Workshops## $L$-functions of exponential sums over finite fields

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Shaofang Hong

2012-01-16

16:00:00 - 16:50:00

101 , Mathematics Research Center Building (ori. New Math. Bldg.)

Let ${\bf F}_q$ be the finite field of $q$ elements with characteristic $p$ and ${\bf F}_{q^m}$ its extension of degree $m$. Fix a nontrivial additive character $\psi$ of ${\bf F}_p.$ For any Laurent polynomial $f(x_1, ..., x_n)\in {\bf F}_q[x_1^{\pm 1}, ..., x_n^{\pm 1}]$, we form the exponential sum $$ S_m(f):=\sum_{(x_1,...,x_n)\in ({\bf F}_{q^m})^n}\psi (\hbox {Tr}_{{\bf F}_{q^m}/ {\bf F}_p}(f(x_1,...,x_n))). $$ The corresponding $L$-function is defined by $$ L(f,t):=\hbox {exp}(\sum^{\infty}_{m=0}S_m(f){\frac {t^m} {m}}). $$ The well-known Dwork-Grothendieck theorem says that the $L$- function $L(f,t)$ is a rational function. Write $$L(f,t)=\frac{\prod^{d_1}_{i=1}(1-\alpha _it)}{\prod^{d_2}_{j=1}(1-\beta _jt)},$$ where $\alpha _i (1\le i\le d_1)$ and $\beta _j (1\le j\le d_2)$ are algebraic integers. Equivalently, we have $$S_m(f)=\sum^{d_2}_{j=1}\beta _j^m-\sum^{d_1}_{i=1}\alpha _i^m.$$ Thus finding sharp $p$-adic estimates for the sum $S_m(f)$ are reduced to determining the $p$-adic absolute values of the reciprocal roots $\alpha _i(1\le i\le d_1)$ and the reciprocal poles $\beta _j(1\le j\le d_2).$ This question can best be described in terms of Newton polygons. In this talk, we mainly speak about the $p$-adic Newton polygons of $L$- function $L(f,t)$. We will review some old results due to Dwork, Katz, Sperber, Adolphson and Wan. We will also report some new results obtained by the speaker et al.For material related to this talk, click here.