Epsilon Factors for Meromorphic Connections and Gauss Sums

TL;DR

This paper establishes a connection between epsilon factors for rank one meromorphic connections and Gauss sums, revealing a deeper link with Galois epsilon factors through explicit calculations.

Contribution

It provides an explicit formula for epsilon factors of rank one connections using Gauss sums, bridging De Rham and Galois epsilon factors.

Findings

Epsilon factors for rank one connections can be computed explicitly via Gauss sums.

The formula indicates a deeper relationship between De Rham and Galois epsilon factors.

The work extends the understanding of irregular Riemann-Hilbert correspondence and local epsilon factors.

Abstract

Let $E$ is be vector bundle with meromorphic connection on $\proj^1/k$ for some field $k \subset \cplx$, and let $\mathbf{E}$ be the sheaf of horizontal sections on the analytic points of $X$. The irregular Riemann-Hilbert correspondence states that there is a canonical isomorphism between the De Rham cohomology of $L$ and the `moderate growth' cohomology of $\mathbf{L}$. Recent work of Beilinson, Bloch, and Esnault has shown that the determinant of this map factors into a product of local `$\epsilon$-factors' which closely resemble the classical $\epsilon$-factors of Galois representations. In this paper, we show that $\epsilon$-factors for rank one connections may be calculated explicitly by a Gauss sum. This formula suggests a deeper relationship between the De Rham $\epsilon$-factor and its Galois counterpart.