The p-parity conjecture for elliptic curves with a p-isogeny
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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 III (E/K) [p∞] for a prime p this is equivalent to the p-parity conjecture: the global root number matches the parity of the Z[subscript p]-corank of the p∞-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 ≤ 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 III (E/K) [p∞] is infinite, it must contain (Q[subscript p]/Z[subscript p])².
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Journal für die reine und angewandte Mathematik (Crelles Journal)
Walter de Gruyter
Česnavičius, Kęstutis. “The p-Parity Conjecture for Elliptic Curves with a p-Isogeny.” Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal) 2016, 719 (January 2016): 45-73 © 2016 De Gruyter
Final published version