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Local Zeta Functions over p-Adic Fields
Cameron, Stephen P
Cameron, Stephen P
Abstract
A $p$-adic field $K$ is a finite extension of the $p$-adic numbers $\Q_p$. The ring of integers $O_K$ is the integral closure of the $p$-adic integers $\Z_p$, with unique maximal ideal $\pi_K O_K$. We can define a non-Archimedean absolute value $|\cdot|_K$ on $K$ such that $O_K = \{ x \in K : |x|_K \leq 1\}$ and $\pi_K O_K = \{ x \in K : |x|_K < 1\}$. The field $K$ is locally compact, and has a unique Haar measure $\mu_K$ normalized such that $\mu_K(O_K) = 1$. For any polynomial $f \in O_K[x_1,x_2,\ldots, x_n]$, we define the local zeta function of $f$ as $$Z(f,s) = \displaystyle\int\limits_{O_K^n} | f( {\bf x} )|_K^s d\mu_K^n({\bf x}).$$ Igusa's theorem states that this is a rational function of $q^{-s}$, where $q =$card($O_K/\pi_KO_K$). Given a polynomial in $f\in O_K[x_1,x_2,\ldots, x_n]$, the Poincar\`e series of $f$ is the infinite series $$Q(f,t) = \displaystyle\sum\limits_{m=0}^\infty N_m t^m,$$ where $N_m = $card($\{{\bf x}\in(O_K/\pi_K^mO_K)^n: f({\bf x}) \in \pi_K^m O_K\}$) is the number of zeroes of $f$ mod $\pi_K^m$. These two functions are related by $$Q(f,q^{-n}t) = \displaystyle\frac{tZ(f,s)-1}{t-1},$$ where we take $t =q^{-s}$. Thus calculating the zeta function of $f$ allows us to count the number of zeroes of $f$ mod $\pi_K^m$. This thesis is broken up into two main sections. In section 1, we construct the $p$-adic numbers $\Q_p$ and prove necessary properties of $p$-adic fields. In section 2, we consider the Haar measure on $K^n$. We then define local zeta functions and state necessary theorems for the calculation of them. We then define Poincar\'e series, and do some basic calculations with them. For a special case, we derive a recursive relation for the coefficients of the Poincar\'e series using the relation between the local zeta function and the Poincar\'e series. In the appendix, we show how local zeta functions may be considered on $K$-analytic manifolds, ending with a proof of Serre's theorem.
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Date
2014-05-01
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Mathematics