Vol. 3, No. 5, 2009

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T-adic exponential sums over finite fields

Chunlei Liu and Daqing Wan

Vol. 3 (2009), No. 5, 489–509

We introduce T-adic exponential sums associated to a Laurent polynomial f. They interpolate all classical pm-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 pm-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.

$T$-adic sum, exponential sum, $L$-function, Newton polygon
Mathematical Subject Classification 2000
Primary: 11T23
Secondary: 11G25
Received: 3 March 2008
Revised: 8 January 2009
Accepted: 9 April 2009
Published: 9 November 2009
Chunlei Liu
Department of Mathematical Sciences
Shanghai Jiao Tong University
Shanghai 200240
Daqing Wan
Department of Mathematics
University of California
Irvine, CA 92697-3875
United States