Table Of ContentAspects of Mathematics
Wilfred W. J. Hulsbergen
Conjectures in
Arithmetic Algebraic
Geometry
A Survey
Second Edition
Wilfred W. j. Hulsbergen
Conjectures in Arithmetic
Algebraic Geometry
Asped~f
Mathematic~
Edited by Klas Diederich
Vol. E 2: M. Knebusch/M. Kolster: Wittrings
Vol. E 3: G. Hector/U. Hirsch: Introduction to the Geometry of Foliations, Part B
Vol. E 5: P. Stiller: Automorphic Forms and the Picard Number of an Elliptic Surface
Vol. E 6: G. Failings/G. Wustholz et al.: Rational Points*
Vol. E 7: W. Stoll: Value Distribution Theory for Meromorphic Maps
Vol. E 9: A. Howard/P.-M. Wong (Eds.): Contribution to Several Complex Variables
Vol. E 10: A. J. Tromba (Ed.): Seminar of New Results in Nonlinear Partial Differential
Equations*
Vol. E 13: Y. Andre: G-Functions and Geometry*
Vol. E 14: U. Cegrell: Capacities in Complex Analysis
Vol. E 15: J.-P. Serre: Lectures on the Mordeii-Weil Theorem
Vol. E 16: K. lwasaki/H. Kimura/S. Shimomura/M. Yoshida: From Gauss to Painleve
Vol. E 17: K. Diederich (Ed.): Complex Analysis
Vol. E 18: W. W. J. Hulsbergen: Conjectures in Arithmetic Algebraic Geometry
Vol. E 19: R. Rocke: Lectures on Nonlinear Evolution Equations
Vol. E 20: F. Hirzebruch, Th. Berger, R. Jung: Manifolds and Modular Forms*
Vol. E 21: H. Fujimoto: Value Distribution Theory of the Gauss Map of Minimal
Surfaces in Rm
Vol. E 22: D. V. Anosov/ A. A. Bolibruch: The Riemann-Hilbert Problem
Vol. E 23: A. P. Fordy/J. C. Wood (Eds.): Harmonic Maps and Integrable Systems
Vol. E 24: D. S. Alexander: A History of Complex Dynamics
*A Publication of the Max-Pianck-lnstitut fur Mathematik, Bonn
Volumes of the German-language subseries ·Aspekte der Mathematik" are listed at the end af the book.
Conjectures in Arithmetic
Algebraic Geollletry
J.
Wilfred W. Hulsbergen
A Survey
Second Revised Edition
II
Vleweg
Wilfred W. J. Hulsbergen
KMA, NL-4800 RG Breda
The N etherlands
AMS subject classification: llG, llM, 14C, 14G, 14H, 14K, 190, 19E, 19F.
First Edition 1992
Second Revised Edition 1994
Ali rights reserved
© Springer Fachmedien Wiesbaden 1994
Originally published by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH. BraunschweiglWiesbaden in 1994
No part of this publication may be reproduced, stored in a retrieval system or
transmitted, mechanical, photocopying or otherwise without prior permission
of the copyright holder.
