The constant in the functional equation and derived extrior powers

TL;DR

This paper proposes a conjecture relating the constant in the functional equation of a scheme's zeta-function to Euler characteristics of derived exterior powers of differentials, and proves it for low-dimensional cases.

Contribution

It introduces a conjecture linking the zeta-function's constant to derived exterior powers and confirms it for schemes of dimension 1 and 2.

Findings

Conjecture proven for dimension 1 schemes.

Conjecture proven for dimension 2 schemes.

Supports compatibility of special value formulas with the functional equation.

Abstract

Let X be a regular scheme, projective and flat over the integers. Let A be the constant in the conjectured functional equation for the zeta-function of X. We give a conjecture computing A in terms of Euler characteristics of derived exterior powers of the sheaf of Kahler differentials on X, and prove this conjecture when the dimension of X is 1 or 2. This conjecture essentially says that the formulas for the special values of the zeta-function of X given by the author in a previous preprint are compatible with the functional equation.