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The remaining cases of the Kramer–Tunnell conjecture

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  29 July 2016

Kęstutis Česnavičius  and
Naoki Imai
Kęstutis Česnavičius
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA email kestutis@berkeley.edu
Naoki Imai
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan email naoki@ms.u-tokyo.ac.jp
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Abstract

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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 $E$.

Keywords

elliptic curvelocal fieldroot numberWeierstrass equation

MSC classification

Primary: 11G07: Elliptic curves over local fields
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 11 , November 2016 , pp. 2255 - 2268
Copyright
© The Authors 2016 

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