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- W2017444792 abstract "For an elliptic curve $E$ over a number field $K$, one consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number matches the parity of the Mordell-Weil rank. Assuming finiteness of $mathrm{Sha}(E/K)[p^infty]$ for a prime $p$ this is equivalent to the $p$-parity conjecture: the global root number matches the parity of the $mathbb{Z}_p$-corank of the $p^infty$-Selmer group. We complete the proof of the $p$-parity conjecture for elliptic curves that have a $p$-isogeny for $p > 3$ (the cases $p le 3$ were known). T. and V. Dokchitser have showed this in the case when $E$ has semistable reduction at all places above $p$ by establishing respective cases of a conjectural formula for the local root number. We remove the restrictions on reduction types by proving their formula in the remaining cases. We apply our result to show that the $p$-parity conjecture holds for every $E$ with complex multiplication defined over $K$. Consequently, if for such an elliptic curve $mathrm{Sha}(E/K)[p^infty]$ is infinite, it must contain $(mathbb{Q}_p/mathbb{Z}_p)^2$." @default.
- W2017444792 created "2016-06-24" @default.
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- W2017444792 date "2012-07-02" @default.
- W2017444792 modified "2023-10-16" @default.
- W2017444792 title "The $p$-parity conjecture for elliptic curves with a $p$-isogeny" @default.
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