite ﬁelds of characteristic
ptogether: this leads to
Fp, the algebraic closure ofF p.Inthiscaseweof
course do not say that we study it locally, but simply overFp.
Let us give simple but typical examples of all this. Consider ﬁrst the Dio-
phantine equationx
2
+y
2
= 3 to be solved in rational numbers or equivalently,
the Diophantine equationx
2
+y
2
=3z
2
to be solved in rational integers. We
may assume thatxandyare coprime (exercise). Looking at the equation
modulo 3, i.e., in the ﬁeldF
3, we see that it has no solution (x
2
andy
2
are
congruent to 0 or 1 modulo 3; hencex
2
+y
2
is congruent to 0 modulo 3 if
and only ifxandyare both divisible by 3, excluded by assumption). Thus,
our initial Diophantine equation does not have any solution.
We are here in the case of aquadraticDiophantine equation. It is crucial
to note that this type of equation canalwaysbe solved by local methods.
In other words, either we can ﬁnd a solution to the equation (often helped
by the local conditions), or it is possible to prove that the equation does not
have any solutions using positivity conditions together with congruences as
above (or equivalently, real andp-adic solubility). This is the so-calledHasse

6 1. Introduction to Diophantine Equations
principle, a nontrivial theorem (see Theorem 5.3.3) that is valid for asingle
homogeneous quadraticDiophantine equation, but is in general not true for
higher-degree equations or for systems of equations.
Consider now the Diophantine equationx
3
+y
3
= 1 to be solved in nonzero
rational numbers, or equivalently, the Diophantine equationx
3
+y
3
=z
3
to
be solved in nonzero rational integers. Once again we may assume thatx,y,
andzare pairwise coprime. It is natural to consider once more the problem
modulo 3. Here, however, the equation has nonzero solutions (for example
1
3
+1
3
≡2
3
(mod 3)). We must go up one level, and consider the equation
modulo 9 = 3
2
to obtain a partial result: since it is easily checked that an
integer cube is congruent to−1, 0, or 1 modulo 9, if we exclude the possibility
thatx,y,orzis divisible by 3 we see immediately that the equation does not
have any solution modulo 9, hence no solution at all. Thus we have proved
that ifx
3
+y
3
=z
3
, then one ofx,y,andz is divisible by 3. This is called
solving the ﬁrst case of FLT for the exponent 3. To show that the equation
has no solutions at all, even withx,y,orzdivisible by 3, is more diﬃcult
andcannotbe shown by congruence conditions alone (see Sections 6.4.5 and
6.9). Indeed Proposition 6.9.11 tells us that the equationx
3
+y
3
=z
3
has a
solution withxyzΛ= 0 in everyp-adic ﬁeld, hence modulop
k
for any prime
numberpand any exponentk(and it of course has real solutions). Thus,
the Hasse principle clearly fails here since the equation does not have any
solution in rational integers withxyzΛ= 0. When this happens, it is necessary
to use additionalglobalarguments, whose main tools are those of algebraic
number theory developed by Kummer et al. in the nineteenth century, and in
particular class and unit groups, which are objects of a strictlyglobalnature.
1.1.3 Dimensions
An important notion that has come to be really understood only in the
twentieth century is that ofdimension. It is not our purpose here to deﬁne
it precisely,
2
but to give a feeling of its meaning. We stick to the algebraic
and/or arithmetic case, since the topological or analytic case is simpler.
Consider ﬁrst the classical (algebraic) situation, say over the complex
numbersC.Apointis clearly of dimension 0, and more generally a ﬁnite
set of points deﬁned by a system of algebraic equations has dimension 0.
Similarly, a curve (for example deﬁned by a single equation in two variables
f(x, y) = 0 in aﬃne coordinates) has dimension 1 (note however that a
complexcurve has dimension 1 overCbut has dimension 2 overR), and so
on with surfaces which have by deﬁnition dimension 2, or arbitrary varieties
of higher dimension.
Consider now thearithmeticsituation, say over the integersZ.Iff(x, y)
is a polynomial in two variables with integer coeﬃcients, we can of course
2
in the language of schemes, it is the maximal length of an ascending chain of
irreducible subschemes.

1.1 Introduction 7
consider the curvef(x, y) = 0 as deﬁning a complex curve of dimension 1.
But when we consider the Diophantine equationf(x, y) = 0, then as we have
seen, it is essential to consider it also modulopand more generally in thep-
adic ﬁeldsQ
pfor every primep(including the prime “at inﬁnity,” which gives
the ﬁeldR). Thus, as a Diophantine equation,f(x, y) = 0 should not be seen
as a curve (i.e., of dimension 1), but in fact as a surface, called anarithmetic
surface. In other words, the ringZmust be considered to be of dimension
1 (its points being the prime numbersptogether with 0 corresponding to
the prime at inﬁnity), and any system of equations considered as a system
of Diophantine equations overZshould be considered to have one additional
dimension compared to its ordinary complex dimension. See Exercise 4 for
an illustration.
One of the goals of the modern theory ofarithmetic geometryis to ex-
tend to arithmetic surfaces and more generally to arithmetic varieties of any
dimension results known for ordinary surfaces and varieties.
Using these notions, we can quite naturally put a hierarchy on the objects
that naturally occur in algebraic number theory.
– Finite ﬁelds. These are the simplest objects, not only because they are
ﬁnite (ﬁnite rings and groups are extremely diﬃcult to study; see Exercise
3) but because they have a very simple structure, which we will recall in
detail in the text. They occur asresidue ﬁelds(we will see the meaning of
this later, butZ/pZ is a typical example).
– Local ﬁelds. Local ﬁelds of characteristic 0 are the p-adic ﬁeldsQ
p,the
real numbersR, and their ﬁnite extensions, which are thep-adic ﬁeldsK
p
and the ﬁeldC. There are also local ﬁelds of nonzero characteristic, which
we will not consider in this book.
– Global ﬁelds, and rings of dimension1. Global ﬁelds are the ﬁeld
of rational numbersQ, its ﬁnite extensions (i.e., number ﬁelds), and in
nonzero characteristic the ﬁeldsF
q(X) and their ﬁnite extensions, which
are the function ﬁelds of curves. The corresponding rings of integers of
these global ﬁelds (Z, Z
K,Fq[X], etc.) are of dimension 1.
–Any object of higher dimension will be called a curve, surface, etc. Be
careful with the terminology: when we speak of acurve, it usually means
a variety of dimension 1 over the base ﬁeld, but if we consider it overZ,it
then becomes an arithmeticsurface, hence of dimension 2 = 1+1. Another
possible confusion is that a complex curve is a real manifold of dimension
2, i.e., a surface, here because 2 = 2·1.
The reason and necessity of using this language cannot be clearly under-
stood without a course in modern algebraic geometry, but nevertheless it is
a good thing to have in mind, since it explains the utmost importance of the
objects that we are going to study.

8 1. Introduction to Diophantine Equations
1.2 Exercises for Chapter 1
1. The following problem seems similar to the congruent number problem, but is
much simpler. Show that for any integernthere exists a (not necessarily right-
angled) triangle with rational sidesa,b,andc and arean(recall that the area
is given byn
2
=s(s−a)(s−b)(s−c), wheres=(a+b+c)/2). Try to give
a complete parametrization of such triangles (I do not know the answer to this
latter question).
2. Let us say that a triangle isalmost equilateralif it satisﬁes the following two
conditions: its vertices have integer coordinates, and the lengths of its sides are
three consecutive integersa−1,a,anda+ 1. Show that such a triangle exists
if and only ifahas the forma=(2+
√
3)
k
+(2−
√
3)
k
forkα1 (you will ﬁrst
need to know the solution to the Pell–Fermat equation, see Proposition 6.3.16).
3. This exercise is mainly to emphasize that ﬁnite ﬁelds are really simple objects.
LetG(n)(respectivelyR(n),F(n)) be the number of groups (respectively rings
with nonzero unit, ﬁelds) of ordernup to isomorphism. ComputeG(n)for
1φnφ11 (you will need a little group theory for this),F(n) for 1φnφ100
(using the theory recalled in the next chapter), andR(n) for as many consecutive
values ofnstarting atn= 1 as you can. In the same ranges compute the number
G
a(n) of abelian groups, andR c(n) the number of commutative rings.
4. The goal of this exercise is to illustrate the fact thatZ[X] has dimension 2.
(a) LetIbe a nonzero ideal ofZ[X]. Prove that there exists a primitive polynomial
G(X)∈Z[X] such thatQI=G(X)Q[X], and thatI⊂G(X)Z[X]. We set
J=I/G(X), which is again a nonzero ideal ofZ[X], and it containsI.
(b) Prove that there existsn∈Z
α1such thatJ∩Z=nZ(you must prove that
nε=0).
(c) From now on, assume thatIis a prime ideal. Prove that eitherJ=Z[X], or
thatJis a prime ideal.
(d) IfJ=Z[X]provethatG(X ) is irreducible inQ[X], and conversely that if
G(X) is irreducible thenI=G(X)Z[X] is a prime ideal.
(e) Finally, we assume from now on thatJε=Z[X], or equivalently, thatG(X)/∈I,
hence thatJis a prime ideal. Prove thatJ⊂I,hencethatI=J, and deduce
thatG(X)=1.
(f) Prove that the integerndeﬁned above is a prime numberp, and in particular
thatnε=1.
(g) Prove that there exists a polynomialH(X)∈Z[X] such thatI=pZ[X]+
H(X)Z[X], and that the reduction
H(X)inF p[X] is either 0 or is irreducible in
F
p[X]. In particular, you must show that
H(X) cannot be a nonzero constant.
Conversely, ifH(X) is irreducible inF p[X] prove that the above idealIis a
prime ideal.
(h) Conclude that the (nonzero) prime ideals ofZ[X] have three types: ﬁrst the
idealsI=G(X)Z[X]withG(X)∈Z[X] irreducible and primitive, second the
idealsI=pZ[X]forp prime, and third the prime idealspZ[X]+H(X)Z[X]
withH(X) irreducible inF p[X].
The ascending chains of prime ideals
{0}⊂pZ[X]⊂pZ[X]+H(X)Z[X]and{0}⊂H(X)Z[X]⊂pZ[X]+H(X)Z[X]
mean that the dimension ofZ[X] is equal to 2.

Part I
Tools

2. Abelian Groups, Lattices, and Finite Fields
This chapter introduces a number of necessary tools for the rest of the book,
at diﬀerent levels. The theory of ﬁnitely generated abelian groups, including
the elementary divisor theorem and the structure theorem, as well as the
theory of ﬁnite ﬁelds, should be known at the undergraduate level, but expe-
rience shows that this is not always the case, so for completeness we will give
all the important proofs. Note that the theory of ﬁnitely generated abelian
groups extends almost completely verbatim to ﬁnitely generated modules over
a principal ideal domain; we will have the occasion to use this more general
setting over the ring ofp-adic integersZ
p.
At a deeper level in this chapter we will also describe important results
on the number of solutions of systems of polynomial equations over ﬁnite
ﬁelds, culminating with the Weil conjectures proved by Deligne. Finally, we
also include a section on lattices, seen mainly from the point of view of the
LLL algorithm, which will be the main tool that we will use in applications
to Diophantine equations.
2.1 Finitely Generated Abelian Groups
AsetG is an abelian group if and only if it is aZ-module. We will use
indiﬀerently both terms, but we will usually use abelian group when we want
to emphasize group-theoretic properties, while we will useZ-module when
considering bases or generating families.
2.1.1 Basic Results
Lemma 2.1.1.LetGbe a ﬁnitely generated torsion-free abelian group gen-
erated byx
1,...,xn, and assume thatGcannot be generated by fewer thann
elements. Then there is no nontrivial relation
Λ
1ΘiΘn
aixi=0witha i∈Z.
Proof.Assume the contrary, and among all sets ofngenerators and all
such relations on them, choose one for which
Λ
1ΘiΘn
|ai|is the smallest. We
distinguish two cases:

