p-ADIC VARIATION OF L FUNCTIONS OF ONE VARIABLE EXPONENTIAL SUMS, II

Let d ≥ 3 be an integer and p a prime coprime to d. Let Q and Qp be the algebraic closure of Q and Qp respectively. Let Zp be the ring of integers of Qp. Suppose f(x) is a degree-d polynomial in (Q ∩ Zp)[x]. Let Q(f) be the number field generated by coefficients of f in Q. Let P be a prime ideal in the ring of integers OQ(f) of Q(f) lying over p, with residue field Fq for some p-power q. Let A be the dimension-d affine space over Q, identified with the space of coefficients of degree-d monic polynomials. Let NP(f(x) mod P) denote the p-adic Newton polygon of L(f(x) mod P;T ). Let HP(A) denote the p-adic Hodge polygon of A, that is, the lower convex hull in R2 of the points (n, n(n+1) 2d ) for 0 ≤ n ≤ d − 1. We prove that there is a Zariski dense open subset U defined over Q in A such that for every geometric point f(x) in U(Q) we have lim p→∞ NP(f(x) mod P) = HP(A), where P is any prime ideal of OQ(f) lying over p.