Cyclic isogenies of elliptic curves over a fixed quadratic field

Abstract

Building on the work of Mazur on prime degree isogenies, Kenku determined in 1981 all the possible cyclic isogeny degrees of elliptic curves over Q. Although more than 40 years have passed, the possible isogeny degrees have not been determined over a single number field since.In this paper we determine such a classification of all possible cyclic isogeny degrees for the quadratic field Q( √ 213), assuming the Generalised Riemann Hypothesis. Along the way we determine all of the finitely many quadratic points on the modular curves X 0 (125) and X 0 (169).

Computing points on bielliptic modular curves over fixed quadratic fields

“…For certain (but not all) integers d, the results of Theorem 1 could be achieved by applying [13,Theorem 1.1] or some of the techniques described in [2]. In Section 3, we compare (for N = 53) our results with those one can obtain by applying [13,Theorem 1.1], and use this to provide an example of a curve that violates the Hasse principle.…”

Computing Points on Bielliptic Modular Curves Over Fixed Quadratic Fields

“…There are precisely 10 values of N such that the modular curve X 0 (N) is bielliptic with an elliptic quotient of positive rank [3, pages 26-28]. For two of these values of 2 P. Michaud-Jacobs [2] N, namely 37 and 43, the methods we present will not work (see Remark 2.2), and so we will consider the remaining eight values of N, which are N ∈ N := {53, 61, 65, 79, 83, 89, 101, 131}.…”

“…For certain (but not all) integers d, the results of Theorem 1.1 could be achieved by applying [13,Theorem 1.1] or some of the techniques described in [2]. In Section 3, we compare (for N = 53) our results with those one can obtain by applying [13, Theorem 1.1], and use this to provide an example of a curve that violates the Hasse principle.…”

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.…”