T-adic exponential sums over finite fields

Liu, Chunlei; Wan, Daqing

doi:10.2140/ant.2009.3.489

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T-adic exponential sums over finite fields

Chunlei Liu and Daqing Wan

Vol. 3 (2009), No. 5, 489–509

DOI: 10.2140/ant.2009.3.489

Abstract

We introduce $T$ -adic exponential sums associated to a Laurent polynomial $f$ . They interpolate all classical $p^{m}$ -power order exponential sums associated to $f$ . We establish the Hodge bound for the Newton polygon of $L$ -functions of $T$ -adic exponential sums. This bound enables us to determine, for all $m$ , the Newton polygons of $L$ -functions of $p^{m}$ -power order exponential sums associated to an $f$ that is ordinary for $m = 1$ . We also study deeper properties of $L$ -functions of $T$ -adic exponential sums. Along the way, we discuss new open problems about the $T$ -adic exponential sum itself.

Keywords

$T$-adic sum, exponential sum, $L$-function, Newton polygon

Mathematical Subject Classification 2000

Primary: 11T23

Secondary: 11G25

Milestones

Received: 3 March 2008

Revised: 8 January 2009

Accepted: 9 April 2009

Published: 9 November 2009

Authors

Chunlei Liu
Department of Mathematical Sciences Shanghai Jiao Tong University Shanghai 200240 China

Daqing Wan
Department of Mathematics University of California Irvine, CA 92697-3875 United States

Vol. 3, No. 5, 2009