Cyclic isogenies of elliptic curves over fixed quadratic fields

Abstract

Building on Mazur’s 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over Q \mathbb {Q} . Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised. In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields Q ( d ) \mathbb {Q}(\sqrt {d}) with | d | > 10 4 |d| > 10^4 we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over 19 19 quadratic fields, including Q ( 213 ) \mathbb {Q}(\sqrt {213}) and Q ( − 2289 ) \mathbb {Q}(\sqrt {-2289}) . To make this procedure work, we determine all of the finitely many quadratic points on the modular curves X 0 ( 125 ) X_0(125) and X 0 ( 169 ) X_0(169) , which may be of independent interest.

On the effective version of Serre's open image theorem

“…Generalizing Mazur's result to elliptic curves over arbitrary number fields appears to be challenging. Nonetheless, assuming GRH, if $$K$$ is among a certain finite set of quadratic fields $$K$$, Banwait [3] and Banwait, Najman, and Padurariu [4], building on the earlier work of David [16], Larson and Vaintrob [26], and Momose [35], proved an analog of Mazur's cyclic isogeny theorem for $$E/K$$. Thus, it is promising that one may be able to extend our work to give an explicit open image theorem for elliptic curves defined over these quadratic fields.…”