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Zeta functions of totally ramified p-covers of the projective line

Hanfeng Li·Hui June Zhu
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Abstract

In this paper we prove that there exists a Zariski dense open subset U defined over the rationals Q in the space of all one-variable rational functions with arbitrary k poles of prescribed orders, such that for every geometric point f in U(Qbar)$, the L-function of the exponential sum of f at a prime p has Newton polygon approaching the Hodge polygon as p approaches infinity. As an application to algebraic geometry, we prove that the p-adic Newton polygon of the zeta function of a p-cover of the projective line totally ramified at arbitrary k points of prescribed orders has an asymptotic generic lower bound.

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Hanfeng Li, Hui June Zhu. 2005-01-30. Zeta functions of totally ramified p-covers of the projective line. https://arxiv.org/abs/math/0312423

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