Eisenstein congruences and Euler systems

Abstract: Let $f$ be a cuspidal eigenform of weight two, and let $p$ be a prime at which $f$ is congruent to an Eisenstein series. Beilinson constructed a class arising from the cup-product of two Siegel units and proved a relationship with the first derivative of the $L$-series of $f$ at the near central point $s=0$. I will motivate the study of congruences between modular forms at the level of cohomology classes, and will report on a joint work with Victor Rotger where we prove two congruence formulas relating the Beilinson class with the arithmetic of circular units. The proofs make use of delicate Galois properties satisfied by various integral lattices and exploits Perrin-Riou's, Coleman's and Kato's work on the Euler systems of circular units and Beilinson-Kato elements and, most crucially, the work of Fukaya-Kato.

algebraic geometrynumber theory

Audience: researchers in the topic

( paper | slides )

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Rendez-vous on special values and periods

Series comments: The main objective of this conference is to gather together young researchers interested in special values of L-functions and periods. These objects are at the crossroads of many recent important developments in arithmetic geometry, such as Euler systems or the theory of motives. The different talks will portray the variety of viewpoints with which L-functions and periods are studied at present.

Registration is free and mandatory, to get access to the livestream and recording of the talks.

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