12 2. Abelian Groups, Lattices, and Finite Fields
–If at least two of thea iare nonzero, then permuting subscripts and chang-
ing signs if necessary we may assume thata
1ﬀa2>0. Clearlyx 1,x1+x2,
x
3,...,xnstill generateG, and the relation between these generators is
(a
1−a2)x1+a2(x1+x2)+···+a nxn= 0, and the corresponding sum
of absolute values of coeﬃcients is thus strictly smaller than the preceding
one, a contradiction.
–If only one of thea
i,saya 1, is nonzero, the relation isa 1x1=0,and
sinceGis torsion-freex
1= 0; henceGis generated by then−1 elements
x
2,...,xn, a contradiction. ιβ
Corollary 2.1.2.Any ﬁnitely generated torsion-freeZ-module is free.
Proof.Indeed, choose a generating system (x
i) having the smallest number
of elements. By the lemma, it is aZ-basis ofG. ιβ
Theorem 2.1.3 (Elementary divisor theorem I).LetGbe a ﬁnitely gen-
erated torsion-free(hence free)abelian group, and letHbe a subgroup ofG.
There exists a basisx
1,...,xnofGand strictly positive integersm 1,...,mr
for somerﬃnsuch thatm i|mi+1for1ﬃiﬃr−1and such that them ixi
for1ﬃiﬃrform a basis forH. In addition, ifHhas ﬁnite index inGthen
r=n.
Proof.We can assumeHnontrivial, otherwise we can chooser=0.
For the moment lety
1,...,ynbe any basis ofG. For any nonzeroh=
ﬀ
1ﬃiﬃn
aiyi∈H,setd(h) = gcd(a 1,...,an). I claim that this does not
depend on the chosen basis, but only onh: indeed, any otherZ-basis is given
in terms of the initial one by ann×nintegral matrixPwhose inverse is also
integral, in other words that has determinant±1 (the group of such matrices
is denoted by GL
n(Z)). IfAis the column vector of the (a i), the new coeﬃ-
cients are given by the vectorP
−1
A. Clearly the GCD of the coeﬃcients of
Adivides that ofP
−1
A, and sinceA=PP
−1
A, the converse is also true;
hence the GCD is the same, proving our claim.
Choose forha nonzero element ofHfor whichd(h) is as small as pos-
sible, and choose a basisy
1,...,ynofGfor which the corresponding sum
ﬀ
1ﬃiﬃn
|ai|is as small as possible. If two of thea iwere nonzero, as in the
proof of Lemma 2.1.1, we could decrease
ﬀ
1ﬃiﬃn
|ai|by modifying the basis,
a contradiction. Thus only onea
iis nonzero, and after permuting subscripts
we may assume thata
1=m 1>0 is the only nonzero coeﬃcient.
Now letz=
ﬀ
1ﬃiﬃn
biyibe any element ofH. Then we obtain succes-
sively
–m
1|b1since otherwise 0<b 1−cm1<m1for the Euclidean quotientcof
b
1bym 1would gived(z−ch)ﬃb 1−cm1<m1=d(h), a contradiction.
–
ﬀ
2ﬃiﬃn
biyi=z−(b 1/m1)h∈Hclearly.
–For eachi,m
1|bi, for otherwiset=m 1y1+
ﬀ
2ﬃiﬃn
biyi∈Hwould have
d(t)<m
1=d(h).

2.1 Finitely Generated Abelian Groups 13
Finally, letG 1be the group generated byy 2,...,yn,letH 1=G 1∩H,
and choosex
1=y1.Wehaveprovedthat
G=Zx
1⊕G1andH=Zm 1x1⊕H1,
and that the coeﬃcients of the elements ofH
1ony 2,...,yn, hence on any
Z-basis ofG
1, are divisible bym 1.
We now repeat the process onG
1andH 1instead ofGandH,andby
the last remark, we obtainm
1|m2, and we continue untilHis exhausted,
proving the main part of the theorem. Ifr<n, then themx
nform∈Z
belong to distinct cosets ofH,soH does not have ﬁnite index inG,proving
the last point. 
Corollary 2.1.4.With the notation of the theorem, if we denote by
xthe
class of an element ofGinG/H, we have
G/H=
∈
1ﬃiﬃr
(Z/m iZ)
xi⊕
∈
r<iﬃn
Z
xi.
Proof.Clear. Note that this is anequality, not only an isomorphism.
Corollary 2.1.5.Any subgroup of a ﬁnitely generated free abelian group is
a ﬁnitely generated free abelian group of lower dimension.
Proof.Also clear,Hbeing free on them
ixifor 1ﬃiﬃrandrﬃn.
In a diﬀerent direction we have the following result, which can also be
proved directly (see Exercise 1).
Corollary 2.1.6.LetV∈Z
n
be a column vector ofnglobally coprime in-
tegers. There exists an integral matrixA∈GL
n(Z)(in other words with
determinant±1)havingVas ﬁrst column.
Proof.In the proposition, we letG=Z
n
andH=ZV. There exists a basis
A
1,...,AnofGandd∈Z ﬀ1such thatdA 1is a basis ofH. In particular,
V∈dkA
1for somek∈Z, and since the coeﬃcients ofVare globally coprime
it follows thatd=1andk=±1; hence±V=A
1is the ﬁrst column of the
matrix of theA
i, which is in GLn(Z). 
We now come to the general structure theorem for ﬁnitely generated
abelian groups.
Theorem 2.1.7 (Elementary divisor theorem II).LetGbe a ﬁnitely
generated abelian group. There exist elementsx
1,...,xnofGand positive in-
tegersm
1,...,mrfor somerﬃnsuch thatm i>1for1ﬃiﬃr,m i|mi+1
for1ﬃiﬃr−1,m ixi=0for1ﬃiﬃr, and such that every element of
Gcan be written uniquely in the form
ﬀ
1ﬃiﬃn
aixiwith0ﬃa i<miwhen
1ﬃiﬃr. Furthermore,n,r, and them
iare unique.

14 2. Abelian Groups, Lattices, and Finite Fields
Proof.To prove existence, lety 1,...,yNbe any generators ofG, and let
G
∗
=
⊥
1∗i∗N
ZYiγZ
N
be the free abelian group onNgeneratorsY i.
There is a natural surjection fromG
∗
toGsendingY itoyifor alli,andif
H
∗
is its kernel, we have a natural isomorphismGγG
∗
/H
∗
, so under this
isomorphism we can identifyGwithG
∗
/H
∗
. SinceG
∗
is free, we can apply
the above theorem and corollary toG
∗
andH
∗
, obtaining generatorsX iand
integersM
ifor 1∗i∗R. If we denote byx ithe image ofX iinG,wethus
have
G=
→
1∗i∗R
(Z/M iZ)xi⊕
→
r<i∗N
Zxi.
We can evidently suppress from this equality all the components withM
i=1,
and if we callm
ithe remainingM i(in the same order), all the conditions of
the theorem are satisﬁed, proving existence.
To prove uniqueness, assume that we have a second such representation,
where we add
χ
to all the letters. We ﬁrst prove thatn=n
χ
. Indeed, assume
for instance thatn>n
χ
, and letpbe a prime dividingm 1ifr>0, and letp
be any prime otherwise. Using the ﬁrst representation, we have a surjection
fromGto (Z/pZ)
n
sending
∞
a ixito the vector ofa imodulop, which makes
sense sincep|m
ifor alli. Since the (x
χ
i
) generateG, it follows that (Z/pZ)
n
must be generated by the images of thex
χ
i
, which is absurd since there exist
at mostn
χ
<nsuch images.
Now for anym>0, consider the groupmG={mx, x∈G}.Wecan
obtain a representation as above by replacing thex
iby themx i,miby
m
i/gcd(m i,m), and deleting themx ifor whichm i|m. It follows that for
aﬁxedi, m
iis uniquely deﬁned by the property thatm iis the smallest
m>1 for which a canonical representation ofmGas above uses at most
n−igenerators. 
As in Corollary 2.1.4, we can restate the existence part of the theorem by
writing
G=
→
1∗i∗r
(Z/m iZ)xi⊕
→
r<i∗n
Zxi.
Corollary 2.1.8.Any subgroup of a ﬁnitely generated abelian group is ﬁnitely
generated.
Proof.Once again, we use a ﬁnitely generated free abelian groupG
∗
and
a surjective map fromG
∗
toG.IfHis a subgroup ofG, denote byH
∗
the
inverse image ofHby this map. By Corollary 2.1.5,H
∗
is ﬁnitely generated,
and the images of a ﬁnite set of generators ofH
∗
generateH. 
We now easily deduce the structure theorem for ﬁnite abelian groups:
Theorem 2.1.9.LetGbe a ﬁnite abelian group. There exist unique integers
m
i>1for1∗i∗ksuch thatm i|mi+1for1∗i<k, and nonunique
elementsg
i∈Gsuch that

2.1 Finitely Generated Abelian Groups 15
G=
∈
1ﬃiﬃk
(Z/m iZ)gi,
so that in particularGγ
λ
1ﬃiﬃk
(Z/m iZ).
Proof.Indeed, ifGis ﬁnite it is ﬁnitely generated. We have seen above as
a consequence of Theorem 2.1.7 that any such group can be written
G=
∈
1ﬃiﬃr
(Z/m iZ)gi⊕
∈
r<iﬃk
Zgi,
for someg
i∈Gandm i>1 such thatm i|mi+1for 1ﬃi<r.Ifr<k , then
Gcontains copies ofZ; hence it is inﬁnite. Thus, ifGis ﬁnite we must have
r=k, proving the theorem. ιβ
Finally, note that there is a matrix version of the elementary divisor the-
orem, called theSmith normal form. Recall that a matrix isunimodularif
it is an element of GL
k(Z), i.e., an integral square matrix with determinant
equal to±1 (not only +1).
Theorem 2.1.10 (Smith normal form). LetAbe a square integral ma-
trix with nonzero determinant. There exist two unimodular matricesUandV
and a diagonal integral matrixDwith strictly positive diagonal entries such
thatD=UAV,andifD =(d
i,j)thend i,i|di+1,i+1 for alliﬃk−1.
Proof.We apply Theorem 2.1.3 toG=Z
k
andHthe group ofZ-linear
combinations of columns ofAconsidered as elements ofZ
k
. We leave to the
reader to check that we thus obtain the present theorem (Exercise 2).ιβ
Note thatDis unique butUandVare not (for instance ifAis the
identity matrixI, thenD=Iand we can take any matrixUandV=U
−1
).
To ﬁnish this section, recall the following.
Deﬁnition 2.1.11.LetGbe a group andg∈G.ThesetEof elements
e∈Zsuch thatg
e
=1is of the formkZfor a uniquekﬀ0.Ifk=0we say
thatghas inﬁnite order, otherwise we callktheorderofginG. It is thus
characterized by the following:g
k
=1,andg
n
=1if and only ifk|n.
Proposition 2.1.12.LetGbe an abelian group and letg∈Gbe an element
of ﬁnite orderk=k
1k2withk 1andk 2coprime. There existg 1andg 2inG
of respective ordersk
1andk 2such thatg=g 1g2.
Proof.Sincek
1andk 2are coprime there exist integersu 1andu 2such
thatu
1k1+u2k2=1.Wesetg 1=g
u2k2
andg 2=g
u1k1
. It is clear that
g
1g2=g, and furthermore by deﬁnitiong
n
1
= 1 if and only ifg
u2k2n
=1if
and only ifk|u
2k2nif and only ifk 1|u2n, and sinceu 1k1+u2k2=1,k 1
andu 2are coprime, hencek 1|n, so thatg 1has orderk 1, and similarlyg 2
has orderk 2. ιβ

