Slopes for higher rank artin–schreier–witt towers

Rufei Ren; Daqing Wan; Liang Xiao; Myungjun Yu

doi:10.1090/tran/7162

Slopes for higher rank artin–schreier–witt towers

Rufei Ren, Daqing Wan, Liang Xiao, Myungjun Yu

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

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_pℓ-Artin–Schreier–Witt tower defined by^¯f(x); this is a tower of curves · · · → C_m → C_m−1 →· · · →C₀ = A¹, whose Galois group is canonically isomorphic to Z_pℓ, 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_pℓ-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
Pages (from-to)	6411-6432
Number of pages	22
Journal	Transactions of the American Mathematical Society
Volume	370
Issue number	9
DOIs	https://doi.org/10.1090/tran/7162
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

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10.1090/tran/7162

Cite this

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title = "Slopes for higher rank artin–schreier–witt towers",

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 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.",

author = "Rufei Ren and Daqing Wan and Liang Xiao and Myungjun Yu",

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: {\textcopyright} 2018 American Mathematical Society.",

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doi = "10.1090/tran/7162",

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journal = "Transactions of the American Mathematical Society",

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AU - Ren, Rufei

AU - Wan, Daqing

AU - Xiao, Liang

AU - Yu, Myungjun

N1 - 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.

PY - 2018

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N2 - 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.

AB - 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.

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Slopes for higher rank artin–schreier–witt towers

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