Mordell–Weil Lattices (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics)

436 Pages·2019·2.769 MB·English

by  Matthias Schütt, Tetsuji Shioda

Ergebnisse der Mathematik und ihrer Grenzgebiete.
3. Folge / A Series of Modern Surveys in Mathematics 70
Matthias Schütt
Tetsuji Shioda
Mordell–Weil
Lattices
Ergebnisse der Mathematik und ihrer
Grenzgebiete. 3. Folge / A Series of Modern
Surveys in Mathematics
Volume 70
Series Editors
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More information about this series at http://www.springer.com/series/728
ü
Matthias Sch tt Tetsuji Shioda
(cid:129)
–
Mordell Weil Lattices
123
Matthias Schütt Tetsuji Shioda
Institut für Algebraische Geometrie Department ofMathematics
LeibnizUniversität Hannover RikkyoUniversity
Hannover, Niedersachsen,Germany Tokyo,Japan
ISSN 0071-1136 ISSN 2197-5655 (electronic)
ErgebnissederMathematikundihrerGrenzgebiete.3.Folge/ASeriesofModernSurveys
in Mathematics
ISBN978-981-32-9300-7 ISBN978-981-32-9301-4 (eBook)
https://doi.org/10.1007/978-981-32-9301-4
MathematicsSubjectClassification(2010): 14J27,14J20,14J26,14J28,11G05,11G07,11G25,11R09,
11G50,11H31,06B15
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Dedicated to Yukiko
Preface
In this book, we will study the Mordell–Weil lattice of an elliptic curve over a
function field and discuss various applications.
Theideaforthebookaroseintheearly1990s.Yetittookmorethan20yearsto
finishit.Atleast,thisdelaycamewiththeincidentaladvantagethatwewereableto
cover several recent developments building on Mordell–Weil lattices. Let us
highlight here only the notion of (multiplicative) excellent families and the impact
of Mordell–Weil lattices on the theory of K3 surfaces in recent years. Once again,
these applications display the powerful interactions which Mordell–Weil lattices
fosterinaseeminglyelementarywaybetweenfoundationaltopicssuchasalgebraic
equations, elliptic curves and lattices.
One characteristic of the Mordell–Weil lattice is the fact it is equipped with a
clear-cut height pairing, based on the intersection theory of the associated elliptic
surface.Asaconsequence,wearenaturallyledtosomebeautifullatticessuchasE
8
(the positive-definite unique even unimodular lattice of rank eight) among the
Mordell–Weil lattices.
When writing the present book, we had a fairly broad readership in mind, from
novicetoadvancedresearchersrangingthroughdifferentfieldssuchasclassicaland
algebraic geometry and number theory, reflecting the diverse input into the theory
of Mordell–Weil lattices.To keep themanuscript as self-contained as possible, we
decided to include concise reviews of all the background material needed (notably
lattices, elliptic curves, and algebraic surfaces). Necessarily, this means that some
readers will be familiar with parts of the early chapters, and that the level of the
material presented rises steadily as the book advances. Numerous examples are to
befoundthroughoutthebook,somebuildingexplicitlyoneachothersothereader
canseetheideasofthesubjectsunderconsiderationunfoldoneaftertheotheruntil
the full impact and connections of Mordell–Weil lattices have been revealed.
Hannover, Germany Matthias Schütt
Tokyo, Japan Tetsuji Shioda
June 2019
vii
Acknowledgements
This book, and some of the developments described in it, would have been
impossible without Jean-Pierre Serre and Friedrich Hirzebruch, whose advice and
encouragementwereinvaluabletous.Wearenolessindebtedtoourcolleaguesand
co-authorsthroughouttheyears,ofwhomweshouldliketomentionNoboruAoki,
Bert van Geemen, Torsten Ekedahl, Noam D. Elkies, Klaus Hulek, Toshiyuki
Katsura,RemkeKloosterman,ShigeyukiKondo,MasatoKuwata,AbhinavKumar,
Keiji Oguiso, Ichiro Shimada, Tomohide Terasoma, Jaap Top, and Hisashi Usui.
SpecialthanksgotoSimonBrandhorstformanyvaluablecommentswhichhelped
us improve the manuscript.
We are indebted to the referees for their numerous suggestions which helped
streamline the exposition and for providing many additions and corrections. Our
thanks go to everyone at Springer who helped to make this book possible, espe-
cially to Reinhold Remmert.
Wethanktheinstitutionswhichhavesupported,oreveninfluenced,thegenesis
ofthis book inthelast 25years, notablyRikkyoUniversity, MPIM Bonn,Leibniz
Universität Hannover, University of Tokyo, University of Copenhagen, and RIMS
Kyoto.
Funding from various sources is gratefully acknowledged. Among others we
received generous support from JSPS under Grant-in-Aid for Scientific Research
No. 17540044, (C) 20540051, (C) 25400052, from ERC under StG 279723
(SURFARI), and from DFG in the framework of RTG 1463.
Most importantly, we would like to thank our families for their patience and
support throughout all these years.
Hannover, Germany Matthias Schütt
Tokyo, Japan Tetsuji Shioda
June 2019
ix
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Generalities on Lattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Sphere Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Root Lattices and Their Dual Lattices . . . . . . . . . . . . . . . . . . . 19
2.3.1 Auxiliary Lattice Lr;s . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.2 Ar and A_r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.3 Dr and D_r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.4 Er and Er_ ðr ¼6;7;8Þ. . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.5 Discriminant Groups. . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.6 Invariant Theory of the Weyl Groups, I. . . . . . . . . . . . 33
2.3.7 Invariant Theory of the Weyl Groups, II . . . . . . . . . . . 35
3 Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 The Group Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Mordell–Weil Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 Degenerate Plane Cubics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.1 Case (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4.2 Case (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4.3 Further Degenerations. . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4.4 The First Elliptic Surface . . . . . . . . . . . . . . . . . . . . . . 53
3.4.5 An Elliptic Surface with Constant Moduli . . . . . . . . . . 53
