Symmetric bilinear forms and local epsilon factors of isolated singularities in positive characteristic

TL;DR

This paper relates the local epsilon factor of vanishing cycles to a symmetric bilinear form for isolated singularities in positive characteristic, refining Milnor's formula and generalizing the Arf invariant in characteristic 2.

Contribution

It provides a new formula linking epsilon factors to bilinear forms and extends invariants like the Arf invariant to broader classes of singularities in positive characteristic.

Findings

The local epsilon factor sign is determined by the bilinear form's discriminant.

A formula expressing the epsilon factor in terms of the bilinear form is established.

A generalization of the Arf invariant for non-degenerate quadratic singularities in characteristic 2 is introduced.

Abstract

Let $f\colon X\to\mathbb{A}^1_k$ be a morphism from a smooth variety to an affine line with an isolated singular point. For such a singularity, we have two invariants. One is a non-degenerate symmetric bilinear form (de Rham), and the other is the vanishing cycles complex (\'etale). In this article, we give a formula which expresses the local epsilon factor of the vanishing cycles complex in terms of the bilinear form. In particular, the sign of the local epsilon factor is determined by the discriminant of the bilinear form. This formula can be thought as a refinement of the Milnor formula, which compares the total dimension of the vanishing cycles and the rank of the bilinear form. In characteristic $2$, we find a generalization of the Arf invariant, which can be regarded as an invariant for non-degenerate quadratic singularities, to general isolated singularities.