Beilinson–Flach elements, Stark units and 𝑝-adic iterated integrals

Rivero, Óscar; Rotger, Víctor

doi:10.1515/forum-2018-0281

Cited by 2 publications

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References 18 publications

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“…We can attach in a natural way two canonical differentials ω gα and η gα to the weight one modular form g, as it is recalled in [RiRo1,§2 and §3]. The reinterpretation of the main results of [KLZ] in [RiRo1] and [RiRo2] establishes the existence of cohomology classes…”

Section: The Exceptional Zero Phenomenon: Beilinson-flach Elements An...mentioning

confidence: 99%

“…with C an explicit constant involving u ψ 1 , u gα and the periods of [RiRo2], and u 2 = u ψ 2 u ′ ψ 2 . An interesting observation is the case where the Euler system of elliptic units presents an exceptional zero never arises in the setting of [RiRo1], due to the regularity assumptions which are assumed in loc.…”

Section: The Exceptional Zero Phenomenon: Beilinson-flach Elements An...mentioning

confidence: 99%

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The exceptional zero phenomenon for elliptic units

Óscar¹

2019

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The exceptional zero phenomenon has been widely studied in the realm of padic L-functions, where the starting point lies in the foundational work of Mazur, Tate and Teitelbaum. This phenomenon also appears in the study of Euler systems, which comes as no surprise given the interaction between these two settings. When this occurs, one is led to study higher order derivatives of the Euler system in order to extract the arithmetic information which is usually encoded in the explicit reciprocity laws. In this work, we focus on the elliptic units of an imaginary quadratic field and study this exceptional zero phenomenon, proving an explicit formula relating the logarithm of a derived elliptic unit either to special values of Katz's two variable p-adic L-function or to its derivatives. Further, we interpret this fact in terms of an L-invariant, and relate this result to other approaches to the exceptional zero phenomenon concerning Heegner points and Beilinson-Flach elements.Contents 10 3.3. Eisenstein series and improved p-adic L-functions 11 3.4. A reciprocity law for elliptic units 12 4. Exceptional zeros and elliptic units 13 4.1. Specialization at the character ψ 13 4.2. Specialization at the character ψN 14 4.3. A particular case coming from the theory of circular units 17 5. The exceptional zero phenomenon: Beilinson-Flach elements and elliptic units 17 5.1. Elliptic units and Beilinson-Flach elements 18 5.2. Beilinson-Flach elements and exceptional zeros 20 References 22

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Section: The Exceptional Zero Phenomenon: Beilinson-flach Elements An...mentioning

confidence: 99%

Section: The Exceptional Zero Phenomenon: Beilinson-flach Elements An...mentioning

confidence: 99%

The exceptional zero phenomenon for elliptic units

Óscar¹

2019

Preprint

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“…The results presented at this chapter are the content of the research article [RR19], which is a joint work with Victor Rotger.…”

Section: Beilinson-flach Elements Stark Units and P-adic Iterated Int...mentioning

confidence: 97%

“…The results presented at this chapter are the content of the research article [Ri20c], which is currently a work in progress and will be available soon.…”

Section: Cyclotomic Derivatives Of Beilinson-flach Classes and A New ...mentioning

confidence: 99%

“…However, pairs (g, g * ) present a completely different scenario, as we already hinted at above. Not only κ(g α , g * 1/β ) = 0, but it also turns out that the derived class κ (g α , g * 1/β ) is not crystalline at p, as opposed to the set-up of [DR16] and [RR19] (the latter contained as part of Chapter 4). Yet it is still possible to formulate a conjecture analogous to loc.…”

Section: Introductionmentioning

confidence: 99%

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Arithmetic applications of the Euler systems of Beilinson-Flach elements and diagonal cycles

