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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie


IMPACT FACTOR 2018: 1.859

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1435-5345
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Volume 2016, Issue 719

Issues

The p-parity conjecture for elliptic curves with a p-isogeny

Kęstutis Česnavičius
Published Online: 2014-06-11 | DOI: https://doi.org/10.1515/crelle-2014-0040

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] for a prime p this is equivalent to the p-parity conjecture: the global root number matches the parity of the 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 p3 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] is infinite, it must contain (p/p)2.

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About the article

Received: 2012-09-03

Revised: 2014-03-10

Published Online: 2014-06-11

Published in Print: 2016-10-01


Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2016, Issue 719, Pages 45–73, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2014-0040.

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