16 2. Abelian Groups, Lattices, and Finite Fields
2.1.2 Description of Subgroups
Given a ﬁnite abelian groupG, it is often useful to enumerate the subgroups
ofG. This can easily be done using theHermite normal formof a matrix.
Recall the following deﬁnition.
Deﬁnition 2.1.13.Ann×nmatrixMis said to be in(upper triangu-
lar)Hermite normal form(HNF for short)ifMis upper triangular with non-
negative integral entries, the diagonal entriesm
i,iofMare strictly positive,
and the nondiagonal entriesm
i,jwithj>iare such that0Θm i,j<mi,i.
Note that this deﬁnition can easily be generalized to nonsquare matrices
(see Deﬁnition 6.2.1), but here we do not need this generality.
It is easy to show that ifAis an integral matrix with nonzero determinant,
there exists a unimodular matrixUsuch thatH=AUis in HNF, andH
(and thereforeU) is unique. More generally, ifAis ann×kmatrix withnΘk
of maximal rankn, there exists a unimodular matrixUand a matrixHin
HNF such thatAU=(H|0), concatenation ofHwithk−nzero columns,
andHis unique (but notUifk>n); see Exercise 3 and Proposition 6.2.2.
The HNF is useful in many contexts, essentially of algorithmic nature. Its
relevance here is the following result.
Theorem 2.1.14.LetGbe a ﬁnite abelian group, and using the notation of
Theorem 2.1.9, write
G=
π
1ΘiΘk
(Z/m iZ)gi.
Denote byDbe thek×kdiagonal matrix whose diagonal entries are the
integersm
iand byEthe row vector whose entries are the generatorsg i.The
subgroupsG
Ω
ofGare in one-to-one correspondence with left divisorsMof
Din HNF, i.e., integral matricesMin HNF such thatM
−1
Dhas integral
entries. The correspondence is as follows:
(1)IfM=(m
i,j)1Θi,jΘk is such an HNF matrix, the subgroupG
Ω
is generated
byE
Ω
=EMwith relations given by the columns of the matrixM
−1
D.
(2)Conversely, ifG
Ω
isasubgroupofGgenerated by a row of elementsE
Ω
,
there exists an integer matrixPsuch thatE
Ω
=EP, and the corre-
sponding HNF matrixMis the HNF of the matrix(P|D)obtained by
concatenation of the matricesPandD.
(3)In this correspondence, we have|G
Ω
|=|G|/det(M), or equivalently,
|G/G
Ω
|=[G:G
Ω
] = det(M ).
Proof.By deﬁnition, the following sequence is exact:
1−→
k
π
i=1
miZ−→Z
k
φ
−→G−→1,
where

2.1 Finitely Generated Abelian Groups 17
φ(x1,...,xk)=

1ﬃiﬃk
xigi.
Let (ε
i)1ﬃiﬃk be the canonical basis ofZ
k
, and let Λ be the subgroup ofZ
k
deﬁned by Λ =
λ
i
miεi(this is alattice, see Section 2.3.1). We thus have a
canonical isomorphismGγZ
k
/Λ, obtained by sending theith generatorg i
ofGto the class ofε i.
Subgroups ofZ
k
/Λ have the form Λ
χ
/Λ, where Λ
χ
is a lattice such that
Λ⊂Λ
χ
⊂Z
k
. By existence and uniqueness of the HNF of a matrix of maximal
rank, such a lattice Λ
χ
can be uniquely deﬁned by a matrixMin HNF such
that the columns of this matrix express aZ-basis of Λ
χ
on theε i. The con-
dition Λ
χ
⊂Z
k
means thatMhas integer entries, and the condition Λ⊂Λ
χ
means thatM
−1
Dalso has integer entries, since it is the matrix that ex-
presses the given basis of Λ in terms of that of Λ
χ
. In terms of generators,
this correspondence translates into the equalityE
χ
=EM. Furthermore, if
0
Gdenotes the unit element ofGthenE
χ
X=0 Gif and only ifEMX=0 G;
henceMX=DY,orX=M
−1
DY,andsoifG
χ
is the subgroup ofG
corresponding to Λ
χ
/Λ, it is given in terms of generators and relations by
(EM,M
−1
D), proving (1).
For (2), we note that the entries ofEDare equal to 0
G; hence ifE
χχ
=
E(P|D), we have simply added some 1
G
ﬀ’s to the generators ofG
χ
.Thus,
the group can be deﬁned by the generatorsE
χχ
and the matrix of relations
of maximal rank (P|D), hence also by (E
χχ
,M), whereMis the HNF of this
matrix.
For (3), we know thatM
−1
Dexpresses a basis of Λ in terms of a basis of
Λ
χ
; hence
|G
χ
|=|Λ
χ
/Λ|= det(M
−1
D)=|G|/det(M).

Example.The matrixMcorresponding to the subgroup{0
G}ofGisM=
D, and the matrix corresponding to the subgroupGofGisM=I
k,the
k×kidentity matrix.
Remark.Thanks to the above theorem the algorithmicenumerationof sub-
groups of a ﬁnite abelian group is reduced to the enumeration of the integral
left divisors of a diagonal matrix. This is considerably more technical, and
since the present book is not primarily algorithmic in nature we refer to
[Coh1] Section 4.1.10 for complete details on the subject.
2.1.3 Characters of Finite Abelian Groups
First, an important notation. Here and in the rest of this book we use the sym-
bolζ
nfor aprimitiventh root of unity, either viewed as an (abstract) alge-
braic number (see Chapter 3), or as an element ofC(for instance exp(2iπ/n)),
or sometimes of other ﬁelds such asp-adic ﬁelds. Ifd|n, it is understood
that we chooseζ
dsuch thatζ d=ζ
n/d
n
.

18 2. Abelian Groups, Lattices, and Finite Fields
Deﬁnition 2.1.15.LetGbe a ﬁnite abelian group. AcharacterofGis
a group homomorphism fromGto the multiplicative groupC
∗
of nonzero
complex numbers. The group of characters ofGis called thedual groupof
Gand denoted by

G. The character sending all elements ofGto1,whichis
the unit element of

G, is called the trivial character.
Letχ∈

G.If|G|=n,thenforeveryg∈Gwe haveχ(g)
n
=χ(g
n
)=
χ(1) = 1. It follows that any character takes values in the unit circle of
complex numbers of modulus 1, more precisely in the group ofnth roots of
unity, which we denote byµ
n.
Proposition 2.1.16.LetGbe a ﬁnite abelian group. The dual group

Gis
noncanonically isomorphic toG(hence has the same cardinality).
Proof.By the structure theorem for ﬁnite abelian groups (Theorem 2.1.9)
we know that
G=
π
1φiφk
(Z/m iZ)giγ
π
1φiφk
(Z/m iZ)
for certain integersm
iandg i∈G. On the other hand, we clearly have
φG
1⊕G2γ

G1⊕

G2. It follows that to prove the ﬁrst part of the proposition it
is suﬃcient to prove it for ﬁnite cyclic groups. But such a group is isomorphic
toZ/mZ for somem, and characters ofZ/mZ are simply determined by the
image of the class of 1, which can be anymth root of unity. Thus we have
(canonically)
φ
Z/mZ γµ
m, and (noncanonically)µ mγZ/mZ, proving the
result. ιβ
Remark.It follows from the proof that characters of a ﬁnite abelian group
Gcan be described very concretely. We writeG=
λ
1φiφk
(Z/m iZ)gias in
Theorem 2.1.9, and for eachiwe choose somea
i∈Z/m iZ. We then deﬁne
χ
a1,...,ak


γ
1φiφk
xigi

=

1φiφk
ζ
aixi
mi
,
where theζ
mi
are ﬁxed primitivem ith roots of unity inC. Even more explic-
itly, since all them
idividem k=m,weﬁxζ =ζ mand chooseζ mi
=ζ
m/mi
,
so that
χ
a1,...,ak


γ
1φiφk
xigi

=ζ
S
withS=
γ
1φiφk
aixi(m/mi),
so the value of the characterχcan be represented by the integerS.
Corollary 2.1.17.LetGbe a ﬁnite abelian group andHasubgroupofG.
Any character ofHcan be extended to exactly[G:H]characters ofG.In
particular, the natural restriction map from

Gto

His surjective.

2.1 Finitely Generated Abelian Groups 19
Proof.Letfbe the above restriction map. The kernel offis the group
of charactersχofGthat are trivial onH, in other words the characters of
G/H. It follows by the proposition that the cardinality of the image offis
equal to
|

G|
|
Ω
G/H|
=
|G|
|G/H|
=|H|=|

H|;
hencefis surjective, as claimed, and the number of preimages of a character
by the restriction map is equal to|Ker(f )|=[G:H]. ιβ
Remarks.(1) Since we have proved surjectivity using a counting argument,
the above reasoning cannot be applied to inﬁnite abelian groups. See
Proposition 4.4.43 for a generalization using Zorn’s lemma.
(2) The importance of the above corollary is not so much the exact number
of extensions of a character, but the simple fact that such extensions
exist. For instance:
Corollary 2.1.18.Ifgis not the unit element ofGthere existsχ∈

Gsuch
thatχ(g)Λ=1.
Proof.Ifn>1 is the order ofgwe setχ(g
k
)=ζ
k
n
, which deﬁnes a
character such thatχ(g)Λ= 1 on the subgroupHofGgenerated byg,and
we extendχtoGusing the above corollary. ιβ
Corollary 2.1.19.The natural mapaνω(χνωχ(a))gives a canonical iso-
morphism fromGto the dual of its dual.
Proof.By the preceding corollary this map is injective, and since both
groups have the same cardinality it is an isomorphism. ιβ
One of the most important properties of characters is their orthogonality
properties as follows.
Proposition 2.1.20.LetGbe a ﬁnite abelian group and letKbe a commu-
tative ﬁeld.
(1)Ifχ
1andχ 2aredistinctgroup homomorphisms fromGtoK
∗
then
γ
g∈G
χ1(g)χ
−1
2
(g)=0,
or equivalently, ifχis not the constant homomorphism equal to1then
γ
g∈G
χ(g)=0.

20 2. Abelian Groups, Lattices, and Finite Fields
(2)In the special caseK=Cthen ifg 1andg 2aredistinctelements ofG
we have
γ
χ∈

G
χ(g1g
−1
2
)=0,
or equivalently, ifgis not the unit element ofGthen
γ
χ∈

G
χ(g)=0.
Proof.The two statements of (1) are clearly equivalent, as are those of
(2). SetS=
∞
g∈G
χ(g). Leth∈Gsuch thatχ(h)∞= 1. Then
χ(h)S=
γ
g∈G
χ(h)χ(g)=
γ
g∈G
χ(hg)=
γ
g
∞
∈G
χ(g
χ
)=S,
henceS= 0 sinceχ(h)∞=1.
In the special caseK=C,ifgis not the unit element ofG, Corollary
2.1.18 shows that there existsψ∈

Gsuch thatψ(g)∞= 1. The reasoning we
have just presented is thus applicable (we setS=
∞
χ
χ(g) and show that
ψ(g)S=S). ιβ
Note that ifK∞=C(more precisely ifKis not an algebraically closed
ﬁeld of characteristic 0 or of characteristic not dividing|G|), then (2) is not
necessarily true, see Exercise 4.
2.1.4 The Groups (Z/m Z)
∗
We ﬁrst recall that for any commutative ringR,R
∗
denotes the group of
unitsofR, i.e., invertible elements inR. This is equal toR\{0}if and only
ifRis a ﬁeld.
Lemma 2.1.21.Letv∞2be an integer anda∈Z.
(1)Ifpis an odd prime number, the following statements are equivalent:
(a)a
p
≡1 (modp
v
).
(b)a≡1 (modp
v−1
).
(c)For anyw∞v−1there existsb∈Zcoprime topsuch that
a≡b
(p−1)p
v−2
(modp
w
).
In particular,v
p(a
p
−1) =v p(a−1) + 1whenv p(a−1)∞1.
(2)Ifp=2anda≡1 (mod 4)the following statements are equivalent:
(a)a
2
≡1 (mod 2
v
).
(b)a≡1 (mod 2
v−1
).

