The First Slope Case of Wan's Conjecture

Abstract

Let d ≥ 2 and p a prime coprime to d. For f(x) ∈ (Zp ∩Q)[x], let NP1(f mod p) denote the first slope of the Newton polygon of the L-function of the exponential sums ∑ x∈F p` ζ TrF p` /Fp (f(x)) p . We prove that there is a Zariski dense open subset U in the space Ad of degree-d monic polynomials over Q such that for all f(x) ∈ U we have limp→∞ NP1(f mod p) = 1 d . This is a “first slope case” of a conjecture of Wan. Let d ≥ 2 be an integer and p a prime coprime to d. Let A be the set of all degreed monic polynomials over Q. For any f(x) = x+ad−1x+ . . .+a0 ∈ (Zp∩Q)[x] and for any integer ` ≥ 1 let S`(f) := ∑ x∈F p` ζ TrF p` /Fp (f(x)) p . The L function of f(x) mod p is defined by L(f mod p;T ) = exp (∑∞ `=1 S`(f) T ` ` ) . It is a theorem of Dwork-Bombieri-Grothendieck that L(f mod p;T ) = 1 + b1T + . . . + bd−1T d−1 ∈ Z[ζp][T ] for some p-th root of unity ζp in Q. Define the Newton polygon of f mod p, denoted by NP(f mod p), as the lower convex hull of the points (`, ordpb`) in R for 0 ≤ ` ≤ d − 1 where we set b0 = 1. It is exactly the p-adic Newton polygon of the polynomial L(f mod p;T ). Let NP1(f mod p) denote its first slope. Define the Hodge polygon HP(f) as the convex hull in R of the points (`, `(`+1) 2d ) for 0 ≤ ` ≤ d − 1. It is proved that the Newton polygon is always lying above the Hodge polygon ( see [3] [6] and [2]). The following conjecture was proposed by Wan in the Berkeley number theory seminar in the fall of 2000, a general form of which will appear in [7, Section 2.5]. Conjecture 1 (Wan). There is a Zariski dense open subset U in A such that for all f(x) ∈ U we have limp→∞NP(f mod p) = HP(f). The cases d = 3 and 4 are proved in [6] and [4], respectively. It is also known that if p ≡ 1 mod d then NP(f mod p) = HP(f) for all f ∈ A (see [1]). In this paper we use an elementary method to prove the “first slope case” of this conjecture. For any real number r let dre denote the least integer greater than or equal to r. For any integer N and for any Laurent polynomial g(x) in one variable, we use [g(x)]xN to denote the x -coefficient of g(x). Theorem 2. Let d ≥ 2 and p a prime coprime to d. Let f(x) be a degree-d monic polynomial in (Zp ∩Q)[x]. Suppose [ f(x)d p−1 d e ] xp−1 6≡ 0 mod p. If p > d2 + 1 then NP1(f mod p) = ⌈ p−1 d ⌉ /(p− 1). Date: February 1, 2002. 1991 Mathematics Subject Classification. 11L, 14H, 14M.