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

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 Tn 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.

exponential sums, $A$-hypergeometric functions
Mathematical Subject Classification 2010
Primary: 11T23
Received: 7 December 2010
Revised: 28 March 2011
Accepted: 8 May 2011
Published: 5 July 2012
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