padic variation of Lfunctions of exponential sums, I
Abstract
For a polynomial $f(x)$ in $(\mathbb{Z}_p\cap \mathbb{Q})[x]$ of degree $d>2$ let $L(f \bmod p;T)$ be the $L$function of the exponential sum of $f \bmod p$. Let $\mathrm{NP}(f \bmod p)$ denote the Newton polygon of $L(f \bmod p;T)$. Let $\mathrm{HP}(f)$ denote the Hodge polygon of $f$, which is the lower convex hull in the real plane of the points $(n,n(n+1)/(2d))$ for $0\leq n\leq d1$. We prove that there is a Zariski dense subset $\mathcal{U}$ defined over $\mathbb{Q}$ in the space $\mathbb{A}^d$ of degree$d$ monic polynomials over $\mathbb{Q}$ such that for all $f$ in $\mathcal{U}(\mathbb{Q})$ we have $\lim_{p\rightarrow\infty} \mathrm{NP}(f \bmod p) = \mathrm{HP}(f)$. Moreover, we determine the $p$adic valuation of every coefficient of $L(f \bmod p;T)$ for $p$ large enough and $f$ in $\mathcal{U}(\mathbb{Q})$, and that of $L(x^d+a x \bmod p;T)$ for all $a\neq 0$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 2001
 arXiv:
 arXiv:math/0111194
 Bibcode:
 2001math.....11194Z
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Number Theory
 EPrint:
 This is the closest (and final) LaTeX version to the published version in American Journal of Mathematics (2003). Among many differences from v1, the published Section 6 under the title "Generic Newton Polygon for x^d+ax" gives its padic Newton polygon explicitly