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Arithmetic Applications of the Euler Systems of Beilinson–Flach Elements and Diagonal Cycles

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Arithmetic Applications of the Euler Systems of Beilinson–Flach Elements and Diagonal Cycles Universitat Politecnica` de Catalunya Programa de doctorat de matematica` aplicada Facultat de Matematiques` i Estad´ıstica Arithmetic applications of the Euler systems of Beilinson{Flach elements and diagonal cycles Author: Advisor: Oscar´ RIVERO SALGADO Victor ROTGER CERDA` PhD dissertation submitted for the degree of Doctor in Mathematics Barcelona, February 2021 2 Aos meus pais, Alberto e Josefa, por quererme sempre de xeito incondicional. \Tra bufalo e locomotiva la differenza salta agli occhi: la locomotiva ha la strada segnata, il bufalo pu`oscartare di lato e cadere." Francesco De Gregori. Contents Abstract 5 Resum 7 Acknowledgements and funding 9 Introduction 11 The Birch and Swinnerton-Dyer conjecture and beyond . 11 A taste of the main results: a p-adic Gross{Stark formula . 15 Exceptional zeros of Euler systems . 18 Artin formalism and Euler systems . 19 Organization of the thesis . 20 1 Background material I 21 1.1 BSD-type conjectures . 21 1.2 Euler systems . 26 1.3 p-adic L-functions . 33 2 Background material II 43 2.1 Gross{Stark units . 43 2.2 The exceptional zero phenomenon . 53 2.3 Congruences between modular forms . 58 3 Beilinson{Flach elements and exceptional zeros 65 3.1 Introduction . 65 3.2 Preliminary concepts . 72 3.3 Derived Beilinson{Flach elements . 76 3.4 Derivatives of Fourier coefficients via Galois deformations . 83 3.5 Proof of main results . 88 3.6 Darmon{Dasgupta units and p-adic L-functions . 91 4 Beilinson{Flach elements and Stark units 95 4.1 Introduction . 95 4.2 The main conjecture of Darmon, Lauder and Rotger . 99 4.3 Beilinson{Flach elements and the main conjecture . 100 4.4 The self-dual case . 105 4.5 Particular cases of the conjecture . 109 3 4 CONTENTS 5 Cyclotomic derivatives and a Gross{Stark formula 113 5.1 Introduction . 113 5.2 Analogy with the case of circular units . 116 5.3 Beilinson{Flach elements . 118 5.4 Cyclotomic derivatives and proof of the main theorem . 121 5.5 A reinterpretation of the special value formula . 123 5.6 A conjectural p-adic L-function . 125 6 Exceptional zeros and elliptic units 129 6.1 Introduction . 129 6.2 Circular units . 132 6.3 Elliptic units . 137 6.4 Exceptional zeros and elliptic units . 142 6.5 Beilinson{Flach elements and beyond . 148 7 Generalized Kato classes and exceptional zeros 155 7.1 Introduction . 155 7.2 Preliminaries . 161 7.3 Derived diagonal cycles and an explicit reciprocity law . 166 7.4 Derivatives of triple product p-adic L-functions . 172 7.5 Applications to the Elliptic Stark Conjecture . 175 8 Congruences between units and Kato elements 181 8.1 Introduction . 181 8.2 Modular curves, modular units and Eisenstein series . 185 8.3 First congruence relation . 186 8.4 Second congruence relation . 196 9 Summaries in Catalan and Galician 201 9.1 Resum extens en catal`a . 201 9.2 Resumo extenso en galego . 208 9.3 Tra bufalo e locomotiva: una visi´opersonal . 215 Abstract In this thesis we study several applications of the theory of Euler systems and p-adic L-functions, with an emphasis on special value formulas, exceptional zeros, and congruence relations. The first chapters deal with different kinds of exceptional zero phenomena. The main result we obtain is the proof of a conjecture of Darmon, Lauder and Rotger on special values of the Hida{Rankin p-adic L-function, which may be regarded both as the proof of a Gross{Stark type conjecture, or as the determination of the L-invariant corresponding to the adjoint representation of a weight one modular form. The proof recasts to Hida theory and to the ideas developed by Greenberg{Stevens, and makes use of the Galois deformation techniques introduced by Bella¨ıche and Dimitrov. We further discuss a similar exceptional zero phenomenon from the Euler system side, leading us to the construction of derived Beilinson{Flach classes. This allows us to give a more conceptual proof of the previous result, using the underlying properties of this Euler system. We also discuss other instances of this formalism, studying exceptional zeros at the level of cohomology classes both in the scenario of elliptic units and diagonal cycles. The last part of the thesis aims to start a systematic study of the Artin formalism for Euler systems. This relies on ideas regarding factorizations of p-adic L-functions, and also recasts to the theory of Perrin-Riou maps and the study of canonical periods attached to weight two modular forms. We hope that these results could be extended to different settings concerning the other Euler systems studied in this memoir. Keywords: exceptional zeros, Euler systems, p-adic L-functions, Elliptic Stark Conjecture, Eisenstein congruences. MSC2010: 11F67, 11F80, 11G40, 11F33. 