Cover design: Wolfgang Nieger, Wiesbaden
Printed on acid-free paper
ISSN 0179-2156
ISBN 978-3-663-09507-1 ISBN 978-3-663-09505-7 (eBook)
DOI 10.1007/978-3-663-09505-7
Contents
Introduction 1
1 The zero-dimensional case: number fields 5
1.1 Class Numbers ...... . 5
1.2 Dirichlet L-Functions ... . 8
1.3 The Class Number Formula 11
1.4 Abelian Number Fields ... 12
1.5 Non-abelian Number Fields and Artin L-Functions 15
2 The one-dimensional case: elliptic curves 21
2.1 General Features of Elliptic Curves 21
2.2 Varieties over Finite Fields ........ . 25
2.3 L-Functions of Elliptic Curves ...... . 27
2.4 Complex Multiplication and Modular Elliptic Curves 30
2.5 Arithmetic of Elliptic Curves . 35
2.6 The Tate-Shafarevich Group 39
2. 7 Curves of Higher Genus . . . 44
2.8 Appendix . . . . . . . . . . . 45
2.8.1 B & S-D for Abelian Varieties 46
2.8.2 Bloch's Version of B & S-D 47
2.8.3 1-Motives, Mixed Motives and B & S-D. 50
3 The general formalism of L-functions, Deligne
cohomology and Poincare duality theories 55
3.1 The Standard Conjectures .. 55
3.2 Deligne-Beilinson Cohomology 58
3.3 Deligne Homology . . . . . 64
3.4 Poincare Duality Theories 67
3.5 Gillet's Axioms ..... . 71
4 Riemann-Roch, K-theory and motivic cohomology 79
4.1 Grothendieck-Riemann-Roch . 79
4.2 Adams Operations . . . . . . . . . . . . . . . . . . . 81
4.3 Riemann-Roch for Singular Varieties ...... . 83
4.4 Higher Algebraic /{-Theory . . . . . . . . . . . . 83
4.5 Adams Operations in Higher Algebraic K-Theory 91
4.6 Chern Classes in Higher Algebraic /{.Theory . 93
4.7 Gillet's Riemann-Roch Theorem . 96
4.8 Motivic Cohomology . . . . . . . . . . . . . . 98
5 Regulators, Deligne's conjecture and Beilinson's
first conjecture 101
5.1 Borel's Regulator ........... , 101
5.2 Beilinson's Regulator ......... . 103
5.3 Special Cases and Zagier's Conjecture . 105
5.4 Zagier's conjecture and mixed motives 112
5.5 Riemann Surfaces .. 116
5.6 Models over Spec(Z) ... . 120
5.7 Deligne's Conjecture ... . 121
5.8 Beilinson's First Conjecture 124
5.9 Numerical evidence ..... 127
6 Beilinson's second conjecture 131
6.1 Beilinson's Second Conjecture 131
6.2 Hilbert Modular Surfaces ... 133
7 Arithmetic intersections and Beilinson 's third
conjecture 137
7.1 The Intersection Pairing ... 137
7.2 Beilinson's Third Conjecture . 141
8 Absolute Hodge cohomology, Hodge and Tate
conjectures and Abel-Jacobi maps 145
8.1 The Hodge Conjecture .... 145
8.2 Absolute Hodge Cohomology . 148
8.3 Geometric Interpretation 151
8.4 Abel-Jacobi Maps ... 153
8.5 The Tate Conjecture . 155
8.6 Absolute Hodge Cycles 158
8. 7 Motives . . . . . ... 160
8.8 Grothendieck's Conjectures 165
8.9 Motives and Cohomology .. 168
9 Mixed realizations, mixed motives and Hodge and
Tate conjectures for singular varieties 173
9.1 Tate Modules ..................... . . . . . . . . 173
9.2 Mixed Realizations .... . 176
9.3 Weights .......... . 179
9.4 Hodge and Tate Conjectures 182
9.5 The Homological Regulator 185
10 Examples and Results 187
10.1 B & S-D revisited .. 187
10.2 Deligne's Conjecture 192
10.3 Artin and Dirichlet Motives 195
10.4 Modular Curves ..... . 198
10.5 Other Modular Examples . 203
10.6 Linear Varieties ..... . 204
11 The Bloch-Kato conjecture 207
11.1 The Bloch-Kato Conjecture 207
11.2 Work of Fontaine & Perrin-Riou . 218
Bibliography 229
Index 241
Introduction
In this expository text we sketch some interrelations between several famous
conjectures in number theory and algebraic geometry that have intrigued math
ematicians for a long period of time.
Starting from Fermat's Last Theorem one is naturally led to introduce L
functions, the main, motivation being the calculation of class numbers. In partic
ular, Kummer showed that the class numbers of cyclotomic fields play a decisive
role in the corroboration of Fermat's Last Theorem for a large class of exponents.
