Vol. 6, No. 3, 2012

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On unit root formulas for toric exponential sums

Alan Adolphson and Steven Sperber

Vol. 6 (2012), No. 3, 573–585
Abstract

Starting from a classical generating series for Bessel functions due to Schlömilch, 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 ${\mathbb{T}}^{n}$ 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.

Keywords
exponential sums, $A$-hypergeometric functions
Primary: 11T23
Milestones
Received: 7 December 2010
Revised: 28 March 2011
Accepted: 8 May 2011
Published: 5 July 2012
Authors
 Alan Adolphson Department of Mathematics Oklahoma State University Stillwater, OK 74078 United States Steven Sperber School of Mathematics University of Minnesota Minneapolis, MN 55455 United States