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 Zpℓ-Artin–Schreier–Witt tower defined by¯f(x); this is a tower of curves · · · → Cm → Cm−1 →· · · →C0 = A1, whose Galois group is canonically isomorphic to Zpℓ, the degree ℓ unramified extension of Zp, which is abstractly isomorphic to (Zp)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 Zpℓ-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.
Bibliographical noteFunding 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.
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All Science Journal Classification (ASJC) codes
- Applied Mathematics