4 Algebraic Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 Divisors and Picard Group . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Néron–Severi Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3 Intersection Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4 Hodge Index Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
xi
xii Contents
4.5 Blow-Ups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.6 Minimal Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.7 Invariants of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.7.1 Relation with Picard Number . . . . . . . . . . . . . . . . . . . 71
4.8 Enriques–Kodaira Classification. . . . . . . . . . . . . . . . . . . . . . . . 72
4.9 Castelnuovo’s Criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.10 Non-smooth Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5 Elliptic Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1 Definition of an Elliptic Surface. . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Sections Versus Rational Points. . . . . . . . . . . . . . . . . . . . . . . . 81
5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.4 Singular Fibres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.5 Connection with Dynkin Diagrams . . . . . . . . . . . . . . . . . . . . . 90
5.6 The Kodaira–Néron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.6.1 Group Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.7 The Weierstrass Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.8 Tate’s Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.8.1 Multiplicative Reduction. . . . . . . . . . . . . . . . . . . . . . . 98
5.8.2 Additive Reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.9 Singular Fibres Versus Discriminant and j-Invariant . . . . . . . . . 101
5.9.1 Quadratic Twists . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.9.2 Base Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.9.3 Dokchitsers’ Refinement. . . . . . . . . . . . . . . . . . . . . . . 106
5.10 Minimal Weierstrass Models . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.11 Canonical Divisor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.12 Euler Characteristic and Euler Number. . . . . . . . . . . . . . . . . . . 111
5.13 Differential Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6 Mordell–Weil Lattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.1 The Trivial Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2 The Néron–Severi Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.3 Horizontal and Vertical Divisors . . . . . . . . . . . . . . . . . . . . . . . 118
6.4 Essential Lattice and Frame. . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.5 Mordell–Weil Lattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.6 Torsion Sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.6.1 Determinant Formula . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.6.2 Frame Versus Singular Fibres . . . . . . . . . . . . . . . . . . . 133
6.7 Narrow Mordell–Weil Lattice . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.8 The Case of a Unimodular Néron–Severi Lattice . . . . . . . . . . . 137
6.9 Functorial Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Contents xiii
7 Rational Elliptic Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.3 The Case of High Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.4 Weierstrass Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.5 Cubic Pencils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.6 General Cubic Pencil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8 Rational Elliptic Surfaces and E8-Hierarchy. . . . . . . . . . . . . . . . . . 161
8.1 Singular Fibres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.2 Structure of Mordell–Weil Lattice . . . . . . . . . . . . . . . . . . . . . . 163
8.3 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
8.4 Torsion Sections on Rational Elliptic Surfaces . . . . . . . . . . . . . 169
8.5 Proof of Classification Theorem 8.8. . . . . . . . . . . . . . . . . . . . . 170
8.5.1 r(cid:2)4 (Nos. 1–14). . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8.5.2 r ¼3 (Nos. 15–24) . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.5.3 r ¼2 (Nos. 25–42) . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.5.4 r ¼1 (Nos. 43–61) . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.5.5 r ¼0 (Nos. 62–74) . . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.6 Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.7 Integral Sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8.8 Extremal Rational Elliptic Surfaces . . . . . . . . . . . . . . . . . . . . . 180
8.9 Existence of Types of Rational Elliptic Surfaces. . . . . . . . . . . . 181
8.10 Maximal Singular Fibres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.10.1 T ¼A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8
8.10.2 T ¼D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8
8.10.3 T ¼E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
8
8.10.4 Comment on K3 Surfaces and Beyond . . . . . . . . . . . . 187
8.10.5 Connection with Polynomial Sections . . . . . . . . . . . . . 187
9 Galois Representations and Algebraic Equations . . . . . . . . . . . . . . 191
9.1 Galois Representations Arising From Mordell–Weil
Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
9.2 Algebraic Equations Arising From Mordell–Weil Lattices. . . . . 194
9.3 The Specialization Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
9.4 Galois Representation and Algebraic Equation
of Type E ;E or E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
6 7 8
9.4.1 Galois Representations on Hexagonal Elliptic
Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
9.4.2 Set-Up for the Additive Type Er ðr ¼6;7;8Þ . . . . . . . 202
9.4.3 Generic Galois Representation and Algebraic
Equation of Additive Type Er ðr ¼6;7;8Þ . . . . . . . . . 205
9.5 Application to Number Theory . . . . . . . . . . . . . . . . . . . . . . . . 211
9.5.1 Exceptional Type Erðr ¼6;7;8Þ Versus Classical
Type An(cid:3)1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211