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(OX) - H1(OX).


Additional Information

Print ISSN
pp. 503-522
Launched on MUSE
Open Access
Back To Top

This website uses cookies to ensure you get the best experience on our website. Without cookies your experience may not be seamless.