Computing canonical heights on Jacobians [Elektronische Ressource] / von Jan Steffen Müller

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• Mathematics

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• Publié par: universitat_bayreuth
• Publié le: 01 janvier 2010
• Nombre de lectures: 38
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Computing canonical heights on
Jacobians
Von der Universita¨t Bayreuth
zur Erlangung des Grades eines
Doktors der Naturwissenschaften (Dr. rer. nat.)
genehmigte Abhandlung
von
Jan Steffen Mu¨ller
aus Gießen
1. Gutachter: Prof. Dr. Michael Stoll
2. Gutachter: Prof. Dr. Victor Flynn
Tag der Einreichung: 6.10.2010
Tag des Kolloquiums: 16.12.2010iiiii
Abstract
Thecanonical height isan indispensabletoolfor thestudyofthe
arithmeticofabelianvarieties. Inthisdissertationweinvestigate
methodsfor theexplicit computation of canonical heights on Ja-
cobians of smooth projective curves. Building on an existing
algorithm due to Flynn and Smart with modifications by Stoll
we generalize efficient methods for the computation of canonical
heights on elliptic curves to the case of Jacobian surfaces. The
main tools are the explicit theory of the Kummer surface asso-
ciated to a Jacobian surface which we develop in full generality,
buildingon earlier work dueto Flynn, and a careful studyof the
local N´eron models of the Jacobian.
As a first step for a further generalization to Jacobian three-
folds of hyperelliptic curves, we completely describe the asso-
ciated Kummer threefold and conjecture formulas for explicit
arithmetic on it, based on experimental data. Assuming the va-
lidityofthisconjecture, manyoftheresultsforJacobiansurfaces
can be generalized.
Finally, we use a theorem due to Faltings, Gross and Hriljac
which expresses the canonical height on the Jacobian in terms
of arithmetic intersection theory on a regular model of the curve
to develop an algorithm for the computation of the canonical
heightwhichisapplicableinprincipletoanyJacobian. However,
it uses several subroutines and some of these are currently only
implemented in the hyperelliptic case, although the theory is
available in general.
Amongthepossibleapplicationsofthecomputationofcanon-
ical heights are the determination of generators for the Mordell-
Weil group of theJacobian and the computation of its regulator,
appearingforinstance inthefamousBirch andSwinnerton-Dyer
conjecture. We illustrate our algorithm with two examples: The
regulator of a finite index subgroup of the Mordell-Weil group
of the Jacobian of a hyperelliptic curve of genus 3 and the non-
archimedean part of the regulator computation for the Jacobian
ofanon-hyperellipticcurveofgenus4,wheretheremainingcom-
putations can be done immediately once the above-mentioned
implementations are available.iv
Acknowledgements
First I would like to thank my advisor Michael Stoll for sug-
gesting this research project to me, for many useful ideas and
discussions and for answering a lot of questions, occasionally
several times.
I would like to thank my parents for always supporting me
and believing in me, especially during those times when I found
it hard to do so myself.
I wish to acknowledge financial support from Jacobs Univer-
sity Bremen (2006–2007) and from Deutsche Forschungsgemein-
schaft (DFG-Grant STO 299/5-1, 2007–2010).
I would like to thank my colleagues Brendan Creutz and
TzankoMatevforinterestingmathematical andnon-mathemati-
cal discussions in Bremen, Bayreuth and in other places; spe-
cial thanks are due to Tzanko for providing me with a proof of
Proposition 4.1. I would also like to thank Elvira Rettner and
Axel Kohnert for helping me with many practical problems in
Bayreuth.
I was very fortunate to have the opportunity to visit several
mathematical institutionsduringmyworkonthisthesis; Iwould
like to thankthe following mathematicians for either inviting me
to their institutions, helping me with my research and/or sup-
porting my visits financially: Samir Siksek and David Holmes
at the University of Warwick, Victor Flynn at the University of
Oxford, Ulf Ku¨hn and Vincenz Busch at the Universita¨t Ham-
burg, Kiran Kedlaya and Jen Balakrishnan at MIT and Sylvain
Duquesne at the Universit´e Rennes I.
ThanksarealsoduetoSteveDonnellyoftheMagmagroupat
the University of Sydney for writing the RegularModel package
in MagmawhichlargepartsofthealgorithmpresentedinChapter
5 rely on, taking into account my (rather long) wish list.