2.1 Finitely Generated Abelian Groups 21
If in additionvα4, they are also equivalent to the following statement:
c) For anywαv−1there existsb∈Zodd such that
a≡b
2
v−3
(mod 2
w
).
Proof.(1). To prove that a) is equivalent to b) we seta=1+b.By
Fermat’s theorem we havea
p
≡a(modp); hence we may assume thatp|b,
so that
a
p
=1+pb+
γ
2∗j∗p−1
χ
p
j
ω
b
j
+b
p
.
Sincep|
ε
p
j
∆
it follows that all the terms with 2∗j∗p−1 are divisible
bypb
2
. Furthermore, sincepα3andp|b,wealsohavepb
2
|b
p
. It follows
that all the terms afterpbhavep-adic valuation strictly greater than that of
pb; hencev
p(a
p
−1) =v p(pb)=1+ v p(a−1), proving that a) and b) are
equivalent.
By the Euler–Fermat theorem, ifbis coprime topwe haveb
(p−1)p
v−2
=
b
φ(p
v−1
)
≡1 (modp
v−1
); hence c) implies b). To prove the converse, we
consider the mapffrom (Z/p
w−v+2
Z)
∗
to the subgroupGof elements of
(Z/p
w
Z)
∗
congruent to 1 modulop
v−1
induced byyνωy
(p−1)p
v−2
.Bythe
equivalence of a) and b),y≡1 (modp
w−v+2
) impliesy
p
v−2
≡1 (modp
w
);
hence the mapfis well deﬁned. Sincey
(p−1)p
v−2
≡1 (modp
v−1
) its image lies
inG. Furthermore,f(
y) = 1 if and only ify
(p−1)p
v−2
≡1 (modp
w
) if and only
ify
p−1
≡1 (modp
w−v+2
), again by the equivalence of a) and b). Since we will
prove below that (Z/p
w−v+2
Z)
∗
is a cyclic group, the number of
yof order
dividingp−1 in that group is equal top−1, so that|Ker(f)|=p−1. It follows
that|Im(g)|=φ(p
w−v+2
)/p=p
w−v+1
, and since clearly|G|=p
w−v+1
, this
means thatfis surjective, proving the equivalence of b) and c).
(2). Ifa=1+bwith 2
v−1
|bthena
2
=1+2b+b
2
≡1 (mod 2
v
) since
vα2. Conversely, ifa
2
≡1 (mod 2
v
), then 2
v
|(a−1)(a+1), and sincea≡1
(mod 4),v
2(a+ 1) = 1, and therefore 2
v−1
|a−1, proving the equivalence
of the ﬁrst two conditions.
Ifbis odd we haveb
2v−3
=(b
2
)
2
v−4
≡1 (mod 2
v−1
) by what we have
just shown andb
2
≡1 (mod 8). Conversely, we consider as above the map
ffrom (Z/2
w−v+3
)
∗
to the subgroupGof elements of (Z/2
w
Z)
∗
congruent
to 1 modulo 2
v−1
induced byyνωy
2
v−3
. Sincew−v+3α2, by what we
have just showny≡1 (mod 2
w−v+3
) impliesy
2
v−3
≡1 (mod 2
w
); hencef
is well deﬁned, and as abovey
2
v−3
≡1 (mod 2
v−1
) so the image offlies in
G.Wehavef(
y) = 1 if and only ify
2
v−3
≡1 (mod 2
w
) if and only ify
2
≡1
(mod 2
w−v+4
) by what we have just shown. Writingy
2
−1=(y+ 1)(y −1)
and noting thatw−v+4α3 we see that this is equivalent toy≡±1
(mod 2
w−v+3
); hence|Ker(f )|= 2. It follows that|Im(g)|=φ(2
w−v+3
)/2=
2
w−v+1
, and since clearly|G|=p
w−v+1
, this again means thatfis surjective,
ﬁnishing the proof. ιβ

22 2. Abelian Groups, Lattices, and Finite Fields
Note also the following generalization, which we will need later.
Lemma 2.1.22.Letpbe a prime number,san integer such thats≡1
(modp),andletn∈Z
>0.Whenp=2, assume that eithers≡1 (mod 4)or
nis odd. Then
v
p(s
n
−1) =v p(s−1) +v p(n).
Proof.Writen=p
v
mwithpΛm. We prove the lemma by induction on
v. Assume ﬁrst thatv= 0, so thatn=m. By the binomial theorem, we have
s
m
−1=
γ
1ΘkΘm
Ω
m
k
∆
(s−1)
k
.
SincepΛm,wehave
v
p
ΩΩ
m
1
∆
(s−1)
1
∆
=v
p(m(s−1)) =v p(s−1),
while for 2ΘkΘmwe have
v
p
ΩΩ
m
k
∆
(s−1)
k
∆
Λkv
p(s−1)Λ2v p(s−1).
Sincev
p(s−1)Λ1 by assumption it follows thatv p(s
m
−1) =v p(s−1) as
claimed.
WhenvΛ1 we apply Lemma 2.1.21 (1) toa=s
p
v−1
m
≡1 (modp) (since
s≡1 (modp)), hencev
p(a
p
−1) =v p(a−1), so the result forpodd follows
by induction onv. Similarly the result forp= 2 follows by induction from
Lemma 2.1.21 (2). 
Corollary 2.1.23.Ifp=2ands≡1 (mod 2)then for alln∈Z
>0we have
v
p(s
n
−1) =
∂
v
p(s−1) ifnis odd,
v
p(s
2
−1) +v p(n/2) ifnis even.
Proof.The casenodd is given by the lemma. Whennis even, sinces
2
≡1
(mod 4) the lemma givesv
p(s
n
−1) =v p((s
2
)
n/2
−1) =v p(s
2
−1) +v p(n/2).

Proposition 2.1.24.LetmΛ2be an integer, and letm=
Θ
1ΘiΘg
p
vi
i
be
its decomposition into a product of powers of distinct primes. The abelian
group structure of(Z/mZ)
∗
is given as follows:
(1)We have
(Z/mZ)
∗
γ

1ΘiΘg
(Z/p
vi
i
Z)
∗
.

2.1 Finitely Generated Abelian Groups 23
(2)IfpΛ3andvΛ1, we have
(Z/p
v
Z)
∗
γZ/(p
v−1
(p−1))Z;
in other words the group(Z/p
v
Z)
∗
is cyclic.
(3)Ifp=2andvΛ3, we have
(Z/2
v
Z)
∗
γZ/2
v−2
Z×Z/2Z .
In addition, if desired we can always take the class of5as generator of
the groupZ/2
v−2
Z,and−1 as generator ofZ/2Z.
(4)Ifp=2andvΘ2, then(Z/2Z)
∗
is the trivial group and(Z/4Z)
∗
γ
Z/2Z.
Proof.(1). I ﬁrst claim that ifm=m
1m2with gcd(m 1,m2) = 1, then
(Z/mZ)
∗
γ(Z/m 1Z)
∗
×(Z/m 2Z)
∗
. Indeed, there exist integersuandv
such thatum
1+vm 2= 1. Denoting bya+nZthe class inZ/nZ of an
integeramodulo any integern(which is in fact the correct notation), we
consider the mapf
1from (Z/mZ)
∗
to (Z/m 1Z)
∗
×(Z/m 2Z)
∗
deﬁned by
f
1(a+mZ)=( a+m 1Z,a+m 2Z), and the mapf 2in the other direction
deﬁned byf
2(b+m 1Z,c+m 2Z)=cm 1u+bm 2v+mZ. Sincem=m 1m2
these maps are clearly well deﬁned, and we immediately check that they are
group homomorphisms which are inverse to one another, proving my claim.
By induction ong, this proves (1).
(2). For any integersmanda, we will say thatais aprimitive root
modulomif the class ofamodulomgenerates (Z/mZ)
∗
(so that in particular
(Z/mZ)
∗
is cyclic and gcd(a, m ) = 1). By Corollary 2.4.3 below we know that
(Z/pZ)
∗
is cyclic; in other words there existsg∈Zthat is a primitive root
modulop. By Fermat’s theorem, i.e., the fact that|(Z/pZ)
∗
|=p−1, we know
thatg
p−1
≡1 (modp). Assume ﬁrst thatg
p−1
Λ1 (modp
2
). I claim thatgis
a primitive root modulop
v
for anyvΛ1. Indeed, otherwise there would exist
a prime divisorqofφ(p
v
)=p
v−1
(p−1) such thatg
φ(p
v
)/q
≡1 (modp
v
).
Sincea
p
≡a(modp) for alla,ifq|p−1wehaveg
p
v−1
(p−1)/q
≡g
(p−1)/q
≡1
(modp), which is absurd sincegis a primitive root modulop.IfqΛp−1
thenq=p, and since by Lemma 2.1.21,a
p
≡1 (modp
k
)fork Λ2 implies
thata≡1 (modp
k−1
), the congruenceg
p
v−2
(p−1)
≡1 (modp
v
) implies that
g
p−1
≡1 (modp
2
), contrary to our assumption.
Assume now thatg
p−1
≡1 (modp
2
). Theng+pis also a primitive root
modulop, and sincepΛ3,
(g+p)
p−1
≡g
p−1
+(p−1)pg
p−2
≡1−pg
p−2
Λ1 (modp
2
),
so it follows from what we have just proved thatg+pis a primitive root
modulop
v
for allvΛ1, proving (2).
(3). LetHdenote the subgroup of (Z/2
v
Z)
∗
formed by the classes of
integers congruent to 1 modulo 4. Since any odd integer is congruent to

24 2. Abelian Groups, Lattices, and Finite Fields
±1 modulo 4, we clearly have (Z/2
v
Z)
∗
γH×Z/2Z, so that in particular
|H|=φ(2
v
)/2=2
v−2
. I claim thatHis cyclic, generated by the class of 5.
Since the only prime dividing 2
v−2
is 5, it is enough to show that 5
2
v−3
Λ1
(mod 2
v
). If we assume the contrary, then since 5≡1 (mod 4), using once
again Lemma 2.1.21 we would obtain that 5≡1 (mod 2
3
), a contradiction.
(4) is trivial. ιβ
Corollary 2.1.25.FormΛ2, the group(Z/mZ)
∗
is cyclic if and only if
m=2,4,p
k
,or2p
k
forpan odd prime andkΛ1.
Proof.Note that in a cyclic group the number of elements of order dividing
2 is less than or equal to 2. From the proposition it follows that the number of
such elements is exactly equal to 2
ωo(m)+ω 2(m)
, whereω o(m) is the number of
distinct odd prime divisors ofmandω
2(m) = 0, 1, or 2 according to whether
v
2(m)∗1,v 2(m) = 2, orv 2(m)Λ3 respectively. The corollary follows from
the inequalityω
o(m)+ω 2(m)∗1. ιβ
Remark.The proofs made in this subsection sometimes use forward refer-
ences, so we must be careful to check that we do not use a circular argument.
Assume for instancepodd, the remark is the same forp= 2. The correct
order of proof (which would be less practical for presentation) is as follows:
the equivalence of (a) and (b) of Lemma 2.1.21 (1), as well as the cyclicity
of (Z/pZ)
∗
which follows from Corollary 2.4.3, are proved directly, without
any reference to results of this subsection. From these two results we deduce
by induction as in the proof of Proposition 2.1.24 (2) that (Z/p
k
Z)
∗
is cyclic
for allkΛ1. Using this, we can ﬁnally prove the equivalence of (b) and (c)
of Lemma 2.1.21 (1).
When working in a group (Z/p
v
Z)
∗
withpand odd prime andvΛ2,
it is tempting to use the existence of a primitive rootgmodulop
v
, since
all elementsacan simply be written asa=g
x
for somexuniquely deﬁned
moduloφ(p
v
). However, this is not always a good idea. For future reference,
we note the following lemma, which usually gives a better representation.
Lemma 2.1.26.Letpbe an odd prime, letvΛ2,andletg be a primitive
root modulop
v
. For anyacoprime topthere existxandysuch that
a≡g
p
v−1
x
(1 +p)
y
(modp
v
),
andxis unique modulop−1,yis unique modulop
v−1
.
Proof.Sincegis a primitive root, we can writea≡g
x
(modp
v
), so that
a
p
v−1
≡g
p
v−1
x
(modp
v
), and sinceghas orderp
v−1
(p−1), it is clear thatx
is unique modulop−1. Sincea
p
v−1
−1
≡1 (modp) by Fermat’s little theorem,
we must simply show that for anyb≡1 (modp) there existsysuch that
b≡(1 +p)
y
(modp
v
). Indeed, the mapy∃∈(1 +p)
y
is clearly a group homo-
morphism from the additive groupZ/p
v−1
Zto the multiplicative subgroup of

