Brauer groups, Tamagawa measures, and rational points on algebraic varieties

280 Pages·2014·1.821 MB·English

by  Jorg Jahnel

Mathematical
Surveys
and
Monographs
Volume 198
Brauer Groups,
Tamagawa Measures,
and Rational Points on
Algebraic Varieties
Jörg Jahnel
American Mathematical Society
Brauer Groups,
Tamagawa Measures,
and Rational Points on
Algebraic Varieties

Mathematical
Surveys
and
Monographs
Volume 198
Brauer Groups,
Tamagawa Measures,
and Rational Points on
Algebraic Varieties
Jörg Jahnel
American Mathematical Society
Providence, Rhode Island
EDITORIAL COMMITTEE
Ralph L. Cohen, Chair Benjamin Sudakov
Robert Guralnick Michael I. Weinstein
Michael A. Singer
2010 Mathematics Subject Classification. Primary 11G35,14F22,
16K50, 11-04, 14G25,11G50.
For additional informationand updates on this book, visit
www.ams.org/bookpages/surv-198
Library of Congress Cataloging-in-Publication Data
Jahnel,Jo¨rg,1968–
Brauergroups,Tamagawameasures,andrationalpointsonalgebraicvarieties/Jo¨rgJahnel.
pagescm. —(Mathematicalsurveysandmonographs;volume198)
Includesbibliographicalreferencesandindex.
ISBN978-1-4704-1882-3(alk.paper)
1. Algebraic varieties. 2. Geometry, Algebraic. 3. Brauer groups. 4. Rational points (Ge-
ometry) I.Title.
QA564.J325 2014
516.3(cid:2)53—dc23
2014024341
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Contents
Preface vii
Introduction 1
Notation and conventions 11
Part A. Heights 13
Chapter I. The concept of a height 15
1. The naive height on the projective space over 15
2. Generalization to number fields (cid:0) 17
3. Geometric interpretation 21
4. The adelic Picard group 25
Chapter II. Conjectures on the asymptotics of points of bounded height 35
1. A heuristic 35
2. The conjecture of Lang 38
3. The conjecture of Batyrev and Manin 40
4. The conjecture of Manin 44
5. Peyre’s constant I—the factor α 47
6. Peyre’s constant II—other factors 50
7. Peyre’s constant III—the actual definition 59
8. The conjecture of Manin and Peyre—proven cases 62
Part B. The Brauer group 81
Chapter III. On the Brauer group of a scheme 83
1. Central simple algebras and the Brauer group of a field 84
2. Azumaya algebras 89
3. The Brauer group 93
4. The cohomological Brauer group 94
5. The relation to the Brauer group of the function field 98
6. The Brauer group and the cohomological Brauer group 101
7. The theorem of Auslander and Goldman 103
8. Examples 107
Chapter IV. An application: the Brauer–Manin obstruction 119
1. Adelic points 119
2. The Brauer–Manin obstruction 122
3. Technical lemmata 126
4. Computing the Brauer–Manin obstruction—the general strategy 129
5. The examples of Mordell 132
v
vi contents
6. The “first case” of diagonal cubic surfaces 146
7. Concluding remark 161
Part C. Numerical experiments 163
Chapter V. The Diophantine equation x4+2y4 =z4+4w4 165
Numerical experiments and the Manin conjecture 165
1. Introduction 166
2. Congruences 167
3. Naive methods 169
4. An algorithm to efficiently search for solutions 169
5. General formulation of the method 171
6. Improvements I—more congruences 172
7. Improvements II—adaptation to our hardware 176
8. The solution found 182
Chapter VI. Points of bounded height on cubic and quartic threefolds 185
1. Introduction—Manin’s conjecture 185
2. Computing the Tamagawa number 189
3. On the geometry of diagonal cubic threefolds 193
4. Accumulating subvarieties 195
5. Results 199
Chapter VII. On the smallest point on a diagonal cubic surface 205
1. Introduction 205
2. Peyre’s constant 208
3. The factors α and β 209
4. A technical lemma 211
5. Splitting the Picard group 212
6. The computation of the L-function at 1 216
7. Computing the Tamagawa numbers 219
8. Searching for the smallest solution 221
9. The fundamental finiteness property 222
10. A negative result 233
Appendix 239
1. A script in GAP 239
2. The list 241
Bibliography 247
Index 261
Preface
In this book, we study existence and asymptotics of rational points on algebraic
varieties of Fano and intermediate type. The book consists of three parts. In the
first part, we discuss tosome extentthe concept of aheight andformulate Manin’s
conjecture on the asymptotics of rational points on Fano varieties.
In the second part, we study the various versions of the Brauer group. We explain
why a Brauer class may serve as an obstruction to weak approximation or even to
the Hasse principle. This includes two sections devoted toexplicit computations of
the Brauer–Manin obstruction for particular types of cubic surfaces.
The final part describes numerical experiments related to the Manin conjecture
that were carried out by the author together with Andreas-Stephan Elsenhans.
Prerequisites. We assume that the reader is familiar with some basic mathemat-
ics, including measure theory and the content of a standard course in algebra.
In addition, a certain background in some more advanced fields shall be necessary.
This essentially concerns three areas.
a) We will make use of standard results from algebraic number theory and class
field theory, as well as such concerning the cohomology of groups. The content
of articles [Cas67, Se67, Ta67, A/W, Gru] in the famous collection edited by
