Elliptic Curves, Hilbert Modular Forms and Galois Deformations

257 Pages·2013·1.917 MB·English

by  Laurent Berger, Gebhard Böckle, Lassina Dembélé, Mladen Dimitrov, Tim Dokchitser, John Voight (auth.)

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Advanced Courses in Mathematics
CRM Barcelona
Centre de Recerca Matemàtica
Managing Editor:
Carles Casacuberta
For further volumes:
http://www.springer.com/series/5038
Laurent Berger
Gebhard Böckle
Lassina Dembélé
Mladen Dimitrov
Tim Dokchitser
John Voight
Elliptic Curves,
Hilbert Modular Forms and
Galois Deformations
Editors for this volume:
Henri Darmon (McGill University)
Fred Diamond (King’s College London)
Luis V. Dieulefait (Universitat de Barcelona)
Bas Edixhoven (Universiteit Leiden)
Víctor Rotger (Universitat Politècnica de Catalunya)
Laurent Berger Gebhard Böckle
UMPA-ENS Lyon Interdisciplinary Center for Scientific
Lyon, France Computing (IWR)
Universität Heidelberg
Lassina Dembélé Heidelberg, Germany
Warwick Mathematics Institute
University of Warwick Mladen Dimitrov
Coventry, UK Cité Scientifique, UFR Mathématiques
Université Lille 1
Tim Dokchitser Villeneuve d’Ascq, France

Department of Mathematics

University of Bristol John Voight

Bristol, UK Department of Mathematics and Statistics

University of Vermont
Burlington, VT, USA

ISBN 978-3-0348-0617-6 ISBN 978-3-0348-0618-3 (eBook)
DOI 10.1007/978-3-0348-0618-3
Springer Basel Heidelberg New York Dordrecht London

Library of Congress Control Number: 2013941399

Mathematical Subject Classification (2010): Primary: 11F80, 11G18, 11G05, 11G07; Secondary: 11R23, 11F41