Before Kummer, Dirichlet had already successfully applied his L-functions to the
proof of the theorem on arithmetic progressions. Another prominent appearance
of an L-function is Riemann's paper where the now famous Riemann Hypothesis
was stated. In short, nineteenth century number theory showed that much, if not
all, of number theory is reflected by properties of L-functions.
Twentieth century number theory, class field theory and algebraic geome
try only strengthen the nineteenth century number theorists's view. We just
mention the work of E. H~cke, E. Artin, A. Weil and A. Grothendieck with
his collaborators. Heeke generalized Dirichlet's L-functions to obtain results on
the distribution of primes in number fields. Artin introduced his L-functions
as a non-abelian generalization of Dirichlet's L-functions with a generalization
of class field theory to non-abelian Galois extensions of number fields in mind.
Weil introduced his zeta-function for varieties over finite fields in relation to a
problem in number theory. Finally, with the invention of l-adic cohomology by
Grothendieck, all of the L-functions mentioned above could be incorporated in a
framework which comprises classical number theory as part of modern algebraic
geometry. In terms of schemes a number field is just a zero-dimensional object,
but it is already highly non-trivial, e.g. its etale cohomology can be identified
with its Galois cohomology which plays a very important role in class field theory.
For higher-dimensional varieties over number fields one has a representation
of the Galois group of the number field on the l-adic cohomology of those vari
eties. The inverses of the characteristic polynomials of the (geometric) Frobenius
element of the Galois group acting on the l-adic cohomology of the· various reduc-.
tions of the variety are now the building blocks of the L-function of the variety.
2 Introduction
Unfortunately there are many more conjectures than proven theorems on these L
functions. All the conjectures discussed in this book will be about the behaviour
of the L-functions at special integral values of their arguments, e.g. the order of
a zero or a pole and the value of the first non-zero coefficient of the Taylor series
of the L-function. In a sense these conjectures are suggested by results or other
conjectures in the lowest dimensional cases of number fields and elliptic curves.
Finally one may 'break a variety into pieces' and define L-functions for the
associated motives. As the theory of motives is still not completely satisfactory we
tried to postpone as long as possible any reference to it. So we have stated most
conjectures and results only for varieties, and for simplicity of the statements the
base field will often be the field of rational numbers. In later chapters, however,
we can not avoid the more general situation and then we shall freely use the
language of (Deligne) motives and other base fields. For convenience a few pages
on these motives are included as they play an increasing role in the literature
and, of course, because they are a very useful instrument.
The first two chapters are an expanded version of a lecture given at the Royal
Military Academy at Breda, the Netherlands, for an audience of mostly applied
mathematicians and computer scientists. The main purpose of the lecture was to
demonstrate the use of several mathematical programs for a personal computer
in the calculation of class numbers and rational points on elliptic curves. These
programs are used mainly for educational purposes and we wanted to demonstrate
that their use may lead to numerical results for more advanced problems. The
calculations are not included here as they do no justice to the more advanced
level of the final version of the text. Anyhow, no new numerical results were
obtained.
As the original title of the lecture also refered to general L-functions and
Beilinson's conjectures, which fer lack of time we could not even mention then, we
were asked to edit a written version of the text. The present text was formulated
out of this suggestion. As a quick look at its contents may reveal, it seemed useful
to us to give a lengthy introduction (Chapters 1, 2, 3 and 4), without making
the book completely self-contained. Also, on several places we have anticipated
concepts that are defined only in a later chapter where their precise description
is more important. In particular, the notion of motive pervades the text from an
early stage, while it is defined only in Chapter 8.
To give full proofs of all theorems would have made the book unwieldily
lengthy. In particular, each of the introductury first four chapters contains al
ready enough material to write a book. Besides, many easily accessible references
are given. So, in these introductory chapters there are no proofs at all. In the
later chapters we indicate the line of thought that eventually leads to a theorem,
and again, the reader should consult the references in the bibliography to find
full proofs. It should be mentioned (and will be evident) that we borrowed a lot
from most of the contributions in the beautiful book [RSSJ, the expository paper
