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

For a polynomial f(x) in (ℤ_p ∩ ℚ)[x] of degree d ≥ 3 let L(f ⊗ double-struck F sign_p; T) be the L function of the exponential sum of f mod p. Let NP(f ⊗ double-struck F sign_p) denote the Newton polygon of L(f ⊗ double-struck F sign_p; T). Let HP (double-struck A sign^d) denote the Hodge polygon of double-struck A sign^d, which is the lower convex hull in ℝ² of the points (n, n(n+1)/2d) for 0 ≤ n ≤ d - 1. Let double-struck A sign^d be the space of degree-d monic polynomials parameterized by their coefficients. Let GNP(double-struck A sign^d;double-struck F sign_p) := inf-_{f∈double-struck A sign d (double-struck F sign p} NP(f̄) be the lowest Newton polygon over double-struck F sign_p if exists. We prove that for p large enough GNP (double-struck A sign^d; double-struck F sign_p) exists and we give an explicit formula for it. We also prove that there is a Zariski dense open subset μ defined over ℚ in double-struck A sign^d such that for f ∈ μ(ℚ) and for p large enough we have NP (f ⊗ double-struck F sign_p) = GNP (double-struck A sign^d; double-struck F sign_p); furthermore, as p goes to infinity their limit exists and is equal to HP (double-struck A sign^d). Finally we prove analogous results for the space of polynomials f(x) = x^d + ax with one parameter. In particular, for any nonzero a ∈ ℚ we show that lim_p→∞ NP(x^d + ax) ⊗ double-struck F signF_p) = HP (double-struck A sign^d).