T-adic exponential sums over finite fields

Details T-adic exponential sums over finite fields T-adic exponential sums over finite fields

2008-02-19

arxiv.org/abs/0802.2589v4

Mathematics

Number Theory

new version, 21 pages, title is changed too

Scientific paper

$T$-adic exponential sums associated to a Laurent polynomial $f$ are introduced. They interpolate all classical $p^m$-power order exponential sums associated to $f$. The Hodge bound for the Newton polygon of $L$-functions of $T$-adic exponential sums is established. 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 $f$ which is ordinary for $m=1$. Deeper properties of $L$-functions of $T$-adic exponential sums are also studied. Along the way, new open problems about the $T$-adic exponential sum itself are discussed.

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