Table Of ContentSpringer Proceedings in Mathematics & Statistics 188
David Loeffler
Sarah Livia Zerbes Editors
Elliptic Curves,
Modular Forms
and Iwasawa
Theory
In Honour of John H. Coates’ 70th
Birthday, Cambridge, UK, March 2015
Springer Proceedings in Mathematics & Statistics
Volume 188
Springer Proceedings in Mathematics & Statistics
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fl
David Loef er Sarah Livia Zerbes
(cid:129)
Editors
Elliptic Curves,
Modular Forms
and Iwasawa Theory
’
In Honour of John H. Coates 70th Birthday,
Cambridge, UK, March 2015
123
Editors
DavidLoeffler SarahLivia Zerbes
Mathematics Institute Department ofMathematics
University of Warwick University CollegeLondon
Coventry London
UK UK
ISSN 2194-1009 ISSN 2194-1017 (electronic)
SpringerProceedings in Mathematics& Statistics
ISBN978-3-319-45031-5 ISBN978-3-319-45032-2 (eBook)
DOI 10.1007/978-3-319-45032-2
LibraryofCongressControlNumber:2016948295
MathematicsSubjectClassification(2010): 11F80,11G05,11G40,11R23,11S40,14H52
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Preface
In March 2015, the Dokchitser brothers and the two of us organized two events, a
workshop and a conference, in honour of John Coates’ 70th birthday, in order to
celebrateJohn’sworkandhismathematicalheritage.Amongtheparticipantsofthe
conference were many young mathematicians, and it is clear that John’s work, in
particular on Iwasawa theory, continues to be a great source of inspiration for the
new generation of number theorists. It is therefore a pleasure to dedicate this
volume to him, in admiration for his contributions to number theory and his
influence on the subject via his many students and collaborators.
Warwick, UK David Loeffler
July 2016 Sarah Livia Zerbes
v
vi Preface
JohnattheRoyalSocietyKavliCentre,March2015
Publishedwithkindpermission,©JohnCoates
Contents
Congruences Between Modular Forms and the Birch
and Swinnerton-Dyer Conjecture... .... .... .... .... .... ..... .... 1
Andrea Berti, Massimo Bertolini and Rodolfo Venerucci
p-adic Measures for Hermitian Modular Forms
and the Rankin–Selberg Method... .... .... .... .... .... ..... .... 33
Thanasis Bouganis
Big Image of Galois Representations Associated with Finite
Slope p-adic Families of Modular Forms. .... .... .... .... ..... .... 87
Andrea Conti, Adrian Iovita and Jacques Tilouine
Behaviour of the Order of Tate–Shafarevich Groups
for the Quadratic Twists of X0ð49Þ . .... .... .... .... .... ..... .... 125
Andrzej Dąbrowski, Tomasz Jędrzejak and Lucjan Szymaszkiewicz
Compactifications of S-arithmetic Quotients for the Projective
General Linear Group .. .... ..... .... .... .... .... .... ..... .... 161
Takako Fukaya, Kazuya Kato and Romyar Sharifi
On the Structure of Selmer Groups. .... .... .... .... .... ..... .... 225
Ralph Greenberg
Control of K-adic Mordell–Weil Groups. .... .... .... .... ..... .... 253
Haruzo Hida
Some Congruences for Non-CM Elliptic Curves... .... .... ..... .... 295
Mahesh Kakde
Diophantine Geometry and Non-abelian Reciprocity Laws I. ..... .... 311
Minhyong Kim
On p-adic Interpolation of Motivic Eisenstein Classes .. .... ..... .... 335
Guido Kings
vii
viii Contents
Vanishing of Some Galois Cohomology Groups
for Elliptic Curves.. .... .... ..... .... .... .... .... .... ..... .... 373
Tyler Lawson and Christian Wuthrich
Coates–Wiles Homomorphisms and Iwasawa Cohomology
for Lubin–Tate Extensions... ..... .... .... .... .... .... ..... .... 401
Peter Schneider and Otmar Venjakob
Bigness in Compatible Systems .... .... .... .... .... .... ..... .... 469
Andrew Snowden and Andrew Wiles
Congruences Between Modular Forms
and the Birch and Swinnerton-Dyer
Conjecture
AndreaBerti,MassimoBertoliniandRodolfoVenerucci
Abstract We prove the p-part of the Birch and Swinnerton-Dyer conjecture for
ellipticcurvesofanalyticrankoneformostordinaryprimes.
· ·
Keywords Elliptic curves Birch and Swinnerton-Dyer conjecture Heegner
·
points Shimuracurves
1 Introduction
The theory of congruences between modular forms has turned out to be a crucial
playerinanumberofmomentousresultsinthetheoryofrationalpointsonelliptic
curves.Tomentiononlyafewinstances,werecallhereMazur’stheoryoftheEisen-
steinideal[16],inwhichcongruencesbetweencuspformsandEisensteinserieson
GL are used to uniformly bound the torsion subgroups of elliptic curves over Q.
2
More germane to our setting, the recent work of Skinner–Urban [22] constructs
classes in the p-primary Shafarevich–Tate group of an elliptic curve over Q (and
more generally, over cyclotomic extensions) when p is ordinary and divides (the
algebraicpartof)thevalueoftheassociatedHasse–WeilL-seriesats=1.Thisis
A.Berti
DipartimentodiMatematicaFederigoEnriques,UniversitàdiMilano,
viaC.Saldini50,Milano,Italy
e-mail:[email protected]
B
M.Bertolini( )·R.Venerucci
UniversitätDuisburg-Essen,FakultätFürMathematik,Mathematikcarrée,
Thea-Leymann-Strasse9,45127Essen,Germany
e-mail:[email protected]
R.Venerucci
e-mail:[email protected]
(cid:2)c SpringerInternationalPublishingSwitzerland2016 1
D.LoefflerandS.L.Zerbes(eds.),EllipticCurves,ModularForms
andIwasawaTheory,SpringerProceedingsinMathematics
&Statistics188,DOI10.1007/978-3-319-45032-2_1
