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 $p$ of degree relatively prime to $p$, 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 (over a large subdomain of the weight space). This extends the main result in a 2016 work of Davis, Wan, and Xiao from rank one case $\ell =1$ to the higher rank case $\ell \geq 1$.