p-adic variation of L-functions of exponential sums, I

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 $0\leq n\leq d-1$. 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 $a\neq 0$.

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