Arithmetic applications of the euler systems of beilinson-flach elements and diagonal cycles

Resumo

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.