#### 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
##### 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
Primary: 11T23
Secondary: 11G25