For certain elliptic curves E=Q with E(Q)[2] = Z=2Z, we prove a criterion for prime twists of E 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 E of rank zero (resp. positive rank).
Prime twists of elliptic curves / D. Kriz, C. Li. - In: MATHEMATICAL RESEARCH LETTERS. - ISSN 1073-2780. - 26:4(2019), pp. 1187-1195. [10.4310/mrl.2019.v26.n4.a10]
Prime twists of elliptic curves
D. Kriz
Primo
;
2019
Abstract
For certain elliptic curves E=Q with E(Q)[2] = Z=2Z, we prove a criterion for prime twists of E 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 E of rank zero (resp. positive rank).
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