{'fraction_non_alphanumeric': 0.033739456419868794, 'fraction_numerical': 0.0018744142455482662, 'mean_word_length': 4.366834170854271, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The p-parity conjecture for elliptic curves with a p-isogeny

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 X.E=K/Œp for a prime p this is equivalent to the p-parity conjecture: the global root number matches the parity of the Zp-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 whenE 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 everyE with complex multiplication defined over K. Consequently, if for such an elliptic curve X.E=K/Œp is infinite, it must contain .Qp=Zp/.