##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory Seminar

## Niccol\`{o} Rocchetti

#### UCLA

## A derived Hecke action on the ordinary Hida tower

##### Abstract:

When studying the cohomology of Shimura varieties and arithmetic manifolds, one encounters the following phenomenon: the same Hecke eigensystem shows up in multiple degrees around the middle dimension, and its multiplicities in these degrees resembles that of an exterior algebra.\\ In a series of recent papers, Venkatesh and his collaborators provide an explanation: they construct graded objects having a graded action on the cohomology, and show that under good circumstances this action factors through that of an explicit exterior algebra, which in turn acts faithfully and generate the entire Hecke eigenspace.\\ In this talk, we discuss joint work with Khare where we investigate the p=p situation (as opposed to the l $\neq$ p situation, which is the main object of study of Venkatesh's Derived Hecke Algebra paper): we construct a degree-raising action on the cohomology of the ordinary Hida tower and show that (under some technical assumptions), this action generates the full Hecke eigenspace under its lowest nonzero degree. Then, we bring Galois representations into the picture, and show that the derived Hecke action constructed before is in fact related to the action of a certain dual Selmer group.

Host: Claus Sorensen

### June 4, 2020

### 2:00 PM

### Register at https://kskedlaya.org/nts.cgi

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