Isogenies of CM Elliptic Curves

TL;DR

This work addresses how often the reductions of two CM elliptic curves become m-isogenous at primes p, fixing the isogeny degree m. It develops an analytic framework based on higher Green functions, Gross–Zagier theory, and GKZ relations, and crucially uses a bridge between incoherent and coherent Eisenstein series through level-raising arguments to control Fourier coefficients. A uniform Petersson-norm bound converts partial Fourier data into global growth information, enabling an effective lower bound: for coprime fundamental discriminants D1, D2, the number of primes p with E1, p and E2, p m-isogenous is at least a constant times m^{1/5−ε}. The results connect CM-elliptic curve arithmetic with equidistribution of Hecke orbits and minimal isogeny-degree phenomena, with implications for S-unit questions and potential cryptographic considerations.

Abstract

Given two CM elliptic curves over a number field and a natural number $m$, we establish a polynomial lower bound (in terms of $m$) for the number of rational primes $p$ such that the reductions of these elliptic curves modulo a prime above $p$ are $m$-isogenous. The proof relies on higher Green functions and theorems of Gross-Zagier and Gross-Kohnen-Zagier. A crucial observation is that the Fourier coefficients of incoherent Eisenstein series can be approximated by those of coherent Eisenstein series of increasing level. Another key ingredient is an explicit upper bound for the Petersson norm of an arbitrary elliptic modular form in terms of finitely many of its Fourier coefficients at the cusp infinity, which is a result of independent interest.

Theorems & Definitions (33)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Corollary 1.6
• Proposition 2.1
• proof
• Theorem 2.2
• Lemma 2.3