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July 5, 2025: Rational points on $X_0(N)^*$ when $N$ is non-squarefree

With Sachi Hashimoto and Samuel Le Fourn, we proved integrality properties of rational points on the Atkin–Lehner quotient $X_0(N)^*$ when $N$ is non-squarefree. This is the quotient of the modular curve $X_0(N)$ by the group of all Atkin–Lehner involutions.

Why this matters. The modular curve $X_0(N)$ is a geometric object that classifies elliptic curves together with additional data, namely a cyclic isogeny of degree $N$; so understanding its points means understanding all elliptic curves of this kind. The $\mathbf{Q}$-rational points of every $X_0(N)$ were determined in celebrated work of Mazur and Kenku. Over larger number fields the picture is far less complete: there is no result that, for a fixed degree $d > 1$, describes the points of $X_0(N)$ over all number fields of degree $d$ and all $N$ at once. Our work goes significantly beyond the state of the art by determining certain points over multiquadratic fields.

Elkies proved that the non-cuspidal rational points correspond to $\mathbf{Q}$-curves over multiquadratic field extensions of $\mathbf{Q}$. He conjectured that for $N \gg 0$, all $\mathbf{Q}$-points are cusps or CM points. (Hence this also holds for all quotients between $X_0(N) \to X_0(N)^*$.)

Following the strategy of Bilu–Parent(–Rebolledo) for the case $N = p^k$ a prime power, we prove that for $N \neq 99,125,147$ such that $g(X_0(N)^*) > 0$, the $j$-invariant of a non-cuspidal rational point is integral (in the ring of integers of the multiquadratic field the elliptic curve is defined over) or at least has denominator dividing $(2^3 \cdot 3 \cdot 5^2 \cdot 7^2 \cdot 31)^N$ if for all $p \in \{2,3,5,7,13\}$, $p^2 \nmid N$ (these are the primes $p$ such that $g(X_0(p)) = 0$).

The key property implied by $N$ non-squarefree is that we can degenerate to levels with one of the Atkin–Lehner involutions having sign $-1$. (Note that for $N$ squarefree, from the sign in the functional equation and assuming BSD, $J_0(N)^*$ will never have a rank $0$ quotient.) We prove: $J_0(pq)^{w_p = -1, w_q = +1}$ has a rank $0$ quotient if $q > 23$ prime and $p \in \{2,3,5,7,13\}$.

We also completely compute and classify rational points on $X_0(N)^*$ of genus between $1$ and $5$ and $N$ non-squarefree. To this end, we develop several algorithms of general interest.

We are working on completing the proof of Elkies’ conjecture for $N$ non-squarefree from this integrality result.