The remaining cases of the Kramer-Tunnell conjecture

TL;DR

This paper completes the proof of the Kramer-Tunnell conjecture for local fields of characteristic 2 by reducing the problem to the characteristic 0 case, using approximation techniques from p-adic fields.

Contribution

It extends the Kramer-Tunnell conjecture proof to all cases, including characteristic 2, by reducing to characteristic 0 through approximation methods.

Findings

The conjecture holds in all remaining cases, including characteristic 2.

A reduction technique from positive characteristic to characteristic 0 is established.

The approach uses approximation of local fields of characteristic p by p-adic fields.

Abstract

For an elliptic curve $E$ over a local field $K$ and a separable quadratic extension of $K$, motivated by connections to the Birch and Swinnerton-Dyer conjecture, Kramer and Tunnell have conjectured a formula for computing the local root number of the base change of $E$ to the quadratic extension in terms of a certain norm index. The formula is known in all cases except some when $K$ is of characteristic $2$, and we complete its proof by reducing the positive characteristic case to characteristic $0$. For this reduction, we exploit the principle that local fields of characteristic $p$ can be approximated by finite extensions of $\mathbb{Q}_p$--we find an elliptic curve $E'$ defined over a $p$-adic field such that all the terms in the Kramer-Tunnell formula for $E'$ are equal to those for $E$.