Table Of ContentSiegel· Lectures on the Geometry ofNumbers
Carl Ludwig Siegel
Lectures on the
Geometry of Numbers
Notes by B. Friedman
Rewritten by
Komaravolu Chandrasekharan
with the Assistance of Rudolf Suter
With 35 Figures
Springer-V erlag Berlin Heidelberg GmbH
Komaravolu Chandrasekharan
Mathematik, ETH Ziirich
CH-8092 Ziirich, Switzerland
Mathematics Subject Classification (1980):
OI-XX, Il-XX, 12-XX, 15-XX, 20-XX, 32-XX, 51-XX, 52-XX
Library of Congress Cataloging-in-Publication Data.
Siegel, Cari Ludwig, 1896-1981
Lccturcs on the geometry of numbers / Cari Ludwig Siegel; notes by B. friedman; rewritten
by Komaravolu Chandrasekharan with the assistance of Rudolf Suter. p. cm.
Bibliography: p. Includes index.
(U.S. : alk. paper)
ISBN 978-3-642-08076-0 ISBN 978-3-662-08287-4 (eBook)
DOI 10.1007/978-3-662-08287-4
1. Geometry of numbers. 1. Chandrasekharan, K. (Komaravolu), 1920 -.
II. Suter, RudolC 1963 -. III. Title. QA241.5.S54 1989512'.5 -dcl9 89-5946 CIP
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Preface
Carl Ludwig Siegel lectured on the Geometry of Numbers at New York
University during 1945-46. There were hardly any books on the subject at
that time other than Minkowski's original one. The freshness of his approach
still lingers, and his presentation has its attractions for the aspiring young
mathematician.
When he received a request, many years later, for permission to reissue
the notes of those lectures, transcribed by the late B. Friedman, he declined.
Those who know of Siegel's insistence on the expurgation of the first printing
of his lecture notes on Transeendental Numbers, or of his insistence on the
dissociation of his name from a projected English translation of his lecture
notes on Analytische Zahlentheorie (Göttingen, 1963/64), which had therefore
to be abandoned, need hardly be told that he had his own requirements.
We had occasion to discuss the matter further, and it was agreed that
the notes should be published only after they had been checked, corrected, and
rewritten, and he handed over hispersonal copy to me. I found that the task of
revision required far more attention to detail than I had at first glance thought
necessary. Other avocations prevented me from completing the work until the
summer of 1987, when it happened that Rudolf Suter got actively interested in
what I had been trying to carry out. His helpful and critical comments came
to me as an unexpected stimulus, and the present version is the result. It is a
pleasure for me to acknowledge Suter's assistance.
Admirers of Siegel's style can scarcely fail to notice his uncanny skill, and
perspicuity in argument, with an occasional flash of wit, as they progress with
the reading.
E.T.H. Zürich
31 March 1988 K. Chandrasekharan
Table of Contents
Chapter I
Minkowski's Two Theorems
Lecture I 3
§ 1. Convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
§ 2. Convex hoclies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
§ 3. Gauge function of a convex hocly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
§ 4. Convex hoclies with a centre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Lecture II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
§ 1. Minkowski's First Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
§ 2. Lemma on hounclecl open sets in lR.n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
§ 3. Proof of Minkowski's First Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
§ 4. Minkowski's theorem for the gauge function . . . . . . . . . . . . . . . . . . . . . . . 17
§ 5. The minimum of the gauge function for an arhitrary lattice in lR.n . 18
§ 6. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Lect ure 111 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
§ 1. Evaluation of a volume integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
§ 2. Discriminant of an irreclucihle polynomial . . . . . . . . . . . . . . . . . . . . . . . . . 27
§ 3. Successive minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
§ 4. Minkowski's Seconcl Theorem (Theorem 16) . . . . . . . . . . . . . . . . . . . . . . . 32
Lect ure IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
§ 1. A possihle methocl of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
§ 2. A simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
§ 3. A complicatecl transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
§ 4. Volume of the transformecl hody . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
§ 5. Proof of Theorem 16 (Minkowski's Seconcl Theorem) . . . . . . . . . . . . . . 39
VIII Table of Contents
Chapter II
Linear Inequalities
Lecture V 43
§ 1. Vector groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
§ 2. Construction of a basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
§ 3. Relation between different bases for a lattice . . . . . . . . . . . . . . . . . . . . . . 4 7
§ 4. Sub-lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
§ 5. Congruences relative to a sub-lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
§ 6. The number of sub-lattices with given index . . . . . . . . . . . . . . . . . . . . . . . 51
Lecttue VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
§ 1. Local rank of a vector group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
§ 2. Decomposition of a general vector group . . . . . . . . . . . . . . . . . . . . . . . . . . 53
§ 3. Characters of vector groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
§ 4. Conditions on characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
§ 5. Duality theorem for character groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
§ 6. Kronecker's approximation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Lect ure VII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
§ 1. Periods of real functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
§ 2. Periods of analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
§ 3. Periods of entire functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
§ 4. Minkowski's theorem on linear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Lect ure VIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
§ 1. Completing a given set of vectors to form a basis for a lattice . . . . . . 72
§ 2. Completing a matrix to a unimodular matrix . . . . . . . . . . . . . . . . . . . . . . 73
§ 3. A slight extension of Minkowski's theorem on linear forms . . . . . . . . . 74
§ 4. A limiting case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
