##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Rufei Ren

#### UC Irvine

## Slopes for higher rank Artin--Schreier--Witt towers

##### Abstract:

We fix a monic polynomial $\bar f(x) \in \mathbb{F}_q[x]$ over a finite field of characteristic $p$, and consider the $\mathbb{Z}_{p^{\ell}}$-Artin--Schreier--Witt tower defined by $\bar f(x)$; this is a tower of curves $\cdots \to C_m \to C_{m-1} \to \cdots \to C_0 =\mathbb{A}^1$, whose Galois group is canonically isomorphic to $\mathbb{Z}_{p^\ell}$, the degree $\ell$ unramified extension of $\mathbb{Z}_p$, which is abstractly isomorphic to $(\mathbb{Z}_p)^\ell$ as a topological group. We study the Newton slopes of zeta functions of this tower of curves. This reduces to the study of the Newton slopes of L-functions associated to characters of the Galois group of this tower. We prove that, when the conductor of the character is large enough, the Newton slopes of the L-function asymptotically form a finite union of arithmetic progressions. As a corollary, we prove the spectral halo property of the spectral variety associated to the $\mathbb{Z}_{p^{\ell}}$-Artin--Schreier--Witt tower. This extends the main result of Davis--Wan--Xiao from rank one case $\ell=1$ to the higher rank case $\ell\geq 1$.

Host: Kiran Kedlaya

### November 17, 2016

### 1:00 PM

### AP&M 7321

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