Prime twists of elliptic curves
Daniel Kriz, Chao Li

TL;DR
This paper establishes a criterion based on 2-adic logarithms of Heegner points to determine the analytic rank of prime twists of certain elliptic curves, contributing to understanding their rank distribution.
Contribution
It introduces a new criterion for the analytic rank of prime twists of elliptic curves with specific torsion, linking it to 2-adic logarithms of Heegner points.
Findings
Criteria for prime twists to have rank 0 or 1
New cases supporting Silverman's conjecture
Application to rank distribution of elliptic curve twists
Abstract
For certain elliptic curves with , we prove a criterion for prime twists of to have analytic rank 0 or 1, based on a mod 4 congruence of 2-adic logarithms of Heegner points. As an application, we prove new cases of Silverman's conjecture that there exists a positive proposition of prime twists of of rank zero (resp. positive rank).
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Analytic Number Theory Research · Advanced Algebra and Geometry