§ 5. A theorem about parquets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
§ 6. Parquets formed by parallelepipeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Lecture IX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
§ 1. Products of linear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
§ 2. Product of two linear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
§ 3. Approximation of irrationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
§ 4. Product of three linear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
§ 5. Minimum of positive-definite quadratic forms . . . . . . . . . . . . . . . . . . . . . 91
Table of Contents IX
Chapter III
Theory of Reduction
Lecture X 95
§ 1. The problern of reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
§ 2. Space of all matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
§ 3. Minimizing vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
§ 4. Primitive sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
§ 5. Construction of a reduced basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
§ 6. The First Finiteness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
§ 7. Criteria for reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
§ 8. Use of a quadratic gauge function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
§ 9. Reduction of positive-definite quadratic forms . . . . . . . . . . . . . . . . . . . . . 105
Lecture XI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
§ 1. Space of symmetric matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
§ 2. Reduction of positive-definite quadratic forms . . . . . . . . . . . . . . . . . . . . . 107
§ 3. Consequences of the reduction conditions . . . . . . . . . . . . . . . . . . . . . . . . . 109
§ 4. The case n = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
§ 5. Reduction of lattices of rank two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
§ 6. The case n = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Lect ure XII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
§ 1. Extrema of positive-definite quadratic forms . . . . . . . . . . . . . . . . . . . . . . 118
§ 2. Closest packing of (solid) spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
§ 3. Closest packing in two, three, or four dimensions . . . . . . . . . . . . . . . . . . 122
§ 4. Blichfeldt's method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Lecture XIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
§ 1. The Second Finiteness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
§ 2. An inequality for positive-definite symmetric matrices . . . . . . . . . . . . . 128
§ 3. The space 'PK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
§ 4. Images of R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Lecture XIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
§ 1. Boundary points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
§ 2. Non-overlapping of images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
§ 3. Space defined by a finite number of conditions . . . . . . . . . . . . . . . . . . . . 140
§ 4. The Second Finiteness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
§ 5. Fundamental region of the space of all matrices . . . . . . . . . . . . . . . . . . . 143
X Table of Contents
Lecture XV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
§ 1. Volume of a fundamental region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
§ 2. Outline of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
§ 3. Change of variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
§ 4. A new fundamental region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
§ 5. Integrals over fundamental regions are equal . . . . . . . . . . . . . . . . . . . . . . 150
§ 6. Evaluation of the integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
§ 7. Generalizations of Minkowski's First Theorem . . . . . . . . . . . . . . . . . . . . . 152
§ 8. A lower bound for the packing of spheres . . . . . . . . . . . . . . . . . . . . . . . . . . 154
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Chapter I
Minkowski's Two Theorems
Lectures I to IV
Lecture I
§ 1. Convex sets
Consider an n-dimensional real Euclidean space lR n, n ;::: 1. Assurne that
a reetangular coordinate system with origin at some point 0 is set up in 1R n,
so that the coordinates of any point PE lRn are Xt, ... , Xn· For simplicity we
shall represent the point P by the vector x = (xb ... , Xn)· The origin 0 is
then represented by the zero-vector 0 = (0, ... , 0).
A non-empty set C contained in lR n is called a linear manifold in lR n, if
whenever any two different points P and Q belong to C, the infinite straight
line passing through P and Q belongs to C. Analytically the definition can be
formulated as follows:
Let x be the vector associated with P, and y the vector associated with Q.
Then Cis a linear manifold, if whenever it contains P and Q, it contains every
point represented by a vector of the form .Xx + p,y, where .X, p, are arbitrary real
numbers, such that .X + J-l = 1.
Any linear manifold C has a dimension m, which is an integer not greater
than n, and which can be found as follows.
Let P be a point in .C. If C contains no other point, then m = 0. Otherwise
0
let P be another point in .C. Then all points on the line passing through
1
Po, P1 belong to C. If C contains no point besides those on the line through
P0 and P1, then m = 1. Otherwise let P2 be a point in C which is not on
the line through P0, P1. Then all points of the plane determined by the points
P0, P1, P2 belong to C. If C contains no point outside this plane, then m = 2.
Otherwise let P3 be a point in C outside the plane through Po, P1, P2, and we
can continue this procedure. Since the highest possible value of m is n, it is
clear that this procedure terminates and gives adefinite value of m. Note that
C is completely determined by the rn-dimensional tetrahedron, or m-simplex,
PoP1 ... Pm obtained in the course of the proof. Analytically this means the
following.
If x<il is the vector representing the point P;, then a point P belongs to C
if and only if its vector can be written as .X0x(o) + .X1x(ll + ... + AmX(m), where
.Xo, .Xb ... , Am are arbitrary real numbers, such that .X0 + .X1 + ... +Am = 1.
A non-empty set JC C lR n is called a convex set if whenever P and Q belong
to JC, the segment joining P and Q belongs to JC. Analytically the definition
can be formulated in this way: if P is represented by the vector x, and Q by