2.1 Finitely Generated Abelian Groups 25
(Z/p
v
Z)
∗
of elements congruent to 1 modulop. The groups having the same
cardinality, and the map being injective by Lemma 2.1.22, it follows that it
is bijective, showing the existence ofyand its uniqueness modulop
v−1
.ιβ
Remark.This lemma can be better understood in the context ofp-adic
numbers (see Chapter 4): indeed, 1 +pis naturally called atopological gen-
erator, and the exponent ysuch thatb≡(1 +p)
y
(modp
v
) can be given
explicitly in terms ofp-adic logarithms asy= log
p(b)/log
p(1 +p)modp
v
.
Note also that 1 +pis only one possible choice, and that we could just as
well choose 1 +kpfor anykΓ0 (modp).
2.1.5 Dirichlet Characters
According to Proposition 2.1.16, the group
ﬃ
(Z/mZ)
∗
of characters of (Z/mZ)
∗
is (noncanonically) isomorphic to (Z/mZ)
∗
. It is convenient to extend such a
character to the whole ofZ/mZ by setting it equal to 0 outside of (Z/mZ)
∗
,
and then toZby composing with the natural surjection fromZtoZ/mZ:
Deﬁnition 2.1.27.A Dirichlet character modulomis a mapχfromZto
Csuch that there exists a characterψ∈
ﬃ
(Z/mZ)
∗
such thatχ(n)=0 if
gcd(n, m) >1,whileχ(n)=ψ (
n)otherwise, wherendenotes the class ofn
modulom.
Note thatχis still multiplicative, and thatχ(m+n)=χ(n) for alln,in
other wordsχis periodic of period dividingm. Furthermore, the values ofχ
are either equal to 0 orφ(m)=| (Z/mZ)
∗
|th roots of unity inC. By abuse
of language we will say thatχhasordernif the corresponding character
ψ∈
ﬃ
(Z/mZ)
∗
has ordern, in other words ifnis the positive generator of the
group of integersksuch thatχ
k
is equal to the trivial character modulom
(see the following deﬁnition). Thus the order of a character modulomdivides
φ(m).
Deﬁnition 2.1.28.Letχbe a character modulom.
(1)Ifd|mwe say thatχcan be deﬁned modulodif there exists a Dirichlet
characterχ
dmodulodsuch thatχ(n)=χ d(n)as soon asgcd(n, m)=1 .
(2)Theconductorof a Dirichlet character is the smallest(for divisibil-
ity)positive integerf|msuch thatχcan be deﬁned modulof.
(3)We will say thatχisprimitiveif the conductor ofχis equal tom,in
other words ifχcannot be deﬁned modulo a proper divisor ofm.
(4)Thetrivial character, often denoted by χ
0, is the character deﬁned by
χ(n)=1whengcd(n, m)=1 andχ(n)=0whengcd(n, m) >1.Itis
the unique character modulomof conductor1.
Remarks.(1) It is clear thatχis primitive if and only ifχcannot be deﬁned
modulom/pfor every primep|m.

26 2. Abelian Groups, Lattices, and Finite Fields
(2) Ifχcan be deﬁned modulod|mandχ dis the corresponding character
modulod, it is clear thatχ=χ
0χd.
Proposition 2.1.29.The number of primitive characters modulomis equal
toq(m),where
q(m)=m

pεm
Ω
1−
2
p
∈

p
2
|m
Ω
1−
1
p
∈
2
.
In particular, there are none if and only ifm≡2 (mod 4).
Proof.I refer the reader to Section 10.1 for the elementary techniques used
here. For any integerfdenote byq(f) the number of primitive characters
modulof. By deﬁnition we have
φ(m)=| (Z/mZ)
∗
|=|
∗
(Z/mZ)
∗
|=
γ
f|m
q(f).
In terms of formal Dirichlet series, this means that
ζ(s)
γ
m∞1
q(m)m
−s
=
γ
m∞1
φ(m)m
−s
=
ζ(s−1)
ζ(s)
;
hence
∞
m∞1
q(m)m
−s
=ζ(s−1)/ζ(s)
2
, and the proposition follows by look-
ing at the Euler factor atp. 
Corollary 2.1.30.Letχbe a primitive character modulomwithmeven.
Then for allnwe haveχ(n+m/2) =−χ(n).
Proof.By the proposition we know that 4|m. The result is thus trivial
ifnis even since both sides vanish; otherwise, denoting byn
−1
an inverse of
nmodulomwe have sincenis odd
χ(n+m/2) =χ(n)χ(1 + (m/2)n
−1
)=χ(n)χ(1 +m/2).
We haveχ((1+m/2)
2
)=χ(1+m(m/4+1)) = 1, henceχ(1+m/2) =±1. If we
hadχ(1 +m/2) = 1 then we would haveχ(k)=χ(k+(m/2)k )=χ(k+m/2)
for allkodd and evidently for all evenk,soχ would be deﬁned modulom/2,
a contradiction. Thusχ(1 +m/2) =−1; henceχ(n+m/2) =−χ(n). 
A similar reasoning will lead to a useful characterization of primitive
characters. We ﬁrst need a lemma which is useful in many contexts.
Lemma 2.1.31.Ifgcd(a, b, c)=1there exists an integerksuch that
gcd(a+kb, c)=1.

2.1 Finitely Generated Abelian Groups 27
Proof.Note that this lemma would immediately follow from Dirichlet’s
theorem on primes in arithmetic progression (see Theorem 10.5.30), but it is
not necessary to use such a powerful tool. In fact, we can givekexplicitly: I
claim that
k=

p|c
pΛ(a/gcd(a,b))
p
is a suitable value. Indeed, letpbe a prime dividingc. We must show that
it does not dividea+kb. Assume ﬁrst thatpΛa/gcd(a, b). Thusp|k;
hencepΛa/gcd(a, b)+ kb/gcd(a, b), and since gcd(a, b, c)=1,wehave
pΛa+kbas desired. Assume now thatp|a/gcd(a, b), hencepΛk. Since
a/gcd(a, b) is coprime tob/gcd(a, b) by deﬁnition of the GCD, it follows that
pΛkb/gcd(a, b); hencepΛa/gcd(a, b)+kb/ gcd(a, b) and once againpΛa+kb
as desired. ιβ
The characterization of primitive characters is a consequence of the fol-
lowing lemma:
Lemma 2.1.32.Letχbe a character modulomand letd|m.Thenχcan
be deﬁned modulodif and only if for allasuch thata≡1 (modd)and
gcd(a, m)=1we haveχ(a)=1.
Proof.The condition is clearly necessary: ifχ(a)=χ
d(a) for allasuch
that gcd(a, m) = 1, then if in additiona≡1 (modd)wehaveχ(a)=1.
Conversely, assume the condition satisﬁed, and letabe such that gcd(a, d)=
1. We want to deﬁneχ
d(a). By the preceding lemma, there existsksuch
that gcd(a +kd, m) = 1. We will setχ
d(a)=χ( a+kd). Since gcd(a+
kd, m) = 1, this is nonzero, and furthermore ifk
Ω
is another integer such that
gcd(a+k
Ω
d, m) = 1, thenb=(a+k
Ω
d)(a+kd)
−1
(inverse taken modulom,
which makes sense since gcd(a+kd, m) = 1) is such thatb≡1 (modd). By
assumption it follows thatχ(b) = 1, in other words thatχ(a+k
Ω
d)=χ( a+kd),
so our deﬁnition ofχ
d(a) does not depend on the choice ofk.Itisthen
immediate to check thatχ
dis a character modulodsuch thatχ d(a)=χ(a)
when gcd(a, m )=1. ιβ
Corollary 2.1.33.Letχbe a character modulom,letd|mwithd<m,
and assume thatχcannot be deﬁned modulod.
(1)For al lrwe have
γ
amodm
a≡r (modd)
χ(a)=0.
(2)Iffis a periodic function of period dividingd, then
γ
0Θa<m
χ(a)f(a)=0.

28 2. Abelian Groups, Lattices, and Finite Fields
In particular, ifχis a primitive character, these properties are true for all
d|msuch thatd<m.
Proof.The proof of (1) is identical to that of Proposition 2.1.20: by the
lemma, there existsb≡1 (modd) with gcd(b, m) = 1 and such thatχ(b)∞=
1. The mapa∃∈abis clearly a bijection from the set of integers modulo
mcongruent tormodulomto itself, so that by multiplicativity we have
χ(b)S=S, henceS= 0, whereSis the sum to be computed. For (2) we
writea=qd+rso that
γ
0∗a<m
χ(a)f(a)=
γ
0∗r<d
f(r)
γ
0∗a<m
a≡r (modd)
χ(a)=0
by (1). ιβ
Proposition 2.1.34.Letm∈Z
∞1,letm 1andm 2be two coprime positive
integers such thatm=m
1m2,andletχ be a Dirichlet character modulom.
(1)There exist unique charactersχ
imodulom isuch thatχ=χ 1χ2, in other
words such thatχ(n)=χ
1(n)χ 2(n)for alln.
(2)The order ofχis equal to the LCM of the orders ofχ
1andχ 2.
(3)The characterχis primitive if and only if bothχ
1andχ 2are primitive.
Proof.(1). Since them
iare coprime, there exist integersuandvsuch that
um
1+vm 2= 1. In view of the mapf 2deﬁned in the proof of Proposition
2.1.24 (1), it is natural to setχ
1(x)=χ( xm 2v+m 1u)andχ 2(x)=χ(ym 1u+
m
2v). Since these maps are obtained by composing the homomorphismf 2
with the natural injections of (Z/m iZ)
∗
into (Z/m 1Z)
∗
×(Z/m 2Z)
∗
, it follows
that they are group homomorphisms hence deﬁne Dirichlet characters modulo
m
1andm 2respectively, and it is also clear thatχ=χ 1χ2.
(2). We haveχ(n)
k
= 1 for allncoprime tomif and only ifχ 1(n)
k
=
χ
2(n)
−k
for all suchn, hence if and only if the primitive character equivalent
toχ
−k
1
is equal to the one equivalent toχ
−k
2
. However, the conductor ofχ
−k
i
dividesm i,andm 1andm 2are coprime, so this is possible if and only ifχ
k
1
andχ
k
2
are trivial characters, hence if and only ifkis a multiple of the orders
ofχ
1andχ 2, proving (2).
The proof of (3) is immediate and left to the reader (Exercise 10).ιβ
It follows in particular from this proposition that any Dirichlet characterχ
modulomcan be written in a unique way as a product of Dirichlet characters
modulo the coprime prime powers dividingm.
Corollary 2.1.35.Letm=
∗
p
p
vp(m)
withv p(m)∞1be the decomposition
into prime powers ofm∈Z
∞1. The order of anyprimitivecharacter modulo
mis divisible byh(m)=
∗
p
p
vp(m)−1
,exceptif8 |min which case it is only
divisible byh(m)/2.

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141. De gefopte Gascogner.
Naar de schilderij van Nicolas Lancret (1690–1743), Louvre, Parijs.
De dubbele moraal uit zich in de sexueele opvoeding nog in velerlei andere richting. Zoo
—als uitvloeisel der geschetste opvattingen, waardoor de gemiddelde fatsoensmensch
zich bij zijn sexueele pedagogie laat leiden,—wordt den zoon groote vrijheid gelaten, de
dochter daarentegen wat men noemt kort gehouden. Bordeelbezoek, meisjes verleiden,
een maîtresse hebben, dit alles wordt door den kring van verwanten bij den jonkman
zooal niet onvoorwaardelijk goedgekeurd, dan toch als vergeeflijke lichtzinnigheid der
jeugd aangemerkt en vergoelijkt. Zelfs in kringen waarin men overhelt tot strenge zeden,
wordt voor de jeugdige mannelijke leden een oogje dicht gedaan. De drogredenen
waarmee dit wordt gemotiveerd, zijn alweer legio. Het geldt als een teeken, dat de
jongeman gezond, vurig bloed heeft, levenslustig en krachtig is en niet tot de
droogpruimers behoort. Door al vroeg intiem verkeer te hebben met vrouwen leert de
jongeling zijn onbeholpen bleuheid en verlegenheid in den omgang met de andere sexe
afleggen, hij krijgt op die manier ondervinding, vrijmoedigheid, zekerheid, leert de
wereld kennen en erlangt tallooze andere voordeelen. Intusschen worden de vrouwen,
die zoonlief al deze deugden moeten bijbrengen, en om zoo te zeggen zijn sexueele
opvoeding en zijn geslachtelijke vorming hebben te voltooien, juist in die kringen het
diepst veracht en verafschuwd, iets wat bij zoo belangrijke diensten, den zoon des huizes

