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Zeta functions of totally ramified \(p\)-covers of the projective line. (English) Zbl 1162.11356

Summary: In this paper we prove that there exists a Zariski dense open subset \(U\) defined over the rationals \(\mathbb Q\) in the space of all one-variable rational functions with arbitrary \(k\) poles of prescribed orders, such that for every geometric point \(f\) in \(U(\bar\mathbb Q)\), 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.

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H30 Coverings of curves, fundamental group
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References:

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