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Witt Group of Nondyadic Curves

Nanjun Yang

Abstract

Witt group of real algebraic curves has been studied since Knebusch in 1970s. But few results are known if the base field is non-Archimedean except the hyperelliptic case by works of Parimala, Arason et al.. In this paper, we compute the derived Witt groups of smooth proper curves over nondyadic local fields with $char\neq2$ by reduction, with a general study of the existence of Theta characteristics.

Witt Group of Nondyadic Curves

Abstract

Witt group of real algebraic curves has been studied since Knebusch in 1970s. But few results are known if the base field is non-Archimedean except the hyperelliptic case by works of Parimala, Arason et al.. In this paper, we compute the derived Witt groups of smooth proper curves over nondyadic local fields with by reduction, with a general study of the existence of Theta characteristics.

Paper Structure

This paper contains 8 sections, 31 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Statement of the Results
  3. The Motivic Stable Homotopy Category
  4. Motivic Cohomology and Bockstein Spectral Sequence
  5. The Special Fiber
  6. The Generic Fiber
  7. Witt Group of Elliptic Curves
  8. Theta Characteristics of Hyperelliptic Reductions

Key Result

Theorem 1

Suppose that $K$ is a nondyadic local field with $char(K)\neq 2$ and that $X_K$ has a rational point and is connected. The $4$-torsion group $W^i(X_K)$ ($=0$ for $i=2,3$) satisfies ($l$ is the length function)

Theorems & Definitions (77)

  • Theorem 1
  • Definition 2
  • Proposition 3
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Definition 7
  • ...and 67 more