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Genus growth in $ \mathbb{Z}_p$-towers of function fields


Authors: Michiel Kosters and Daqing Wan
Journal: Proc. Amer. Math. Soc. 146 (2018), 1481-1494
MSC (2010): Primary 11G20, 11R37, 12F05
DOI: https://doi.org/10.1090/proc/13895
Published electronically: November 13, 2017
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Abstract: Let $ K$ be a function field over a finite field $ k$ of characteristic $ p$ and let $ K_{\infty }/K$ be a geometric extension with Galois group $ \mathbb{Z}_p$. Let $ K_n$ be the corresponding subextension with Galois group $ \mathbb{Z}/p^n\mathbb{Z}$ and genus $ g_n$. In this paper, we give a simple explicit formula for $ g_n$ in terms of an explicit Witt vector construction of the $ \mathbb{Z}_p$-tower. This formula leads to a tight lower bound on $ g_n$ which is quadratic in $ p^n$. Furthermore, we determine all $ \mathbb{Z}_p$-towers for which the genus sequence is stable, in the sense that there are $ a,b,c \in \mathbb{Q}$ such that $ g_n=a p^{2n}+b p^n +c$ for $ n$ large enough. Such genus stable towers are expected to have strong stable arithmetic properties for their zeta functions. A key technical contribution of this work is a new simplified formula for the Schmid-Witt symbol coming from local class field theory.


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Additional Information

Michiel Kosters
Affiliation: Department of Mathematics, University of California, Irvine, 340 Rowland Hall, Irvine, California 92697
Email: kosters@gmail.com

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

DOI: https://doi.org/10.1090/proc/13895
Keywords: $\Z_p$-extension, Artin-Schreier-Witt, Schmid-Witt, local field, global field, genus, conductor
Received by editor(s): March 15, 2017
Received by editor(s) in revised form: June 7, 2017
Published electronically: November 13, 2017
Communicated by: Mathew A. Papanikolas
Article copyright: © Copyright 2017 American Mathematical Society

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