Selmer groups, p-adic L-functions and Euler Systems: A Unified Framework.

Abstract: Selmer groups are key invariants attached to p-adic Galois representations. The Bloch—Kato conjecture predicts a precise relationship between the size of certain Selmer groups and the leading term of the L-function of the Galois representation under consideration. In particular, when the L-function does not have a zero at s=0, it predicts that the Selmer group is finite and its order is controlled by the value of the L-function at s=0. Historically, one of the most powerful tools to prove such relationships is by constructing an Euler System (ES). An Euler System is a collection of Galois cohomology classes over ramified abelian extensions of the base field that verify some co-restriction compatibilities. The key feature of ESs is that they provide a way to bound Selmer groups, thanks to the machinery developed by Rubin, inspired by earlier work of Thaine, Kolyvagin, and Kato. In this talk, I will present joint work with C. Skinner, in which we develop a new method for constructing Euler Systems and apply it to build an ES for the Galois representation attached to the symmetric square of an elliptic modular form. I will stress how this method gives a unifying approach to constructing ESs, in that it can be successfully applied to retrieve most classical ESs (the cyclotomic units ES, the elliptic units ES, Kato’s ES…).

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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