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The remaining cases of the Kramer–Tunnell conjecture
- Part of
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- Journal:
- Compositio Mathematica / Volume 152 / Issue 11 / November 2016
- Published online by Cambridge University Press:
- 29 July 2016, pp. 2255-2268
- Print publication:
- November 2016
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- Article
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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 where 
$K$ 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 
$E^{\prime }$ defined over a 
$p$ -adic field such that all the terms in the Kramer–Tunnell formula for 
$E^{\prime }$ are equal to those for 
$E$ .
