## Abstract

We fix a monic polynomial f(x) ∈ Fq [x] over a finite field of characteristic p of degree relatively prime to p, and consider the Z_{p}ℓ-Artin–Schreier–Witt tower defined by^{¯}f(x); this is a tower of curves · · · → C_{m} → C_{m−1} →· · · →C_{0} = A^{1}, whose Galois group is canonically isomorphic to Z_{p}ℓ, the degree ℓ unramified extension of Z_{p}, which is abstractly isomorphic to (Z_{p})^{ℓ} 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 Z_{p}ℓ-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 ℓ = 1 to the higher rank case ℓ ≥ 1.

Original language | English |
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Pages (from-to) | 6411-6432 |

Number of pages | 22 |

Journal | Transactions of the American Mathematical Society |

Volume | 370 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2018 |

### Bibliographical note

Funding Information:Received by the editors June 24, 2016, and, in revised form, November 18, 2016. 2010 Mathematics Subject Classification. Primary 11T23; Secondary 11L07, 11F33, 13F35. Key words and phrases. Artin–Schreier–Witt towers, T-adic exponential sums, slopes of Newton polygon, T-adic Newton polygon for Artin–Schreier–Witt towers, eigencurves. The third author was partially supported by Simons Collaboration Grant #278433 and NSF Grant DMS–1502147.

Publisher Copyright:

© 2018 American Mathematical Society.

## All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics