The Euler numbers of ℓ-adic sheaves of rank 1 in positive characteristic

Abstract

One of the most important themes in ramification theory is the formula for the Euler characteristic of l-adic sheaves. Although we have the Grothendieck-Ogg-Shafarevich formula [G] in one dimensional case, we don’t have a general formula in higher dimension even in the form of a conjecture. However for sheaves of rank 1, K.Kato formulated a conjecture in arbitrary dimension and actually proved it in dimension 2 in [K2]. In this paper, we will prove it in arbitrary dimension under a certain hypothesis, which is hoped to hold when the variety is sufficiently blowed up.