On unit root formulas for toric exponential sums

Adolphson, Alan; Sperber, Steven

doi:10.2140/ant.2012.6.573

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Form

Policies for Authors

Ethics Statement

ISSN: 1944-7833 (e-only)

ISSN: 1937-0652 (print)

Author Index

To Appear

Other MSP Journals

On unit root formulas for toric exponential sums

Alan Adolphson and Steven Sperber

Vol. 6 (2012), No. 3, 573–585

DOI: 10.2140/ant.2012.6.573

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 $T^{n}$ 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

Mathematical Subject Classification 2010

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

Vol. 6, No. 3, 2012