Abstract

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 Ш ⁢ ( E / K ) ⁢ [ p ∞ ] ${\Sha(E/K)[p^{\infty}]}$ for a prime p this is equivalent to the p-parity conjecture: the global root number matches the parity of the ℤ p ${\mathbb{Z}_{p}}$ -corank of the p ∞ ${p^{\infty}}$ -Selmer group. We complete the proof of the p-parity conjecture for elliptic curves that have a p-isogeny for p > 3 ${p>3}$ (the cases p ≤ 3 ${p\leq 3}$ were known). Tim and Vladimir 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 Ш ⁢ ( E / K ) ⁢ [ p ∞ ] ${\Sha(E/K)[p^{\infty}]}$ is infinite, it must contain ( ℚ p / ℤ p ) 2 ${(\mathbb{Q}_{p}/\mathbb{Z}_{p})^{2}}$ .

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