On the 2-part of the Birch-Swinnerton-Dyer conjecture for elliptic curves with complex multiplication

2013/03/21 by John Coates, Minhyong Kim, Coates, John +5 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.1303.5218

Abstract

Given an elliptic curve E over Q with complex multiplication having good reduction at 2, we investigate the 2-adic valuation of the algebraic part of the L-value at 1 for a family of quadratic twists. In particular, we prove a lower bound for this valuation in terms of the Tamagawa number in a form predicted by the conjecture of Birch and Swinnerton-Dyer.

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• A rank zero p-converse to a theorem of Gross--Zagier, Kolyvagin and Rubin

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