Counting elliptic curves with prescribed entanglements

TL;DR

Addresses counting elliptic curves over $\mathbb{Q}$ with prescribed entanglements of division fields when ordered by naive height. The authors construct explicit 1-parameter families from genus $0$ modular curves and combine Davenport's lemma (geometry of numbers) with sieve methods to obtain asymptotic lower bounds for unexplained $(2,3)$- and $(2,5)$-entanglements, namely $N(X) \gg X^{1/9}$ and $N(X) \gg X^{1/12}$. They develop a general counting framework for globally minimal members of a parametric family $\mathcal{E}_{a,b}$ and apply it to two concrete families $\mathcal{F}_1$ and $\mathcal{F}_2$, deriving the relevant local densities and exponents $d=18$ and $d=24$. The results illuminate the distribution of entanglements in arithmetic statistics and motivate extending the sieve-geometry method to cover more subfamilies and entanglement types.

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.

Theorems & Definitions (36)

• Theorem A: Theorem \ref{['(2,3) theorem section 4']} and \ref{['(2, 5) theorem section 4']}
• Theorem B: Theorem \ref{['main thm section 3']}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• Definition 3.1
• Lemma 3.2
• proof
• Lemma 3.3: Principle of Lipschitz/ Davenport's lemma