Speaker:  Marco Sangiovanni Vincentelli (Princeton University)

Title:  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-functionof 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…).

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Masks are incouraged, but not required.