STNB2023(36a edició)

Distribution of Manin-Stevens elliptic curves in twisted isogeny classes over $\mathbf{Q}$


Joan-C. Lario


Let $\mathcal{A}$ be the $\mathbf{Q}$-isogeny class of an elliptic curve $A$ defined over $\mathbf{Q}$. Does~$\mathcal{A}$ contain any distinguished elliptic curve? First, Mazur and Swinnerton-Dyer proposed the so-called $\mathit{strong}$ curve $A_0\in \mathcal{A}$ which is a $\Gamma_0$-optimal quotient of the Jacobian of the modular curve $X_0(M)$, where $M$ is the conductor of $A$. Later, Stevens suggested that it is better to consider the elliptic curve $A_1\in \mathcal{A}$ such that it is a $\Gamma_1$-optimal quotient of the Jacobian of the modular curve $X_1(M)$. In both cases the Manin constant plays a role, and the Stevens proposal seems to be more intrinsically arithmetic due to the intervention of Néron models, étale isogenies, and Parshin-Faltings heights of the elliptic curves involved in the isogeny class. The Manin-Stevens curve $A_1$ is the one with minimal height.

Let $G(\mathcal A)$ be the natural graph attached to the isogeny class $\mathcal A$: a vertex for every elliptic curve $A\in\mathcal A$, and edges correspond to isogenies of prime degree among them. For every square-free integer $d$, we can consider the graph $G({\mathcal A}^d)$ attached to the twisted elliptic curve $A^d$ by the quadratic character of $\mathbf{Q}(\sqrt{d})$. It turns out that $G({\mathcal A})$ and $G({\mathcal A}^d)$ are canonically isomorphic as abstract graphs (the isomorphism identifies the vertices with equal $j$-invariant.)

In this talk we shall discuss the probability distribution of a vertex in $G({\mathcal A}^d)$ to be a Manin-Stevens elliptic curve as $|d|$ grows to infinity.

(This is work in progress in collaboration with Enrique González-Jiménez, UAM).


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