Furthermore,Ihavehadusefulconversations, inpersonorby
email, with a large number of mathematicians. Among those I
would like to thank whose names have not yet appeared in these
acknowledgments are Karim Belabas, Dominique Bernardi, An-
toine Chambert-Loir, Pierre Chr´etien, Brian Conrad, Christian
Curilla, BernardDeconinck, RobinDeJong, StephanElsenhans,
Pierrick Gaudry, Florian Hess, Marc Hindry, Qing Liu, Jean-
Franc¸ois Mestre, Michael Mourao, Fabien Pazuki, C´edric Pepin,
Anna Posingies, Christophe Ritzenthaler, Mohammad Sadek,
Joe Silverman, Damiano Testa, Yukihiro Uchida and Kentaro
Yoshitomi.v
fu¨r ToniaviContents
Introduction xiii
Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
1 Motivation and background 1
1.1 Places and absolute values . . . . . . . . . . . . . . . . . . . . 2
1.2 Heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 N´eron functions . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 N´eron models . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Curves and Jacobians . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Elliptic curves 25
2.1 Heights on elliptic curves . . . . . . . . . . . . . . . . . . . . 26
2.2 Local heights . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Non-archimedean places . . . . . . . . . . . . . . . . . . . . . 30
2.4 Archimedean places . . . . . . . . . . . . . . . . . . . . . . . 35
3 Jacobian surfaces 39
3.1 Jacobian surfaces and Kummer surfaces . . . . . . . . . . . . 40
3.2 Canonical heights on Jacobian surfaces . . . . . . . . . . . . . 43
3.2.1 Global construction . . . . . . . . . . . . . . . . . . . 43
3.2.2 The algorithm of Flynn and Smart . . . . . . . . . . . 46
3.2.3 Stoll’s refinements . . . . . . . . . . . . . . . . . . . . 47
3.2.4 The “kernel” of " . . . . . . . . . . . . . . . . . . . . 49v
3.3 Kummer surfaces for general models . . . . . . . . . . . . . . 52
3.3.1 Embedding the Kummer surface in arbitrary charac-
teristic . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.2 Duplication . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.3 Biquadratic forms . . . . . . . . . . . . . . . . . . . . 58
3.3.4 Translation by a point of order 2 . . . . . . . . . . . . 59
3.4 Local heights for general models . . . . . . . . . . . . . . . . 62
3.4.1 Definitions and first properties . . . . . . . . . . . . . 62
viiviii CONTENTS
3.4.2 The “kernel” of " revisited . . . . . . . . . . . . . . . 68v
3.4.3 Relation to N´eron models . . . . . . . . . . . . . . . . 70
3.4.4 Simplifying the model . . . . . . . . . . . . . . . . . . 73
3.5 Igusa invariants . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.6 Formulas for local error functions . . . . . . . . . . . . . . . . 84
3.6.1 Case (1) . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.6.2 Case (2) . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.6.3 Case (3) . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.6.4 Case (4) . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.6.5 Case (5) . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.7 Archimedean places . . . . . . . . . . . . . . . . . . . . . . . 105
3.7.1 Approximating using a truncated series. . . . . . . 105v
3.7.2 Theta functions . . . . . . . . . . . . . . . . . . . . . . 106
3.7.3 Richelot isogenies . . . . . . . . . . . . . . . . . . . . . 108
4 Jacobian threefolds 111
4.1 Embedding the Kummer variety . . . . . . . . . . . . . . . . 112
4.2 Defining equations for the Kummer variety . . . . . . . . . . 115
4.3 Remnants of the group law . . . . . . . . . . . . . . . . . . . 118
4.4 Canonical local heights on Jacobians . . . . . . . . . . . . . . 127
4.4.1 Non-archimedean places . . . . . . . . . . . . . . . . . 130
4.4.2 Archimedean places . . . . . . . . . . . . . . . . . . . 130
5 Arithmetic intersection theory 133
5.1 Local N´eron symbols . . . . . . . . . . . . . . . . . . . . . . . 134
5.2 Global N´eron symbols and canonical heights . . . . . . . . . . 138
5.2.1 Representing and reducing divisors . . . . . . . . . . . 141
5.2.2 Mumford representation of divisors on hyperelliptic
curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.3 Computing the global N´eron symbol . . . . . . . . . . . . . . 144
5.3.1 Finding suitable divisors of degree zero . . . . . . . . 145
5.3.2 Determining relevant non-archimedean places . . . . . 147
5.3.3 Regular models . . . . . . . . . . . . . . . . . . . . . . 148
5.3.4 Computing non-archimedean intersection multiplicities 149
5.3.5 Computing the correction term . . . . . . . . . . . . . 158
5.3.6 Computing archimedean intersection multiplicities . . 159
6 Examples and timings 163
6.1 Jacobian surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.1.1 Computing heights . . . . . . . . . . . . . . . . . . .