6 Newton Stratification for Polynomials : the Open Stratum

In this paper we consider the Newton polygons of L-functions coming from additive exponential sums associated to a polynomial over a finite field Fq. These polygons define a stratification of the space of polynomials of fixed degree. We determine the open stratum: we give the generic Newton polygon for polynomials of degree d ≥ 2 when the characteristic p is greater than 3d, and the Hasse polynomial, i.e. the equation defining the hypersurface complementary to the open stratum. Let k := F q be the finite field with q := p m elements, and for any r ≥ 1, let k r denote its extension of degree r. If ψ is a non trivial additive character on F q , then ψ r := ψ • Tr kr/k is a non trivial additive character of k r , where Tr kr /k denotes the trace from k r to k. Let f ∈ k[X] be a polynomial of degree d ≥ 2 prime to p; then for any r we form the additive exponential sum S r (f, ψ) := x∈kr ψ r (f (x)). To this family of sums, one associates the L-function L(f, T) := exp   r≥1 S r (f, ψ) T r r  . It follows from the work of Weil on the Riemann hypothesis for function fields in characteristic p that this L-function is actually a polynomial of degree d − 1. Consequently we can write L(f, T) = (1 − θ 1 T). .. (1 − θ d−1 T). Another consequence of the work of Weil is that the reciprocal roots θ 1 ,. .. , θ d−1 are q-Weil numbers of weight 1, i.e. algebraic integers all of whose conjugates have complex absolute q 1 2. Moreover, for any prime ℓ = p, they are ℓ-adic units, that is |θ i | ℓ = 1. A natural question is to determine their q-adic absolute value, or equivalently their p-adic valuation. In other words, one would like to determine the Newton polygon N P q (f) of L(f, T) where N P q means the Newton polygon taken with respect to the valuation v q normalized by v q (q) = 1 (cf. [9], Chapter IV for the link between the Newton polygon of a polynomial and the valuations of its roots). There is an elegant general answer to this problem when …