Elliptic Curves, Modular Forms and Iwasawa Theory: In Honour of John H. Coates' 70th Birthday, Cambridge, UK, March 2015 PDF
Preview Elliptic Curves, Modular Forms and Iwasawa Theory: In Honour of John H. Coates' 70th Birthday, Cambridge, UK, March 2015
Springer 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 This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today. More information about this series at http://www.springer.com/series/10533 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 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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
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