Deligne--Langlands gamma factors in families

TL;DR

This paper demonstrates that gamma factors, combining epsilon and L-factors, can be interpolated across families of l-adic representations of Weil groups, addressing previous limitations in family interpolations.

Contribution

It introduces a method to interpolate gamma factors over families of l-adic representations, extending the understanding of local factors in p-adic and Galois deformation contexts.

Findings

Gamma factors interpolate over families of l-adic representations.

Epsilon and L-factors do not interpolate well in families.

The product defining gamma factors allows for interpolation.

Abstract

Let F be a p-adic field, W_F its absolute Weil group, and let k be an algebraically closed field of prime characteristic l different from p. Attached to any l-adic representation of W_F are local epsilon- and L-factors. There are natural notions of families of l-adic representations of W_F, such as the theory of Galois deformations or, more generally, families over arbitrary Noetherian W(k)-algebras. However, the epsilon and L-factors do not interpolate well in such families. In this paper it is shown that the gamma factor, which is the product of the epsilon factor with a ratio of L-factors, interpolates over such families.