p-ADIC VARIATION OF L FUNCTIONS OF ONE VARIABLE EXPONENTIAL SUMS, I
نویسنده
چکیده
For a polynomial f(x) in (Zp∩Q)[x] of degree d ≥ 3 let L(f⊗Fp;T ) be the L function of the exponential sum of f mod p. Let NP(f⊗Fp) denote the Newton polygon of L(f⊗Fp;T ). Let HP(Ad) denote the Hodge polygon of Ad, which is the lower convex hull in R2 of the points (n, n(n+1) 2d ) for 0 ≤ n ≤ d−1. Let Ad be the space of degree-d monic polynomials parameterized by their coefficients. Let GNP(A;Fp) := inff∈Ad(Fp) NP(f) be the lowest Newton polygon over Fp if exists. We prove that for p large enough GNP(A;Fp) exists and we give an explicit formula for it. We prove that there is a Zariski dense open subset U defined over Q in Ad such that for f ∈ U(Q) and for p large enough we have NP(f ⊗ Fp) = GNP(A;Fp); furthermore, as p goes to infinity their limit exists and is equal to HP(Ad). Finally we prove analogous results for the space of polynomials f(x) = xd + ax with one parameter. In particular, for any nonzero a ∈ Q we show that limp→∞ NP((xd + ax)⊗ Fp) = HP(Ad).
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