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