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Transactions of the American Mathematical Society

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Slopes for higher rank Artin-Schreier-Witt towers


Authors: Rufei Ren, Daqing Wan, Liang Xiao and Myungjun Yu
Journal: Trans. Amer. Math. Soc. 370 (2018), 6411-6432
MSC (2010): Primary 11T23; Secondary 11L07, 11F33, 13F35
DOI: https://doi.org/10.1090/tran/7162
Published electronically: May 3, 2018
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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$.


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Additional Information

Rufei Ren
Affiliation: Department of Mathematics, University of California, Irvine, 340 Rowland Hall, Irvine, California 92697
Address at time of publication: Department of Mathematics, University of Rochester, Hylan Building, 140 Trustee Road, Rochester, New York 14627
Email: rren2@ur.rochester.edu

Daqing Wan
Affiliation: Department of Mathematics, University of California, Irvine, 340 Rowland Hall, Irvine, California 92697
Email: dwan@math.uci.edu

Liang Xiao
Affiliation: Department of Mathematics, University of Connecticut, 341 Mansfield Road, Unit 1009, Storrs, Connecticut 06269-1009
Email: liang.xiao@uconn.edu

Myungjun Yu
Affiliation: Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, Michigan 48109-1043
Email: myungjuy@umich.edu

DOI: https://doi.org/10.1090/tran/7162
Keywords: Artin--Schreier--Witt towers, $T$-adic exponential sums, slopes of Newton polygon, $T$-adic Newton polygon for Artin--Schreier--Witt towers, eigencurves
Received by editor(s): June 24, 2016
Received by editor(s) in revised form: November 18, 2016
Published electronically: May 3, 2018
Additional Notes: The third author was partially supported by Simons Collaboration Grant #278433 and NSF Grant DMS–1502147.
Article copyright: © Copyright 2018 American Mathematical Society

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