142. Landelijke onschuld.
—Ja mijnheer, ik ga gauw naar Parijs bij mijn zuster, om ook
wat te verdienen.
—Zoo zoo, mijn kind, en wat doet ze daar in Parijs, die zuster
van je?
—Ze is daar rijk gemainteneerd, mijnheer.
Teekening van Henry Gerbault.
bewezen, bedenkelijk zweemt naar zwarte ondankbaarheid. Maar ook hier is de minste
schijn of schaduw van de meest elementaire consequentie weer ver te zoeken.
De eigenlijke redenen voor de
groote sexueele vrijheid, die men
de zoons laat in vergelijking met de
angstvallige bezorgdheid voor de
sexueele onbesprokenheid der
dochters, zijn natuurlijk deze, dat
eerstens den jonkman toch niet kan
worden belet te doen wat hem ten
deze goeddunkt en waartoe zijn
ontluikende zinnelijkheid hem
dringt, terwijl verder zijn sexueele
waarde, uitgedrukt in
huwelijkskansen, er in het minst
niet door vermindert. Men laat den
jongen dus wat vrijheid, wijl men
hem dien toch niet op den duur kan
benemen, en omdat de gevolgen
maatschappelijk van geen
beteekenis zijn. Deze motieven zijn
echter niet van dien aard, dat men
er goedschiks openlijk voor uit kan
komen. Daarom bedient men zich
van drogredenen en troost zich
daarbij met de gedachte, dat ieder
ander toch ook zoo doet.
Ten opzichte van het jonge meisje
staat de zaak geheel anders. Door
de minste opspraak reeds daalt zij
in sexueele waarde. De minste smet
op haar sexueel gedrag heeft een
onherstelbare waardevermindering tengevolge. Om dat te voorkomen wordt zij bij elken
stap met argusoogen bewaakt, want ook de minste kans van opspraak wenschen hare
natuurlijke exploitanten—haar ouders—niet te riskeeren. Het meisje moet, om op de
huwelijksmarkt haar volle handelswaarde te behouden, volstrekt onbesproken blijven.
Daarom is men even streng voor de dochters als men toegeeflijk is voor de zoons. Zij
wordt van alle sexueele gevaren geïsoleerd, en tegelijkertijd dat men zoonlief de dolste
streken laat uithalen, zonder zich er zich veel om te bekommeren, wordt aan de dochter
romanlezen, schouwburgbezoek en dergelijke ontzegd, om „haar hoofd niet op hol te

143. De kleine nichtjes.
—Zeg, Annie, heb jij je neefje al eens een zoen
moeten geven?
—Nog nooit, Marie, ik zou het niet willen, hij kon
het wel eens niet toelaten!
Uit: „Sect”, Weenen.
brengen”, d. w. z. haar niet in ongewenschte
avontuurlijk-romantische stemming te
brengen.
Dikwijls komt met koddige duidelijkheid
aan het licht, dat het daarbij in het minst
niet is te doen haar zinnelijk immuun te
maken. Bij bals en dergelijke mag zij zich
vrijelijk half naakt voor een geheel
gezelschap te pronk stellen—mits onder
toezicht en mits anderen hetzelfde doen.
Wat niet in opspraak brengt, is geoorloofd,
al is het in werkelijkheid nog zoo
bedenkelijk; wat wel opspraak en gefluister
verwekt, is niet geoorloofd, al is het op zich
zelf nog zoo onschuldig. Evenmin zoekt de
gemiddelde sexueele opvoeding neiging tot
coquetterie te beteugelen of tegen te gaan.
Integendeel, het coquette meisje verbetert
haar kansen en—men weet zulks bij instinct
—loopt allerminst gevaar zelf tot
onberaden onvoorzichtigheden te vervallen
—zij ontsteekt het vuur, maar blijft zelf
koel. Coquetterie is dus een waardevolle
deugd, een karaktertrek die men eer moet
toejuichen en in de hand werken dan
tegengaan.
Het grondbeginsel van de sexueele
opvoeding is bij de groote massa der
onnadenkenden en gemakzuchtigen:
onkundig houden. Bij de toepassing van dat
grondbeginsel vormt de natuurlijke
kinderlijke nieuwsgierigheid naar allerlei wat met het geslachtsleven in verband staat,
een groote moeielijkheid. Deze nieuwsgierigheid toch leidt tot vragen—de bekende
lastige kindervragen. De antwoorden, die het kind krijgt op zijn lastige vragen, dragen
het karakter van beminnelijke misleiding. Het nieuwe broertje is gebracht door de
ooievaar, of het is geplukt van den boom, of gegroeid in een groote kool. En zoo voort.
Aan den anderen kant is de kinderlijke onwetendheid een bron van vermaak voor de
ouderen. Het is bekend hoe gretig b.v. tantes en ooms en ook eigen ouders kinderen
uithooren betreffende hun sexueele kennis. Men brengt het kind daarover opzettelijk aan

144. Die moeders.
—Je maakt mij niet wijs, dat een fatsoenlijk meisje zulk
ondergoed heeft te dragen.
Satire van J. L. Forain.
het praten, en vermaakt zich met zijn
onwetendheid, met zijn halve kennis,
met zijn twijfel, onderwerpt het aan
een verhoor om te vernemen wat hij
allemaal weet of denkt van die
dingen. Maar om het kind, hoe dan
ook, ernstig over geslachtelijke
onderwerpen te onderhouden, het
met voorzichtig beleid sexueel te
onderrichten en voor te bereiden op
datgene wat het als geslachtswezen
tegemoet gaat, daarvoor is men in
den regel zelf te onkundig en vooral
te onernstig in het sexueele.
Men is zoo gewoon over sexueele
onderwerpen alleen maar te spreken
in min of meer frivolen toon, dat
men niet bij machte is ze ook maar
een oogenblik met ernst te
bespreken. Het kind snapt al heel
gauw, dat zijn opvoeders hem in die
dingen wat op den mouw spelden.
Het zoekt op andere manieren zijn
natuurlijke nieuwsgierigheid te
bevredigen. Zoo wordt dan het nog onbedorven kindergemoed naar onzuivere bronnen
gedreven en het ontvangt de gezochte kennis in een vorm, die in den regel minstens
bedenkelijk is. En het kind leert ook al heel gauw, dat het zijn kennis op sexueel gebied
voor ouderen moet verbergen. Zoo ontstaat het type van het quasi-onnoozele en
onschuldige kind, voor wie in werkelijkheid het geslachtsleven een open boek is en dan
gewoonlijk een pornografisch boek.

145. Die kinderen.
—Pas zestien en nou al een vrijer?
—Eén is toch zeker het minste wat je hebben kan!
Friedrich Schröder. in „Düsseld. Monatshefte”.

146. Privaatles in den tunnel.
Fransche humoreske.
Amor leidt het spel.
Teekening van Gerda Wegener.

147. In het reddingshuis.
—Ja zuster, ik heb net als u mijn buik vol van de wereld.
Karikatuur van Moloch.
Tot de „lastige kindervragen”
behooren in de eerste plaats die naar
de herkomst van het nieuwe broertje.
Ieder kind wil weten waar het
vandaan komt. En de sexueele
opvoeding heeft een heele reeks van
antwoorden op desbetreffende
vragen verzonnen, die door de
ernstige sexueele pedagogiek zoo
niet als misleidend worden
verworpen, dan toch bedenkelijk
worden geacht, ofschoon er dikwijls
een zekere zin voor poëzie in valt op
te merken. Een tweede categorie van
lastige kindervragen betreft de
geslachtskenmerken. Gesprekken als
dit tusschen vijf- tot achtjarigen zijn
geen uitzonderingen: Hebben jullie
een nieuw kindje gekregen, Wim?
—Ja Annie.—Is het een zusje of een
broertje?—Ik geloof een zusje.
—Is het net zoo als ik, met een
boezelaartje en lang haar? dan is het
een zusje.—Hoe men weet of het
nieuwe kindje een zusje of een
broertje is, is het kind een raadsel en
het stelt zijn vragen. Het wordt weer
op dezelfde wijze door zijn
opvoeders met een kluitje in het riet
gestuurd. Merkt het verschil op in zekere organen, dan worden dikwijls de allerkoddigste
verklaringen daarvan opgedischt.
De sexueele opvoeding, die tot grondbeginsel heeft: onwetend houden zoo lang het maar
eenigszins gaat,—de opvoeding wier hoogste wijsheid dus eigenlijk is: niet op te voeden
—mag, gezien de ondervinding van elken dag, tweeërlei resultaat verwachten: of het
kind doet op eigen gelegenheid sexueele kennis op en gaat dan beladen met een
onzuivere phantasie de gevaren der puberteit tegemoet; of het kind blijft inderdaad
onwetend, en is dan in zijn onkunde zooveel weerloozer in de gevaren, die het van het
intreden der puberteit af van alle kanten zullen bedreigen.

148. Sexueele opvoeding.
—Mag ik u mijn parapluie aanbieden?
—Zeker mijnheer, tot aan de hoek daarginds, waar mijn man
mij opwacht.
Uit „Wiener Karikaturen”.
De gevolgen van het stelsel van
onwetend-houden in sexueele zaken
zijn in den regel niet, dat er
werkelijk onwetendheid bestaat,
maar dat er onwetendheid wordt
voorgewend. Onderdehand beijveren
beide sexen zich wat ze kunnen om
sexueele kennis op te doen. Het
meisje ontvangt haar kennis ten deze
van haar vriendinnen en vooral ook
door lectuur. En de onderwijzeressen
der mannelijke jeugd zijn de
prostituees. Zoo moest er een paar
jaar geleden in Frankrijk door een
vereeniging voor zedelijke en
sanitaire voorbehoeding een boekje
worden uitgegeven en op groote
schaal verspreid, dat ook in het
Nederlandsch is verschenen en tot
titel draagt: „Wat volwassen jongens
wel eens mogen weten.” Dit boekje
handelt over de gevaren der
afdwaling van half-volwassenen op
geslachtelijk gebied. Het onwetende
meisje weet ook zeer goed, hoeveel
meer sexueele vrijheid de jongens
„genieten” en welk gebruik deze
daarvan plegen te maken. Een meisje
stelde haar moeder, die haar den
avond voor haar huwelijk overlaadde
met wenken en raadgevingen, de
snibbige vraag of haar Willem dien
avond nu ook al die wijze lessen kreeg van zijn moeder.
De sexueele opvoeding beoogt eigenlijk alleen sexueele waarden zoo voordeelig
mogelijk in exploitatie te brengen. Zij beoogt zelden of nooit in te wijden in ’t wezen
van het geslachtsleven. Zij kan dit niet beoogen, reeds hierom niet, wijl dit voor het
meerendeel der opvoeders zelf een gesloten boek is. Maar dit weten allen, dat de sexe,
en speciaal de vrouwelijke sexe, een artikel van waarde is en dat men bij een eventueel
bod niet te haastig moet zijn met toeslaan, daar er allicht nog een beter bod kan worden
gedaan.

