Zeta elements for Shimura Varieties I: Generalities

Abstract: A well-established technique towards understanding Selmer groups of Galois representations is the construction of an Euler system. One may ask if such systems can be created for Galois representations that arise in the cohomology of a given Shimura variety. For such purposes, it is customary to utilize push-forwards of fundamental cycles or Eisenstein classes from sub-Shimura varieties, and to then establish norm relations between the push-forwarded classes involving certain Hecke operators which compute appropriate automorphic L-factors.

In this talk, I will motivate how classical Euler systems such as Kato's Siegel units and Kolyvagin's Heegner points may be viewed through an axiomatic lens and build up to a general framework in which norm relations for higher dimensional Shimura varieties may be systematically studied. I will highlight some of the computational challenges that arise in higher dimensions and outline a theory of double coset decompositions due to Lansky that allows one to overcome these challenges. In the next talk, I'll apply these techniques to concrete examples of arithmetic interest, some old and some new.

algebraic geometrynumber theory

Audience: researchers in the topic

UCSB Seminar on Geometry and Arithmetic