##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## James Upton

#### UC Irvine

## A trace formula for deformations of $p$-adic representations

##### Abstract:

Let $X$ be a smooth affine variety over a finite field of characteristic $p$. The Dwork-Monsky trace formula is a fundamental tool in understanding the $L$-functions of $p$-adic representations of $\pi_1(X)$. We extend this result to the study of representations valued in a higher-dimensional local ring $R$. The special case $R=\mathbb{Z}_p[[T]]$ arises naturally in the study of \'etale $\mathbb{Z}_p$-towers over $X$. Time permitting, we discuss some spectral-halo type results and conjectures describing the $p$-adic variation of slopes in certain $\mathbb{Z}_p$-towers.

Host: Kiran Kedlaya

### February 6, 2020

### 1:00 PM

### AP&M 7321

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