149. Amor ingewijd in de geheimen van het toilet.
Gravure van Carracci, 17e eeuw.
De zucht om de sexueele waarde van het
jonge meisje op de voordeeligste manier te
exploiteeren, openbaart zich al bijzonder
sterk bij den kleinen middenstand. Hier is
elke dochter een belangrijk stuk materieel
bezit, waarop in den regel groote
verwachtingen worden gebouwd. De
sexueele opvoeding wordt daar, natuurlijk
veelal onbewust, geheel op ingericht. Het
spreekt van zelf, dat de dochter veel te
goed wordt geacht om met een man van
lageren of gelijken stand te trouwen. Ook
in dit opzicht is de minachting voor den
werkenden stand het sterkst bij degenen,
die daar maatschappelijk het naast aan
grenzen. „Mijn dochter is veel te goed
voor een gewonen werkman”, zoo denken
en spreken duizenden moeders, vooral
onder die wier maatschappelijk verschil
met den „gewonen werkman” meer dan
dubieus is. „Voor mijn dochter is toch
zeker wel wat beters te krijgen”, is de
volgende conclusie in dezen
gedachtengang. Vooral ook het meisje zelf
wordt diep van dit gevoelen doordrongen.
Het wordt haar aanhoudend voorgehouden,
dat het maar aan haar zelf ligt, zich een
schitterende positie te verwerven, of, om in
de geijkte terminologie te spreken, een goede partij te doen. Het eerste waar zij op heeft
te passen, is natuurlijk, zich niet te vergooien aan een of anderen armoedzaaier. Zij moet
het ook niet probeeren met zulk een verafschuwd wezen te komen aanzetten. Dat wordt
haar van haar eerste meisjesjaren af zoo diep mogelijk ingeprent.
Ieder manspersoon, die plannen ten opzichte van het sexueele kleinood schijnt te
koesteren, wordt door de zorgzame ouders allereerst gekeurd op zijn vermoedelijken
welstand. Valt de eerste taxatie al niet te gunstig uit, dan worden alle mogelijkheden tot
toenadering onverbiddelijk afgesneden. Voor geen middel wordt daarbij dan
teruggedeinsd. Het eerste der vele middelen die ter beschikking staan is, hem in de
oogen van het meisje zoo diep mogelijk te kleineeren, hem verachtelijk te maken, afkeer
en tegenzin in te boezemen jegens den kalen gelukzoeker, die het wagen durft de oogen
te slaan op een meisje zoover boven zijn stand. Hij moet maar gaan bij de meiden van
zijn soort, maar het niet zoeken bij een meisje van nette familie …. wat denkt zoo’n

150. Liefde te Parijs.
—En dan, ’t is zoo eng, een man.
Satire van J.L. Forain.
151. De reddende modehoed.
goochemerd wel? dat wou er zich
zeker fijn indraaien, maar dan is hij
aan ’t verkeerde kantoor, enz. Van
het meisje zelf wordt natuurlijk
verwacht dat ze zal inzien, dat zoo’n
persoon niets is voor haar. Iedereen
zou haar uitlachen, zij zou in de
heele stad over de tong gaan. En wat
zou juffrouw die en juffrouw die
zich verkneuteren, als ze hoorden
enz. En dan, welk vooruitzicht met
zoo’n snuiter!

Amerikaansche karikatuur van Kepler.
152. Hardhandige sexueele pedagoog.
Illustratie van Holbein in Erasmus’ „Lof der Zotheid”.
Uitgave 1731.

153. Het onwetende bruidje.
—Nu, kind, weet je het voornaamste. Onthoud het nu goed.
—Och mama, dat wist ik allemaal al zoo lang.
Uit „Wiener Caricaturen”.

Het gevoel.
Rijks Prent Cabinet.
Uit de serie: De Vijf Zintuigen, van Abraham Bosse, 18
de
Eeuw.
Valt de aanvankelijke schatting minder ongunstig uit, is de eerste indruk niet
onbevredigend, dan wordt omzichtig getracht zekerheid te erlangen. Er wordt navraag
gedaan, links en rechts geïnformeerd—men gaat niet over één-nachts-ijs! Vallen de zoo
vergaarde inlichtingen min of meer bevredigend uit, dan krijgt het meisje nog tallooze
instructies om zich toch voor onberaden stappen te wachten. Want zoolang het geval nog
min of meer twijfelachtig is, is men nog zeer wantrouwend. En hebben de ouders, meer
speciaal de mama, zelf iets voor de huwbare dochter op het oog, iets waarvan de
financieele kwaliteiten ten volle bekend zijn, dan wordt ook zulk een twijfelachtige
adspirant-minnaar met alle middelen waarover een kwaadsprekende tong kan beschikken
—en dat zijn alle middelen—onverbiddelijk geweerd en weggewerkt.
Zoo bar en vijandig als het onthaal is waarop een vrijer van het ongewenschte soort zich
kan voorbereiden, zoo nederbuigend-lievig en onderdanig-tegemoetkomend is de
houding jegens een partuur, die aan de verwachtingen beantwoordt. Evenals in het eerste
geval eerst alles wordt beproefd om het meisje afkeerig te maken van den

154. Gevaarlijke onnoozelheid of gelegenheid maakt den dief.
Gravure naar een schilderij van Boucher (1703–1770).
verafschuwden
non-valeur, en,
als dat niet baat,
alle
geweldmiddelen
der ouderlijke
macht te baat
worden genomen
om de dreigende
familieramp af te
wenden, evenzoo
wordt in het
tweede geval
alles gedaan om
het meisje er toe
te brengen van de
haar geboden
gelegenheid
gebruik te maken.
Zoo noodig treedt
de zorgzame
moeder zonder
gemoedsbezwaar
op als
koppelaarster. Alles wat er toe leiden kan om het paar zoover te brengen, dat men niet
meer terug kan, wordt met overleg begunstigd; de gelegenheden om zoover te komen
worden desnoods noods expres geschapen. En stuit men op onwil bij het meisje, dan
wordt niets onbeproefd gelaten om dien onwil te breken.
Hoe komisch dit alles den koelen waarnemer ook moge aandoen, is het in werkelijkheid
toch een bron van tallooze stille familiedrama’s en van veel in stilte gedragen leed. Den
humoristen van alle gading levert het intusschen rijke stof tot geestigen spot, vlijmend
sarcasme, bittere ironie of goedmoedigen humor.

155. Kinderen van Flora.

Photo Hanfstaengl, München.
156. Een lustig paar.
Naar de schilderij van Gerard van Honthorst (1590–1656), Gemäldegalerie, Cassel.

VII.

157. Verliefde harten.
Holl. gravure, 17e eeuw.
DE POËZIE VAN HET MINNEN.
Goed vrijen is: zacht pratend hard liegen. Dat is zoo wat de eenige conclusie, die de
erotische humor open laat, als hij het teere thema van het minnen onder handen neemt. Dat
begint al bij den alledaagschen volkshumor, die zich uit in spreuk en spreekwoord: Eeden
van vrijers en beloften van zeelui duren niet langer dan tot de nood voorbij is.
En uit de hoogte boven de groote menigte, vanwaar de denkers en de wijsgeeren
minachtend op het gedoe daar diep in de laagte beneden hen neerzien, en de dingen die hen
bijzonder ergeren overgieten met honend sarcasme, hoort men al weinig andere klanken.
Evenwel, men moet toegeven,
dat verliefden, hoewel ze niet
anders doen dan liegen,
nochthans volkomen oprecht
zijn. Ook moet men toegeven,
dat zoo liegend te minnen
mooier is dan nuchter-waar
zonder liegen te minnen. Want
wat komt er dan van terecht?
Hoe laat Multatuli zijn
Droogstoppel-type, dat toch
ontegenzeggelijk de
meerderheid onder de
stervelingen vertegenwoordigt,
over vrijen en trouwen oreeren?
Natuurlijk als volgt: „Een
meisje is een engel. Wie dit het
eerst ontdekte heeft nooit
zusters gehad. Liefde is een
zaligheid. Men vlucht met het
voorwerp er van naar het einde
der aarde. De aarde heeft geen
einden, en die liefde is ook
gekheid. Niemand kan zeggen,
dat ik niet goed leef met mijn vrouw … zij heeft een sjaal van twee-en-negentig gulden, en
van zulk een malle liefde, die volstrekt aan het einde der aarde wil wonen is toch tusschen
ons nooit sprake geweest. Toen we getrouwd zijn, hebben we een toertje naar den Haag
gemaakt—ze heeft daar flanel gekocht, waarvan ik nog borstrokken draag—en verder heeft
ons de liefde nooit de wereld ingejaagd”.

158. Onder de mistletoe.
Miss Gushington: „Vindt u ook het Kerstfeest met
al zijn oude gewoonten niet heerlijk, mijnheer
Brown?” (Brown schijnt het daar maar matig mee
eens te zijn.)
John Leech, in „Punch”, Londen.

Verliefde harten.
Hollandsche kopergravure van J. Matham (1571–1631), naar de schilderij van Pieter Aartsen bijgenaamd
Langepier (1507–1573).
Waarin bestaat eigenlijk het liegen van verliefden, wat liegen ze dan toch? Eigenlijk alles
wat ze zeggen. Want ze zeggen elkaar alleen mooie dingen en wat mooi is, is niet waar, en
hoe mooier hoe leugenachtiger. De eerste leugen komt in elke liefdescomedie van de
mannelijke partij, als hij in zijn liefdesverklaring zoo hemelhoog opgeeft van zijn
verliefdheid. Daar kan niets van waar zijn; hij kan het zich hoogstens maar verbeelden,
doch hij is onderwijl hij er op los liegt toch oprecht. Is het ijs door den eersten leugen
gebroken, dan is de beurt van onwaarheden te zeggen voorloopig aan de andere partij, die
daarbij, naar men zegt, geheel en al in haar natuurlijk element is. O, wat wordt er bij het
eerste kennismaken, en de heele vrijage door, over en weer gesimuleerd! Al die vleierijtjes
en lieve naampjes, die hoogdravende verzekeringen van hem, dat afwijzen en weigeren en
zich laten naloopen van haar—alles comedie en toch echt. De ouverture van het minnespel
waarbij de een dingen zegt die niet waar kunnen zijn en de ander antwoorden geeft die zij
niet meent, schetst Lovendaal ons in zijn liedje van het Schipperinnetje:
Een scheepje lei al aan de kust
Op ’t deinend vlak te wiegen naar lust;
Daarinne zat een hupsch gezel,

159. De oude coquette.
—De heeren doktoren blijven er zeker geheel onverschillig onder, een vrouw naakt
te zien.
—In den regel wel, maar soms wordt men er toch wel eens naar van.
Humoreske van Abel Faivre, in „Les Maîtres Humoristes”.
Die neuriede lustig
en wel.
Met kwam een
aardig kind
voorbij;
Ik weet niet, wat de
visscher haar zei,
Maar ’t lief zei
lachend „nee, o
nee!
Ik ga in je scheepje
niet mee”.
Doch eer ’t een
maandje verder
was
Daar zeilde een
scheepjen over den
plas,
Het boord was
groen, de vlag woei
uit;
Zij was er de
visscher zijn bruid.
De flinke klant gaf
haar een zoen
En zei: „zeg nog
’reis néé zooals
toen”
Zij gaf er één weer
en zei: „jij plaag!
Ik mocht je van
harte, van harte
toch graag”
Kortom: vrouwen zeggen graag neen om nogmaals gevraagd te worden. Later komt, zoo
heel bij toeval, wel eens uit hoe gemeend die eerste afwijzingen en aarzelingen zijn—wie
veel liegt moet nu eenmaal een zeer sterk geheugen hebben om zich nooit eens te
verspreken. En dan vallen er wel eens ongewilde bekentenissen, zooals in het volgende
idylletje:

—Jong echtgenoot: Weet je nog, schatje, dat wij bij dit boschje door je mama werden
verrast, toen ik je den eersten kus gaf?
—Jong vrouwtje: Ja—en daar heeft die arme moe toen twee uren moeten staan wachten.
Dat het zoo is, dat wisten al de ouderwetsche ulevel-poëeten, blijkens onderstaand
gewrocht van zoo’n ulevel-genie:
De meisjes zeggen neen, maar meenen meestal ja;
Wie net doet of ze vlucht, loopt u in waarheid na.
Het neen antwoorden op verliefde voorstellen is zoo ernstig gemeend, dat dames
verontwaardigd zijn en het als een onvergeeflijke lompheid opnemen, als bedoelde
voorstellen worden voorgedragen op een wijze en onder omstandigheden die haar dwingen,
voor het oogenblik althans, tot een neen, dat werkelijk neen beteekent. Een heer, die twee
dames volgde, kreeg het volgende verwijt te hooren: Het is toch geen manier, mijnheer,
twee dames aan te spreken. Laat ons met rust of zorg dat u ook met u tweeën bent.
160. Duitsche karikatuur uit de 17e eeuw op de molensteenkragen en de Spaansche mode.
Als het minnen om echt poëtisch te zijn leugen noodig heeft, dan heeft het natuurlijk ook
diplomatie noodig. Want diplomatie is leugen door studie volmaakt tot een vak. Waar dus