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Foreword
AresearchprogrammeinArithmeticGeometrywasorganizedattheCentredeRe-
cerca Matem`atica by F. Bars (Universitat Aut`onoma de Barcelona), L. Dieulefait
(Universitat de Barcelona) and V. Rotger (Universitat Polit`ecnica de Catalunya)
in the academic year 2009–2010. It provided an excellent opportunity to bring to-
getherspecialistsandresearchersinthisfieldandgermanelinesofresearch,having
expertise in different but always overlapping aspects of this beautiful branch of
mathematics. The aim of the research programme was to learn about the details
of major developments in the area, to explore new directions and perspectives,
to promote fruitful collaborations among participants and to serve as a valuable
centre for the advancement of young researchers.
The ultimate goal of the research programme was to explore several key ob-
jectsfromthearithmeticofnumberfields/functionfieldswhichconstitutealiving
proof of this fruitful cross-fertilization between the two worlds, such as Drinfeld
and classical modular varieties, classical and p-adic L-functions, the Langlands
programme, the Birch and Swinnerton-Dyer conjecture, Iwasawa theory, and the
Bloch–Kato conjecture.
Alreadyinthe19thcentury,mathematicianshadrealizedthestrikingsimilar-
ities between number fields and function fields; later, prominent number theorists
such as Artin and Weil made essential use of function fields of one variable over
a finite field to reach a deeper understanding of L-functions and the Riemann
hypothesis. Here we can see a first instance of a recurrent theme in the number
fields/function field dialectic: the transposition into one setting of questions first
asked over the other (often a far from trivial task) allows us to gain new insight
and, in many cases, to progress more than in the situation previously considered,
thanks to a richer structure to be exploited. In particular, function fields have
become a standard testing ground to get evidence for hard conjectures proposed
over number fields.
Thefirstandthirdtermsoftheprogrammefocusedonthearithmeticofellip-
ticcurvesandmoregeneralmotives,modularforms,automorphicrepresentations,
L-functions and Shimura varieties over number fields.
During the last few years there have been fundamental breakthroughs in
the subject. Several long-standing conjectures have been settled and fundamen-
tal progress has been achieved towards others. In addition, precise variants and
v
vi Foreword
generalizations of these conjectures have been raised, and these demand refine-
ments of the existing methods as well as completely new ideas in order to be
approached.
Someofthesemajorachievementstaketheformofmodularityresults,which
aim to establish a bridge between algebraic varieties or abstract geometric Galois
representations on one side and modular or automorphic forms on a Shimura
varietyontheother.ProminentinstancesofthisphilosophyareSerre’sModularity
Conjecture and the Fontaine–Mazur Conjecture, among others.
In parallel, the diophantine study of algebraic varieties has received a dra-
matic impulse over the last years provided they can be shown to be modular.
ThankstothecelebratedresultofWilesandhiscollaborators,ellipticcurvesover
thefieldofrationalnumbersprovidethesimplest(buthighlynon-trivial)examples
of this phenomenon, and their arithmetic still offers deep questions which remain
unsolved.Theconstructionofrationalpointsonthem,theBirchandSwinnerton-
DyerConjecture,itsp-adicformulationsandgeneralizationstohigherdimensional
abelianvarietiesovernumberfieldsarecertainlyoneofthem.Sofar,mostinsights
to these problems exploit and are conditional to the fact that they appear as a
factor of the Jacobian of a suitable modular or Shimura curve.
Computation has played a crucial role in the formulation of some of the
mostimportantconjecturesinvolvingmodularformsandellipticcurvesinthelast
decades.Nowadaysawidevarietyofalgorithms,packagesandsoftwareisavailable
and, for this reason, the research programme also focused on the computational
side and explicit approaches to the subject.
Several workshops and advanced courses on the above topics were organized
during this research programme at the Centre de Recerca Matem`atica. This vol-
ume collects the notes of some of the lecture series delivered during the first and
third terms of the programme. What follows is a summary of them.
The notes by L. Berger provide an introduction to p-adic Galois represen-
tations and Fontaine rings. These are useful in particular to describe many local
deformation rings at p considered in Galois deformation theory (see the notes of
B¨ockle’s course).
The notes by G. Bo¨ckle give a comprehensive course on Galois deformation
theory, starting from the foundational results of Mazur and discussing in detail
the theory of pseudo-representations and their deformations, local deformations
atplaces(cid:2)(cid:2)=pandlocaldeformationsatpwithgeometricconditions(inthesense
of the Fontaine–Mazur conjecture). The final chapter of these notes presents the
results of B¨ockle and Kisin on presentations of global deformation rings over local
ones,fromwhichaveryusefullowerboundforglobaldeformationringswithlocal
conditions can be derived.
The notes by M. Dimitrov contain the basics of the arithmetic theory of
Hilbertmodularformsandvarieties,withanemphasisonthestudyoftheimages
oftheattachedGaloisrepresentations,onmodularityliftingtheoremsovertotally
realnumberfieldsandonthecohomologyofHilbertmodularvarietieswithintegral
coefficients.
Foreword vii
The notes by L. Demb´el´e and J. Voight describe methods for performing ex-
plicitcomputationsinspacesofHilbertmodularforms.Thesemethodsdependon
the Jacquet–Langlands correspondence and on computations in spaces of quater-
nionic modular forms, both for the case of definite and indefinite quaternion alge-
bras. The notes contain several examples and include applications to modularity
of Galois representations.
ThenotesbyT.Dokchitserdescribetheproof,obtainedbytheauthorinjoint
work with V. Dokchitser, of the parity conjecture for elliptic curves over number
fieldsundertheassumptionoffinitenessoftheTate–Shafarevichgroup.Thenotes
containthestatementoftheBirchandSwinnerton-Dyerconjectureandadetailed
study of local and global root numbers of elliptic curves and their classification.
Bellaterra, December 2012

Contents
Foreword v
I Galois Deformations 1
On p-adic Galois Representations 3
Laurent Berger
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1 The Galois group of Q . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Ramification of local fields, I . . . . . . . . . . . . . . . . . . . . . 4
3 p-adic representations with (cid:2)(cid:2)=p . . . . . . . . . . . . . . . . . . . 4
4 Rings of periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
5 Galois cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
6 Ramification of local fields, II . . . . . . . . . . . . . . . . . . . . . 7
7 Cyclotomic extensions . . . . . . . . . . . . . . . . . . . . . . . . . 7
8 The cohomology of C . . . . . . . . . . . . . . . . . . . . . . . . . 8
p
9 Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
10 The rings E(cid:2)+ and B(cid:2)+ . . . . . . . . . . . . . . . . . . . . . . . . . 10
11 The field B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
dR
12 De Rham representations . . . . . . . . . . . . . . . . . . . . . . . 11
13 The rings B and B . . . . . . . . . . . . . . . . . . . . . . . . 12
max st
14 Crystalline and semistable representations . . . . . . . . . . . . . . 13
15 Admissible filtered (ϕ,N)-modules . . . . . . . . . . . . . . . . . . 14
16 The groups H1(K,V) . . . . . . . . . . . . . . . . . . . . . . . . . 16
∗
17 A p-adic period pairing . . . . . . . . . . . . . . . . . . . . . . . . 17
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Deformations of Galois Representations 21
Gebhard Bo¨ckle
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1 Deformations of representations of profinite groups . . . . . . . . . 26
1.1 Deformation functors. . . . . . . . . . . . . . . . . . . . . . 26
1.2 A finiteness condition . . . . . . . . . . . . . . . . . . . . . 27
ix