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GOLDFELD’S CONJECTURE AND CONGRUENCES BETWEEN HEEGNER POINTS

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  27 May 2019

DANIEL KRIZ  and
CHAO LI
DANIEL KRIZ
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Memorial Drive, Cambridge, MA 02139, USA; [email protected]
CHAO LI
Affiliation:
Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA; [email protected]

Abstract

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Given an elliptic curve $E$ over $\mathbb{Q}$, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (respectively 1). We show that this conjecture holds whenever $E$ has a rational 3-isogeny. We also prove the analogous result for the sextic twists of $j$-invariant 0 curves. For a more general elliptic curve $E$, we show that the number of quadratic twists of $E$ up to twisting discriminant $X$ of analytic rank 0 (respectively 1) is $\gg X/\log ^{5/6}X$, improving the current best general bound toward Goldfeld’s conjecture due to Ono–Skinner (respectively Perelli–Pomykala). To prove these results, we establish a congruence formula between $p$-adic logarithms of Heegner points and apply it in the special cases $p=3$ and $p=2$ to construct the desired twists explicitly. As a by-product, we also prove the corresponding $p$-part of the Birch and Swinnerton–Dyer conjecture for these explicit twists.

MSC classification

Primary: 11G05: Elliptic curves over global fields
Secondary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e15
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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