161. De argumenten van den verliefden geldschieter.
Naar Picart, 18e eeuw.
162. Vurige liefde.
—Trouw maar gerust met me, ik ben een makkelijk
echtgenoot—tien maanden van het jaar op reis.
vaststaat, dat minnen bestaat
in liegen, daar staat tevens
vast, dat minnen bestaat uit
diplomatie, d.i. de hoogste
vorm van liegen. Die
diplomatie begint al dadelijk
bij de eerste kennismaking.
Wederkeerig wordt er niet
anders dan gediplomateerd,
van de zijde der vrouwelijke
partij het sterkst. Hij doet alles
om een goeden indruk te
maken, zich van den besten
kant te laten zien en het te
laten voorkomen of die beste
kant zijn gewone kant is, hij
doet zijn Zondagsche gezicht
voor en tracht hooge
gedachten te wekken omtrent
zijn persoonlijke waarde, zelfs in de
onbelangrijkste nietigheden, zooals b.v. zijn
maatschappelijke positie, die in engagements-,
verlovings- en dergelijke zaken nu toch
heelemaal geen factor van belang is. Zij
diplomatiseert door niet te begrijpen waar hij
heen wil en niet te zien wat iedereen ziet—hij
mocht anders eens denken dat hij haar laatste
hoop was. Dat gaat dan zoo een tijdje door.
Doordat zij diplomatiek hem ontwijkt, om niet
den schijn op zich te laden van hem na te
loopen, worden de ontmoetingen steeds
talrijker. En zijn uiterlijke uitrusting wordt
steeds statiger. Tot het tenslotte komt tot een
verklaring. Die dan ook weer een interessant
spel vormt van diplomatieke zetten over en
weer. Na het engagement gaat dat zoo door.
Blijkt hij oog te hebben voor nog andere
knappe meisjes, dan veroorzaakt dat alleen bij
in de liefdes-diplomatie zeer slecht onderlegde
vrijsters tranen, verwijten, een gebroken hart
en zoo meer. Een bijdehandte diplomate geeft
toe dat die of die knap is en brengt het gesprek

—Goed, maar wat moet ik in ’s hemelsnaam die
twee overige maanden beginnen?
H. Zasche in „Sect”, Weenen.
163. Vrouwenvereering.
—Waaraan denk je wel, Johan, als je me zoo in je armen
hebt?
—Aan de fooi, freule.
Franz von Reznicek, in „Simplizissimus”.
ongemerkt op wederzijdsche heerenkennissen
—een paar vleiende opmerkingen aan het
adres van zulke kennissen zijn haar
toovermiddel waarmee ze tegelijkertijd
zichzelf wreekt en hem straft en tevens tot
beterschap brengt.
Verlovingsgesprekken van bijvoorbeeld het verloofde meisje met haar moeder, zijn ook al
doortrokken van poëzie. Luisteren wij maar even.
—Ja, moe, aan tante Betsy moeten we
vast een verlovingskaart sturen. ’t
Mensch heeft een paar weken geleden
nog gezegd, dat ik wel zou blijven
zitten. Natuurlijk zei ze dat omdat ‘r
eigen dochters niks kunnen krijgen.
Maar nicht Sophie krijgt geen kaart—‘t
schepsel kijkt ons niet meer aan sinds
haar man van klerk adjunct is
geworden. En oom Willem en zijn
vrouw houden wij er natuurlijk ook
buiten. Sinds vader eenzelfde zaak is
begonnen als hij, hebben ze geen voet
meer over onzen drempel gezet. En u
begrijpt dat ik het niet erg prettig zou
vinden, als mijn aanstaande
schoonouders op de receptie kennis
maakten met zoo’n lomperd als oom
Willem, die van niets weet te praten dan
van den tijd dat hij matroos was. Nog al
wat fijns, matroos! Ik zou me
doodgeneeren. Verbeeld je, als je die
kennis liet maken met mijn
schoonouders. Neef Gerard, daar
denken we natuurlijk ook niet aan om
die uit te noodigen. Die weet nooit zijn
maat te houden, en als hij wat op heeft
slaat hij een taal uit dat je je
doodschaamt voor nette menschen.
Oom Johannes laten we stilletjes thuis,
die wil altijd ceremoniemeester spelen
als er wat te doen is, en dan verveelt hij
het heele gezelschap met zijn geschreeuw van: attentie heeren en dames! en met zijn

flauwe moppen. Neen, die moeten we dezen keer er ook maar buiten laten. Want mijn
aanstaande schoonouders zijn van veel te nette familie. U en vader moeten bij het kennis
maken ook maar het beste beentje voorzetten, dat ik me niet hoef te geneeren. Moe, u zal er
van opkijken, als ze hier komen. Zijn vader is een echte heer, met een deftig voorkomen, en
een ring met zoo ’n diamant. Hij is, geloof ik twee en vijftig, maar je zou zeggen hoogstens
veertig. Op z’n kantoor, ziet u, is hij zooveel als de patroon zelf—procuratiehouder heet
dat; hij teekent, zegt mijn beminde, briefjes van wel duizend gulden. En Frits z’n moeder, u
zal eens zien hoe die er uitziet, hoe ’n deftige dame, heelemaal in ’t zij, o zoo fijn. En Frits
z’n eene zuster krijgt piano-les van een rijksdaalder in het uur en zij is geëngageerd met
een inspecteur van politie. En mijn Frits is binnen het jaar chef-de-bureau, heeft hij gezegd.
U begrijpt, dat we nu ook een beetje om ons fatsoen moeten denken en dat het bij mijn
verlovingsfeest maar niet de zoete inval moet wezen voor iedereen. En bent u nu ook niet
blij en in uw schik met uw dochter, die het zoo getroffen heeft—zoo’n nette familie en een
aanstaande met zoo’n nette positie?….
164. Geen slot noch grendel kan den vurigen minnaar weren!
Fransche gravure, 18e eeuw.

165. De modeduivel.
Middeleeuwsche karikatuur.
En dan al die idyllische vrijages! Een en al poëzie. Het volgende stukje cultuurgeschiedenis
uit Jan Holland’s Darwinia geeft hiervan een te treffend beeld om het hier niet te citeeren.
Karel had een goed oogje op Lina. Want Lina had gevulde vormen en flikkerende oogen.
En Lina was ook lang niet onverschillig voor Karel. Want Karel was een goed gebouwd en
krachtig jonkman.
Karel was op bals in de gelegenheid geweest om
op te merken, dat Lina’s gevulde vormen geen
boerenbedrog waren. Lina had door Karel’s
onvermoeid dansen en door de kracht waarmee
hij haar omvatte en optilde bespeurd, dat hij
stalen spieren bezat.
Derhalve beminden Karel en Lina elkander.
Maar Karel’s middelen veroorloofden hem nog
niet om er een vrouw op na te houden, Lina’s
vader vond, dat hij zijn geld beter kon gebruiken
dan aan z’n dochter een bruidschat te schenken.
Zij had hem buitendien met van alles studeeren
al geld genoeg gekost.
Karel moest dus arbeiden om evenals Jacob z’n
Lea eerst te verdienen. Maar hij ging dat niet
doen door schapen te hoeden. Neen, hij richtte
met eenige kornuiten van goeden naam een
maatschappij van levensverzekering op. En hij
werkte ook niet zeven jaren. Neen, slechts zeven
maanden. Toen ging de maatschappij volgens de regelen der kunst failliet en redden de
oprichters zich met een flink kapitaal.
Karel had zich dus door wijs overleg en energie tot een gelukkig sterveling gemaakt. Nog
in den bloei der jaren kon hij onbekommerd het loon van zijn inspanning genieten.
Dadelijk stuurt hij een uitnoodiging aan Lina tot een onderhoud op zijn kamer over een
voor beider toekomst hoogst gewichtige aangelegenheid. Lina aarzelt niet om aan die
uitnoodiging gehoor te geven, doch is zoo voorzichtig een geladen revolver aan haar
maagdelijken boezem te verbergen, om te voorkomen dat haar geliefde zich vrijheden
veroorlove voordat een deugdelijk contract is gesloten.

166. Haar schatkamer en haar arsenaal.
Le Reverend, in „Les Dessous à travers les âges”.
167. Fransche karikatuur van Boitard (1745) op de hoepelrokken.

Lina, lieve Lina, roept Karel, haar bij het binnentreden toe en wil haar in zijn armen sluiten
en—vlak in de buurt staat de sofa. Doch Lina haalt de revolver voor den dag en zegt: Neen
mijn liefste, zoover zijn we nog niet. Laten we ordentelijk gaan zitten en onze zaken
bespreken; gij daar achter de tafel op de sofa, ik hier over u op een stoel. Wat wilt gij?
U trouwen. Wat anders, voorwerp van al mijn wenschen, zoet beeld mijner droomen?—
Maar ge zijt arm.—Neen, mijn schat, sedert gisteren ben ik rijk.—Laat zien, trouw hart.
Uit de verte toonde Karel haar kostbare papieren, terwijl hij een wantrouwenden blik op de
revolver sloeg. Lina werd verteederd en zeide: Gij hebt mij zoolang een trouwe liefde
toegedragen, dat ik wil aannemen, dat ze niet valsch zijn.—Nu, wat zegt ge? vroeg Karel
zegepralend, zelf verdiend voor u en voor u alleen aangebeden meisje … mag ik nu den
notaris laten komen?—Ja, lispelde Lina, terwijl zij de schuchtere oogen schaamachtig op
de revolver liet neerglijden.
De notaris verscheen. De heer en dame, sprak hij, wenschen door den band des huwelijks
vereenigd te worden? En hij zette zich onmiddellijk aan den arbeid voor het opmaken der
huwelijksche voorwaarden met denzelfden ijver waarmee zulke ambtenaren den uitersten
wil van een reeds stervende plegen op te schrijven.
Zie zoo! zeide hij, uwe namen en uw verlangen in het algemeen staan er. Gelieft mij thans
omtrent de nadere voorwaarden in te lichten. Zoo, bijvoorbeeld—de geachte bruid duide
mij deze onkiesche vraag niet ten kwade—onverhoopt een spruit uit uwe echtelijke
vereeniging mocht voortkomen, wie zal dan de zorg daarvoor op zich
nemen?
Lina’s kiesch gevoel werd pijnlijk door zulk een vraag getroffen. Zij werd beurtelings bleek
en rood en fluisterde nauw hoorbaar: Neen mijnheer de notaris, dat is onmogelijk, dat wil
ik niet en dat gebeurt ook niet.—Ja, mejuffrouw, ik begrijp …, maar de mogelijkheid
bestaat, de natuur is soms sterker dan de kunst. En buitendien, de wet wil, dat
uitdrukkelijke bepalingen op dit punt gemaakt worden.
Ik kan mij met het oog op mijn verdere carrière niet met kinderzorgen belasten, verklaarde
Lina toen op beslisten toon, die sterk in strijd was met haar gewonde kieschheid van
daareven.
Vindt de bruidegom dan goed, dat wij bepalen, dat als het huwelijk van rechtswege
ophoudt te bestaan, hetzij door den dood van een uwer, hetzij door echtscheiding, een
eventueele telg of telgen uit dit huwelijk geheel voor rekening zullen komen van den man?

168. Onnavolgbaar.
—Je krijgt ze niet meer met die onzinnige dingen van tegenwoordig!
A. Willette in „Le Courrier Français”.

Favorietje.
Naar de schilderij van J. H. Fragonard (1732–1806). Collectie Eugène Kraemer, Parijs.
Karel gaf met een benauwd gezicht zijn toestemming, doch werd eenigszins gerustgesteld,
toen de goddelijke Lina hem blozend een bemoedigend knikje toewierp. Nadat nog tal van
zulke kleinigheden geregeld waren en de onderteekening had plaats gehad, ging de notaris
heen. Nauwelijks had de ambtenaar zich verwijderd of Lina, de schuchtere, wierp haar
revolver weg, naderde met een betooverenden glimlach de sofa, sloot Karel in haar armen
en drukte hem een vurigen kus op de lippen, haalde de bundel papieren van waarde uit zijn
binnenzak, bladerde ze door en fluisterde verrukt: Mijn innig geliefd mannetje!
Het meest dichterlijk-idyllische element in de poëzie van het minnen is de bruidschat. Maar
niet ieder heeft oog daarvoor—voor de poëzie daarvan wel te verstaan. Zoo durfde een
Fransch schrijver zich aldus uitlaten: „Het is ongetwijfeld zeker, dat een teef met een tiental
reuen achter zich aan, een minder onverkwikkelijk, althans natuurlijker schouwspel
oplevert dan een Amerikaansche milliardairsdochter met haar sleep van
huwelijkssollicitanten, die zich om haar verdringen met gefingeerde galanterie en
voorgewende verliefdheid”.—Die man had blijkbaar geen oog voor de bruidschatspoëzie—
of behoorde tot degenen die tevergeefs een greep naar zoo’n milliardairsdochter hadden

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