T-adic exponential sums over finite fields

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.

Generic twisted T-adic exponential sums of binomials

“…By a result of Liu [7], the q-adic Newton polygon of L f,u (s, π 1 ) coincides with H ∞ [0,d],u at the point d. By Remark 1, P {1,d},u coincides with (p − 1)H ∞ [0,d],u at the point d, it follows that the π 1 -adic Newton polygon of L f,u (s, π 1 ) coincides with ord p (q)P {1,d},u at the point d. Therefore, it suffices to show that π 1 -adic NP of L f,u (s, π 1 ) = ord p (q)P {1,d},u on [0, d − 1].…”

“…The T -adic exponential sums were first introduced by Liu-Wan [10] and the theory of twisted T -adic exponential sums was developed by Liu [7]. We view L f,u (s, T ) and C f,u (s, T ) as power series in the single variable s with coefficients in the T -adic complete field Q q ((T )).…”

The stable property of Newton slopes for general Witt towers

“…Next, in an analogous manner with [4], we define the T -adic L-function. This function can be seen as a "universal" L-function.…”

“…It is accustomed to call this type of result "Dwork's trace formula", since the basic method was developed by Bernard Dwork. In a similar method, we extend Chunlei Liu and Daqing Wan's work [4] from the unit root case to more general polynomial case, where we meet a new difficulty to deal with the p-orders of the coefficients. In our case, the contribution of p-orders must be counted, so we have to replace T -adic valuation (which is used in [4]) by (p θ , T )-adic valuation for a selected θ > 0.…”

“…Then, we construct the T -adic characteristic function, which interpolates every characteristic function of nontrivial finite conductor. We prove the Dwork's trace formula of Tadic character functions; the calculation process is analogous to [4]. After that, we produce a lower bound of the Newton polygon of the T -adic characteristic function and an upper bound of the π χ -adic Newton polygon of the characteristic function for a finite character χ with a specific conductor.…”