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

Daichi Takeuchi

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 (étale). 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.

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

Abstract

Let 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 (étale). 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 , we find a generalization of the Arf invariant, which can be regarded as an invariant for non-degenerate quadratic singularities, to general isolated singularities.

Paper Structure

This paper contains 18 sections, 57 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. On finite étale algebras over Witt rings
  3. Finite modules on adic rings
  4. On Witt rings
  5. Isolated singularities and symmetric bilinear forms
  6. Reminder on residue symbols
  7. Bilinear forms of isolated singularities
  8. Ordinary quadratic singularities
  9. Deforming isolated singularities
  10. $\mathbb{Z}/2$-coverings associated with isolated singularities
  11. Discriminants of symmetric bilinear forms
  12. The case where $2$ is invertible
  13. The case of characteristic 2
  14. The case of a field of characteristic $2$
  15. Milnor formula for local epsilon factors
  16. ...and 3 more sections

Key Result

Theorem 1.1

(Theorem mainMil.1, $p$ is odd) Let $k$ be a finite field of odd characteristic $p$. Let $X$ be a smooth $k$-scheme of dimension $n$ and let $f\colon X\to \mathbb{A}^1_k$ be a $k$-morphism with an isolated singular point $x\in X$. The singular point $x$ is assumed $k$-rational for simplicity in the where $(\frac{}{k})$ denotes the Legendre symbol and $\tau_\psi$ is the quadratic Gauss sum $\tau_{

Theorems & Definitions (72)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Remark 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Lemma 2.7
  • Lemma 2.8
  • ...and 62 more