On unit root formulas for toric exponential sums

Alan Adolphson; Steven Sperber

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arXiv · 1012.1637

On unit root formulas for toric exponential sums

Alan Adolphson·Steven Sperber

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Abstract

Starting from a classical generating series for Bessel functions due to Schlomilch, we use Dwork's relative dual theory to broadly generalize unit-root results of Dwork on Kloosterman sums and Sperber on hyperkloosterman sums. In particular, we express the (unique) p-adic unit root of an arbitrary exponential sum on the torus in terms of special values of the p-adic analytic continuation of a ratio of A-hypergeometric functions. In contrast with the earlier works, we use noncohomological methods and obtain results that are valid for arbitrary exponential sums without any hypothesis of nondegeneracy.

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Alan Adolphson, Steven Sperber. 2010-12-07. On unit root formulas for toric exponential sums. https://arxiv.org/abs/1012.1637

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