The remaining cases of the Kramer–Tunnell conjecture

Kęstutis Česnavičius, Naoki Imai
2016 Compositio Mathematica  
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$0$.
more » ... aracteristic$0$. 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$E$.
doi:10.1112/s0010437x16007624 fatcat:c2vd2ehouved7fvimuegc456sm