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Local root numbers of elliptic curves over dyadic fields

Naoki Imai

Abstract

We consider an elliptic curve over a dyadic field with additive, potentially good reduction. We study the finite Galois extension of the dyadic field generated by the three-torsion points of the elliptic curve. As an application, we give a formula to calculate the local root number of the elliptic curve over the dyadic field.

Local root numbers of elliptic curves over dyadic fields

Abstract

We consider an elliptic curve over a dyadic field with additive, potentially good reduction. We study the finite Galois extension of the dyadic field generated by the three-torsion points of the elliptic curve. As an application, we give a formula to calculate the local root number of the elliptic curve over the dyadic field.

Paper Structure

This paper contains 4 sections, 16 theorems, 27 equations.

Table of Contents

  1. Elliptic curve
  2. Group theory
  3. Galois group
  4. Root number

Key Result

Proposition 1.1

If $v(j) < 0$, then $E$ has split multiplicative reduction over a quadratic extension of $K$.

Theorems & Definitions (33)

  • Proposition 1.1: cf. SilAEC
  • Proposition 1.2
  • proof
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 23 more