Explicit isogenies of prime degree over number fields

TL;DR

The paper delivers an explicit, algorithmic version of Momose's prime-degree isogeny classification over number fields, with implementations in Sage and PARI/GP. It strengthens and adapts key lemmas to remove the Galois requirement and, under GRH, yields a finite, computable bound MMIB$(k)$ that controls non-Type-3 primes, while unconditional results cover semistable cases. The work combines signature-to-Momose-type analysis, unit- and class-field theory-based pruning, and modular-curve techniques to assemble a comprehensive algorithm (the Combined Algorithm) that outputs a finite superset $S_k$ of potential isogeny primes and determines IsogPrimeDeg$(k)$ in several new settings, including cubic fields. It also provides concrete cubic-field examples demonstrating new isogeny primes under GRH and advances the explicit understanding of isogenies in low-degree number fields through both theory and computational tools.

Abstract

We provide an explicit and algorithmic version of a theorem of Momose classifying isogenies of prime degree of elliptic curves over number fields, which we implement in Sage and PARI/GP. Combining this algorithm with recent work of Box-Gajović-Goodman we determine the first instances of isogenies of prime degree for cubic number fields, as well as for several quadratic fields not previously known. While the correctness of the general algorithm relies on the Generalised Riemann Hypothesis, the algorithm is unconditional for the restricted class of semistable elliptic curves.

Theorems & Definitions (98)

• Theorem 1.1: Larson and Vaintrob, Corollary 2 in larson_vaintrob_2014
• Theorem 1.2: Mazur, Theorem 1 in mazur1978rational
• Theorem 1.3: Banwait
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Remark 2.1