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May 14, 2023: New preprint on quadratic points on $X_0(N)$

Why this matters. A point of $X_0(N)$ over a number field $K$ is the same as an elliptic curve over $K$ with a cyclic $N$-isogeny, so finding such points classifies these curves. After the rational case ($K = \mathbf{Q}$), the first genuinely new problem is that of quadratic points, those defined over a degree-$2$ field. We determine all of them for a large list of modular curves where they were previously unknown, and find that non-cuspidal, non-CM examples are rare.

Let $E$ be an elliptic curve over $\mathbf{Q}$. It is important to know for which primes $p$ the mod-$p$ Galois representation $\rho_{E,p}: G_\mathbf{Q}\to \operatorname{Aut}(E[p](\overline{\mathbf{Q}})) \cong\operatorname{GL}_2(\mathbf{F}_p)$ is reducible. For example, this is the case when there is a non-trivial $p$-torsion point defined over $\mathbf{Q}$. The irreducibility of $\rho_{E,p}$ is an important condition for Euler systems (although we have partially removed this hypothesis in the appendix of [K–Yin 2024]) or in the modular approach to the (generalized) Fermat equation.

In his famous 1977 Eisenstein paper and a subsequent paper, Mazur classified all such primes for elliptic curves over $\mathbf{Q}$. It turns out that the largest “Eisenstein prime”, i.e., a prime with $\rho_{E,p}$ reducible for some $E/\mathbf{Q}$ is $163$, and it is $37$ if $E$ is not CM. The possible “torsion primes” $p$ are $\{2,3,5,7\}$.

The preprint Computing Quadratic Points on Modular Curves $X_0(N)$ is joint work with Nikola Adžaga, Philippe Michaud-Jacobs, Filip Najman, Ekin Ozman, and Borna Vukorepa. We started to work on this at the 2022 MIT Modular curves workshop, whose aim was to extend the LMFDB with data on modular curves.

The $\mathbf{Q}$-rational points on $X_1(N)$ and $X_0(N)$ for all $N$ have been computed by Mazur and Kenku. For the modular curve $X_1(N)$ classifying (generalized) elliptic curves with a point of order $N$, there is a bound $N(d)$ by Merel and Kamienny such that $X_1(N)(K)$ contains only cusps if $N > N(d)$ and $K$ runs through all number fields of degree $d$. This has been used to compute all degree $d$ points for $d \leq 7$ by combined work of Kamienny–Kenku–Momose ($d=2$), Derickx–Etropolski–van Hoeij–Morrow–Zureick-Brown ($d = 3$), and for $4\leq d\leq 7$ and prime values of $N$ by Derickx–Kamienny–Stein–Stoll.

However, there is no $d > 1$ such that one knows all degree-$d$ points on $X_0(N)$ (classifying elliptic curves with a cyclic $N$-isogeny) for all $N$! Conjecturally, the only quadratic points are cusps and CM points for $N \gg 0$.

In our preprint, we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves $X_0(N)$ of genus up to $8$, and genus up to $10$ with $N$ prime, for which they were previously unknown. This extends previous work by Bruin–Najman, Ozman–Siksek, Box, and Najman–Vukorepa. Our methods apply to more than these curves; we restricted to these to make the computations feasible.

We use the following three methods:

1. Going down: If $f\colon X \to Y$ is a finite morphism of curves and $Y(K)$ is known and finite, one can compute $X(K)$ by taking fibers. Problems arise if there are infinitely many quadratic points on $Y$. We use this for $N \in \{58, 68, 76 \}$.

2. Rank $0$: If $J_0(N)(\mathbf{Q}) = J_0(N)(\mathbf{Q})_\mathrm{tors}$ is known (we verify the generalized Ogg conjecture in several cases), one can compute the quadratic points on $X_0(N)$ using a variant of the Mordell–Weil sieve. We use this for $N \in \{ 80, 98, 100 \}$.

3. Atkin–Lehner sieve: This is the most involved method and a variant of the Mordell–Weil sieve, which, if applicable, reduces the problem to considering fixed points of an Atkin–Lehner involution and the rational points on a given Atkin–Lehner quotient. We use this for $N \in \{ 74, 85, 97, 103, 107, 109, 113, 121, 127 \}$.

We also give optimized algorithms to compute models of Atkin–Lehner quotients of $X_0(N)$ and the $j$-invariant morphism.

Note that there can be infinitely many quadratic points on a curve $X$; this happens if and only if $X$ is hyperelliptic (plug in infinitely many values for $x$ in $y^2 = f(x)$) or bielliptic with elliptic curve of rank $> 0$.

We get evidence for the fact that non-cuspidal, non-CM points are rare.

Update (June 22, 2023). Here are my slides for a talk at a conference in Dubrovnik.

Update (August 23, 2023). The article has been accepted for publication in Mathematics of Computation.