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arXiv · 2511.11948

Counting elliptic curves with prescribed entanglements

Zachary Couvillon·Anwesh Ray
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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.

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Zachary Couvillon, Anwesh Ray. 2025-11-14. Counting elliptic curves with prescribed entanglements. https://doi.org/10.1007/s40993-025-00686-y

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