Rivero Salgado

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The main objective of this dissertation is the exploration of certain arithmetic applications of the Euler systems of Beilinson--Flach elements and diagonal cycles. Euler systems have been proved to be a very powerful tool for the study of Iwasawa theory and Selmer groups. Roughly speaking, they are collections of Galois cohomology classes satisfying certain compatibility relations, and are typically constructed using the étale cohomology of algebraic varieties. The genesis of the concept comes from Kolyvagin, who used them to fully prove the Birch and Swinnerton--Dyer conjecture in analytic rank one, and also from Rubin, who proposed a systematic framework to understand this cohomological tool. In the last years, many new constructions and results around these Euler systems have been obtained, and the aim of this thesis is to look at some of their arithmetic applications towards exceptional zeros, special value formulas, and Eisenstein congruences. 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 Bellaïche 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. Aquesta tesi estudia algunes aplicacions a problemes aritmètics dels anomenats sistemes d'Euler, que són classes de cohomologia galoisiana que varien de forma compatible sobre torres de cossos. Seguint una filosofia força general introduïda per Perrin-Riou, la imatge d'aquests sistemes d'Euler sota certs reguladors ens permet recuperar la funció L p-àdica associada a una representació de Galois. En aquesta tesi ens centrem fonamentalment en els sistemes de Beilinson-Flach i de cicles diagonals, tot i que també estudiem altres que comparteixen propietats amb els anteriors i que ens ajuden a una millor comprensió dels mateixos. Recordem que diferents treballs a la darrera dècada ja havíem aconseguit, mitjançant aquests sistemes d’Euler, provar nous casos de la conjectura equivariant de Birch i Swinnerton-Dyer, un dels grans reptes matemàtics dels nostres temps. Les aplicacions aritmètiques que discutim en aquesta monografia són diverses: zeros excepcionals, fórmules de valors especials, resultats de no anul·lació, connexions amb la teoria d’Iwasawa... Els primers capítols de la tesi estudien un fenomen de zeros excepcionals. Recordi’s que les funcions L p-àdiques interpolen, al llarg d’una certa regió, valors de la funció L complexa, llevat de certs factors d’Euler. L’anul·lació d’aquests factors acostuma donar lloc a fenòmens aritmètics interessants. Això, lluny de ser casual, admet una interpretació algebraica en termes de grups de Selmer. Per exemple, la anul·lació en s=0 de la funció de KubotaLeopoldt es relaciona amb el fet que hi hagi una p-unitat extra a la corresponent component del grup d’unitats, i el seu logaritme es relaciona amb la derivada de la funció L p-àdica. Aquesta és una de les conjectures més conegudes de Gross. Aquí comencem estudiant el cas de la representació adjunta d’una forma modular de pes 1. En aquest cas, provem una conjectura de Darmon, Lauder i Rotger que expressa el valor de la derivada de la funció L p-àdica associada en termes d’una combinació de logaritmes de unitats i p-unitats en el cos retallat per la representació. La prova fa servir la teoria de funcions L p-àdiques i funcions L p-àdiques millorades, així com les deformacions de Galois. A més, observem un fenomen que complementa aquest estudi. Les funcions L p-àdiques que hi surten són la imatge del sistema d’Euler de Beilinson-Flach pel morfisme de Perrin-Riou. Aquests zeros excepcionals també s’observen a nivell de sistemes d’Euler, i un pot introduir un concepte de classe derivada que ens permet recuperar l’invariant L que controla l’aritmètica de la representació galoisiana. No només això: amb aquesta noció de derivada podem donar una demostració alternativa del resultat anterior explotant la geometria d’aquests sistemes. Aquesta primera part de la tesi es complementa amb dos capítols on treballem el fenomen de zeros excepcionals tant per a unitats el·líptiques com per a cicles diagonals. Els últims capítols s’endinsen en l’estudi d’altres qüestions al voltant dels sistemes d’Euler, i es comença amb el desenvolupament d’un formalisme d’Artin a nivell de classes de cohomologia. El cas més bàsic passa per considerar una forma pròpia cuspidal de pes 2, de manera que sigui congruent a una sèrie d’Eisenstein. La classe de cohomologia associada a f descompon com la suma de dues components mòdul p. Nosaltres suggerim congruències relacionant cadascuna d’elles amb expressions que involucren unitats circulars. Això fa servir, per una banda, factoritzacions de funcions L p-àdiques i les lleis de reciprocitats; per l’altra, recuperarem alguns resultats de Fukaya-Kato desenvolupats durant la prova de les conjectures de Sharifi.

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Beilinson–Flach elements, Stark units and 𝑝-adic iterated integrals

Cited by 2 publications

References 18 publications

The exceptional zero phenomenon for elliptic units

The exceptional zero phenomenon for elliptic units

Arithmetic applications of the Euler systems of Beilinson-Flach elements and diagonal cycles

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