J.W.S. Cassels and A. Fröhlich shall be more than sufficient.
Here, the most important results that we shall use are the existence of the global
Artin map and, related to this, the computation of the Brauer group of a number
field [Ta67, 11.2].
b) We will use the language of modern algebraic geometry as described in the
textbook of R. Hartshorne [Ha77, Chapter II]. Cohomology of coherent sheaves
[Ha77, Chapter III] will be used occasionally.
c) InChapterIII,wewillmakesubstantialuseofétalecohomology. Thisisprobably
the deepest prerequisite that we expect from the reader. For this reason, we will
formulate its main principles, as they appear to be of importance for us, at the
beginning of the chapter. It seems to us that, in order to follow the arguments, an
understandingofChaptersIIandIIIofJ.Milne’stextbook[Mi]shouldbesufficient.
At a few points, some other background material may be helpful. This concerns,
for example, Artin L-functions. Here, [Hei] may serve as a general reference. Pre-
cise citations shall, of course, be given wherever the necessity occurs.
A suggestion that might be helpful for the reader. Part C of this book describes
experiments concerning the Manin conjecture. Clearly, the particular samples are
vii
viii preface
chosen in such a way that not all the difficulties, which are theoretically possible,
really occur.
ItthereforeseemsthatPartCmight beeasiertoreadthantheothers,particularly
for those readers who are very familiar with computers and the concept of an al-
gorithm. Thus, such a reader could try to start with Part C to learn about the
experiments and to get acquainted with the theory. It is possible then to continue,
in a second step, with Parts A and B in order to get used to the theory in its
full strength.
References and citations. When we refer to a definition, proposition, theorem,
etc., in the same chapter we simply rely on the corresponding numbering within
the chapter. Otherwise, we add the number of the chapter.
For the purpose of citation, the articles and books being used are encoded in the
manner specified by the bibliography. In addition, we mostly give the number of
the relevant section and subsection or the number of the definition, proposition,
theorem, etc. Normally, we do not mention page numbers.
Acknowledgments. I wish to acknowledge with gratitude my debt to Y. Tschinkel.
Most of the work described in this book, which is a shortened version of my Habil-
itation Thesis, was initiated by his numerous mathematical questions. During the
years he spent at Göttingen, he always shared his ideas in an extraordinarily gen-
erous manner.
I further wish to express my deep gratitude to my friend and colleague Andreas-
Stephan Elsenhans. He influenced this book in many ways, directly and indirectly.
It is no exaggeration to say that most of what I know about computer program-
ming I learned from him. The experiments, which are described in Part C, were
carried out together with him as a joint work. I am also indebted to Stephan
for proofreading.
ThecomputerpartoftheworkdescribedinthisbookwasexecutedontheSunFire
V20z Servers of the Gauß Laboratory for Scientific Computing at the Göttingen
MathematicalInstitute.TheauthorisgratefultoY.Tschinkelforpermissiontouse
these machines as well as to the system administrators for their support.
Jörg Jahnel
Siegen, Germany
Spring 2014
Introduction
Here, in the midst of this sad and barren landscape of the Greek ac-
complishmentsinarithmetic,suddenlyspringsupamanwithyouthful
energy: Diophantus. Wheredoeshecomefrom,wheredoeshegoto?
Who were his predecessors, who his successors? We do not know.
It is all one big riddle. He lived in Alexandria. If a conjecture were
permitted, I would say he was not Greek; ... if his writings were
not in Greek, no-one would ever think that they were an outgrowth
of Greek culture ... .
Hermann Hankel (1874, translatedby N. Schappacher)
Diophantine equations have a long history. More than two thousand years ago,
Diophantus of Alexandria considered, among many others, the equations
x2+y2 =z2, (∗)
y(6−y)=x3−x,
and
y2 =x2+x4+x8.
In Diophantus’ book Arithmetica, we find the formula
(cid:2) (cid:3)
(p2−q2)λ,2pqλ,(p2+q2)λ (†)
that generates infinitely many solutions of (∗). For the second and third of the
equations mentioned, Diophantus gives particular solutions, namely (1/36,1/216)
and (1/2,9/16), respectively.
In general, a polynomial equation in several indeterminates, where solutions are
sought in integers or rational numbers, is called a Diophantine equation in honour
of Diophantus. Diophantus himself was interested in solutions in positive integers
or positive rational numbers. Contrary to the point of view usually adopted today,
he did not accept negative numbers.
It is remarkable that algebro-geometric methods have often been fruitful in order
to understand a Diophantine equation. For example, there is a simple geometric
idea behind formula (†).
Indeed, since the equation is homogeneous, it suffices to look for solutions
of X2+Y2 =1in rationals. This equationdefines the unit circle. For every t∈ ,
there is the line “x=−ty+1” going through the point (1,0). An easy calculati(cid:2)on
1