Epsilon Factors for Meromorphic Connections and Gauss Sums

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.