On the ∈-constants of a variety over a finite field

Let X denote an irreducible smooth projective variety over a finite field, which supports an action by a finite group G. For each complex representation V of G, we let ∈(X, G, V) denote the constant in the functional equation of the associated L-function. In this paper we study the constants ∈(X, G, V) when the character of the representation V is real valued.

Our first main result shows that ∈(X, G, V) is positive: either when the dimension, d say, of X is odd and V is orthogonal; or when d is even and V is symplectic. This then generalizes Serre's theorem for the ∈-constants of orthogonal representations for curves.

Our second main result concerns the situation where G acts tamely on X and where the representation V has real valued character, is of even degree and has trivial determinant. We than show that the constants ∈(X, G, V) are determined by certain hermitian Euler characteristics associated to a stratification of X. This result is particularly striking when X is a curve: for then our result shows that the symplectic ∈-constants are all determined by the virtual [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]-module H⁰(O_X) - H¹(O_X).