Counting elliptic curves with prescribed entanglements

TL;DR

This paper establishes lower bounds on the number of elliptic curves over rationals with specific entanglement properties of their division fields, using geometric and sieve techniques, showing growth rates in terms of naive height.

Contribution

It provides the first asymptotic lower bounds for counting elliptic curves with prescribed entanglement of division fields over $\

Findings

Number of elliptic curves with certain entanglements grows at least as fast as a power of the height.

Uses genus 0 modular curves to parametrize families of elliptic curves with prescribed properties.

Applies geometry of numbers and sieve methods to derive growth bounds.

Abstract

We establish asymptotic lower bounds for the number of elliptic curves over $\mathbb{Q}$ with prescribed entanglement of division fields, ordered by naive height. Such elliptic curves are obtained as $1$-parameter families arising from certain genus $0$ modular curves. We apply techniques from the geometry of numbers and sieve methods to prove that the number of elliptic curves with unexplained entanglements $\mathbb{Q}(E[2]) \cap \mathbb{Q}(E[3]) \neq \mathbb{Q}$ and $\mathbb{Q}(E[2]) \cap \mathbb{Q}(E[5]) \neq \mathbb{Q}$ and naive height $\leq X$, grows as $\gg X^{1/9}$ and $\gg X^{1/12}$, respectively.