GOLDFELD’S CONJECTURE AND CONGRUENCES BETWEEN HEEGNER POINTS
Forum of Mathematics, Sigma, Tome 7 (2019)
Voir la notice de l'article provenant de la source Cambridge University Press
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.
@article{10_1017_fms_2019_9,
author = {DANIEL KRIZ and CHAO LI},
title = {GOLDFELD{\textquoteright}S {CONJECTURE} {AND} {CONGRUENCES} {BETWEEN} {HEEGNER} {POINTS}},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {7},
year = {2019},
doi = {10.1017/fms.2019.9},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2019.9/}
}
DANIEL KRIZ; CHAO LI. GOLDFELD’S CONJECTURE AND CONGRUENCES BETWEEN HEEGNER POINTS. Forum of Mathematics, Sigma, Tome 7 (2019). doi: 10.1017/fms.2019.9
Cité par Sources :