$l$ -independence of the trace of monodromy

Abstract

Let X be a proper smooth variety over a local field K of mixed characteristics and let l be a prime number different from the characteristic of the residue field of K. Let IK be the inertia subgroup of GK := Gal(K/K). Our main result is the l-independence of the alternating sum of traces of g ∈ IK on H(X,Ql) and its comparison with the traces on p -adic cohomology. 0.Introduction Let K be a complete discrete valuation field with finite residue field Fph , GK the absolute Galois group of K, IK the inertia subgroup of GK and WK the Weil group of K. Recall that the Weil group is a subgroup of GK defined by the following exact sequence: 0 −→ IK −→ WK u −→ Z −→ Z/hZ −→ 0 ∥ ∩ ∩ ∥ 0 −→ IK −→ GK −→ Gal(Fp/Fp) −→ Gal(Fph/Fp) −→ 0, where the map u is defined by g 7−→ f for the geometric Frobenius f : x 7−→ x in Gal(Fp/Fp) ∼= Ẑ and F denotes the separable closure of F for any field F . We define a subset W K of WK to be W K := {g ∈ WK |u(g) ≥ 0}. Let X be a variety over K (Throughout the paper, a variety X over a field K means a reduced irreducible scheme X separated and of finite type over K). Throughout this paper, X means the scalar extension X ⊗ K K. We denote by l a prime number = p. We consider the traces of the action of elements of W K or WK on the compact support etale cohomology H c(X,Ql) := lim ←− n H c(X,Z/lZ) ⊗ Zl Ql. Let us recall the following classical conjecture. Conjecture([S-T]). For any variety X over K and g ∈ W K , Tr(g∗;Hi c(X,Ql)) is a rational integer which is independent of the choice of l. Remark. If X is a d-dimensional proper smooth variety, the conjecture above holds for i = 0, 1, 2d−1, 2d [SGA7-1]. If X has good reduction, the above conjecture is true for any i due to the Weil conjecture proved by P. Deligne [De]. In this paper, we shall prove the following weak versions of the conjecture: Typeset by AMS-TEX 1