Goldfeld's conjecture and congruences between heegner points
Name
goldfelds_conjecture_and_congruences_between_heegner_points.pdf
Description
Published version
Size
757.59 KB
Format
Adobe PDF
Checksum (MD5)
c4266ed74fb3394ad31de00ea4d6719c
Author(s)
Kriz, D
Li, C
Date Issued
January 1, 2019
Journal
Forum of Mathematics, Sigma
Publisher
Cambridge University Press (CUP)
Version
Final published version
Abstract
© The Author(s) 2019. Given an elliptic curve E over 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 ≫ X= log5/6 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution 4.0 International license
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1017/fms.2019.9
