Rational points on $X_0(N)^*$ when $N$ is non-squarefree

Abstract: The rational points of the modular curve $X_0(N)$ classify pairs $(E,C_N)$ of elliptic curves over $\mathbb{Q}$ together with a rational cyclic subgroup of order $N$. The curve $X_0(N)^*$ is the quotient of $X_0(N)$ by the full group of Atkin-Lehner involutions. Elkies showed that the rational points on this curve classify elliptic curves over the algebraic closure of $\mathbb{Q}$ that are isogenous to their Galois conjugates, and conjectured that when $N$ is large enough, the points are all CM or cuspidal. In joint work with Timo Keller and Samuel Le Fourn, we study the rational points on the family $X_0(N)^*$ for N non-squarefree. In particular we will report on some integrality results for the $j$-invariants of points on $X_0(N)^*$.

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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