ε-factors in the de Rham setting A.Beilinson

The classical theory of ε-factors (Tate, Langlands, Deligne, Laumon..) assigns to every represntation ρ_{v} of the Galois group of a local field F_{v} a function ψ_{v} ↦ ε_{v}(ρ_{v}, ψ_{v}) on the set of non-trivial additive characters of F_{v}. The ε-factors satisfy the key product formula: Let ρ be a Galois representation of a global field F and ε(ρ) the constant in the functional equation for the L-function of ρ; then for every character ψ = (ψ_{v}) of A_{F} /F the product of ε_{v}(ρ_{v}, ψ_{v}) (v runs the set of places of F) equals ε(ρ).

One may hope that in case F = field of rational functions on a curve over a field k there exists a geometric version of the theory of ε-factors which would assigne to every ρ_{v} as above a 1-dimensional local system ε(ρ_{v}) on the moduli of ψ_{v}’s so that in the global setting the tensor product of ε(ρ_{v}) equals the determinant of the global cohomology with coefficients in ρ. The classical theory arises when k is a finite field by considering the Frobenius action on the ε-lines.

In the talk (which is an exposition of a joint work with S.Bloch and H.Esnault) the desired geometric theory is constructed in the de Rham setting. The construction is closely related to Tate’s linear algebra construction of residues: the ε-lines arise as “matrix coefficients” of certain operators in appropriate Clifford algebra.