5 6 ABSTRACT Resum En aquesta tesi estudiem diverses aplicacions de la teoria dels sistemes d'Euler i les funcions L p-`adiques,posant l'`emfasien f´ormules de valors especials, zeros excepcionals i relacions de con- gru`enciesentre formes modulars. Als primers cap´ıtolses consideren diferents fen`omensde zeros excepcionals. El primer resultat que obtenim ´esla prova d'una conjectura de Darmon, Lauder i Rotger al voltant dels valors especials de la funci´o L p-`adicade Hida{Rankin en el cas autoadjunt. Aquest teorema es pot veure, per una banda, com un cas m´esde les conjectures de Gross{Stark, i per l'altra, com la determinaci´ode l'invariant L corresponent a la representaci´oadjunta d'una forma modular de pes 1. La prova fa servir teoria de Hida, aix´ıcom algunes de les idees desenvolupades per Greenberg{Stevens i tamb´e les t`ecniquesde deformacions de Galois introdu¨ıdesper Bella¨ıche i Dimitrov. Discutim despr´esun fenomen semblant de zeros excepcionals des del punt de vista dels sistemes d'Euler, la qual cosa ens porta a la construcci´ode classes de Beilinson{Flach derivades. Aix`oens permetr`adonar una prova m´esconceptual del resultat precedent, fent servir per a aix`oles propietats d'aquest sistema d'Euler derivat. Discutim per ´ultimaltres exemples on s'observen aquests fen`omensal nivell de sistemes d'Euler, centrant-nos en els casos d'unitats el·l´ıptiquesi cicles diagonals. La darrera part de la tesi pret´encomen¸car un estudi sistem`aticdel formalisme d'Artin pels sistemes d'Euler. Aquesta teoria fa servir factoritzacions de les funcions L p-`adiques,i tamb´e requereix un estudi de les aplicacions de Perrin-Riou i dels per´ıodes can`onics associats a les formes modulars de pes 2. Esperem que aquests resultats es puguin estendre a altres casos relatius als sistemes d'Euler estudiats al llarg d'aquesta mem`oria. Paraules claus: zeros excepcionals, sistemes d'Euler, funcions L p-`adiques,conjectures de Stark, congru`enciesd'Eisenstein. MSC2010: 11F67, 11F80, 11G40, 11F33. 7 8 RESUM Acknowledgements and funding My first words of gratitude go to my advisor Victor Rotger, with whom I held many stimulating mathematical discussions and who shared with me his privilege insights into number theory. Along the PhD programme, I did two long-term research visits where I enjoyed the environment of the departments of mathematics at both Princeton University and McGill University. I would like to thank those institutions for their warm hospitality, and specially my hosts there, Chris Skinner and Henri Darmon, with whom I had the privilege to discuss about mathematics and learn from their proficiency. It was also at Princeton University where I had the opportunity to share interesting discussions with Francesc Castell`a,whose help was crucial in some parts of the thesis. This monograph has also benefited from discussions with other number theorists, both during the different conferences I attended or by electronic correspondence. Even at the risk of forgetting some names, I am indebted to Ra´ulAlonso, Daniel Barrera, Denis Benois, Adel Betina, Kazim B¨uy¨ukboduk, Antonio Cauchi, Mladen Dimitrov, Francesca Gatti, Adrian Iovita, Antonio Lei, David Loeffler, Giovanni Rosso, and Preston Wake. I also thank Mart´ıRoset and Guillem Sala for a careful reading of some parts of this dissertation. The environment at the Number Theory group of Barcelona was close to exceptional, having the possibility to learn a lot from the different people with whom I shared seminars and study groups. This includes all the research trainees, postdoctoral researchers and faculties with whom I had the opportunity to coincide. In particular, I would like to thank Francesc Fit´efor his constant support during my first year of PhD. Further, it is a pleasure to mention all these people for the time we have spent together and their constant encouragement: Luis Dieulefait, Daniel Gil, Francesc Gispert, Xavier Guitart, Armando Guti´errez,Valentin Hern´andez,V´ıctor Hern´andez, Marc Masdeu, Santiago Molina, Eduardo Soto, and Carlos de Vera. Apart from them, it is a must to also recognize the work of all those people who are around the STNB activities, with whom it is always a pleasure to share insights and experiences. From the School of Mathematics and Statistics, I would like to thank a few people for their encouragement, and for making the corridors of the office building a more human place. The Dean Jaume Franch represents very well the confidence I have felt from the School, where I had the opportunity to teach several courses; Juanjo Ru´ehas been always supportive, being willing to hear and talk about whatever issue; Santi Boza, Angeles´ Carmona, Andr´esEncinas, Jordi Quer, Natalia Sadovskaia, and Enric Ventura also shared teaching duties with me, and it was enriching to work with them; the administrative stuff coordinating the graduate studies, specially Carme Capdevila, has made our job easier; Marta Altarriba, Carles Checa, and Andreu Tom`astrusted on me for the supervision of their bachelor thesis, and I had the chance to become a better mathematician discussing with them; finally, all my undergraduate students deserve my deepest gratitude.
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