Explorations in math and programming
David Lowry-Duda

I'm happy to announce that a new paper, "Towards a classification of isolated $j$-invariants", now appears on the arxiv. This was done with my collaborators Abbey Bourdon, Sachi Hashimoto, Timo Keller, Zev Klagsbrun, Travis Morrison, Filip Najman, and Himanshu Shukla.

There are many collaborators because this was borne out of a workshop at CIRM from earlier this year, and we all attacked this problem together. This is the first time I have collaborated with any of these collaborators (though several of us are now involved in another project that will eventually appear).

The modular curve $X_1(N)$ is moduli space and an algebraic curve (defined over $\mathbb{Q}$ for us) whose points parametrize elliptic curves with a point of order $N$. We study isolated points, which are morally points on $X_1(N)$ that don't come from infinite families.

Perhaps the simplest form of infinite families comes come from a rational map $f: X_1(N) \longrightarrow \mathbb{P}^1$ of degree $d$. By Hilbert's irreducibility theorem, $f^{-1}(\mathbb{P}^1(\mathbb{Q}))$ contains infinitely many closed points of degree $d$.

Similarly, to any closed point $x$ of degree $d$ one can associate the rational divisor $P_1 + \cdots + P_d$, where $P_j$ are the points in the Galois orbit associated to $x$. This gives a natural map $\Phi_d: X_1(N)^{(d)} \longrightarrow \mathrm{Jac}(X_1(N))$. If $\Phi_d(x) = \Phi_d(y)$ for some point $y$, one can show that there exists a nonconstant function $f : X_1(N) \longrightarrow \mathbb{P}^1$ of degree $d$ again. Thus positive rank abelian subvarieties of the Jacobian also give infinite families of points.

Roughly, we say that a closed point $x$ is isolated if it doesn't come from either of the two constructions ($\mathbb{P}^1$, which we call $\mathbb{P}^1$ isolated — and abelian subvariety, which we call AV-isolated) above. Further, a closed point $x$ of degree $d$ is called sporadic if there are only finitely many closed points of degree at most $d$.

If $x \in X_1(N)$ is an isolated point, we say $j(x) \in X_1(1) \cong \mathbb{P}^1$ is an isolated $j$-invariant. In this paper, we seek to answer a question of Bourdon, Ejder, Liu, Odumodo, and Viray.

Question: Can one explicitly identify the (likely finite) set of isolated $j$-invariants in $\mathbb{Q}$?

Our main result is decision algorithm. Given a $j$-invariant, our algorithm produces a finite list of potential (level, degree) pairs such that one only needs to verify that degree $\mathrm{degree}$ points on $X_1(\mathrm{level})$ are not isolated.

Stated differently, our algorithm has one-sided error. It either reports that an element is not isolated, or it reports that it might be isolated and gives a list of places containing the data where isolated points must come from.

In principle, this sounds like it might be insufficient. But we ran our algorithm on every elliptic curve in the LMFDB and the outputs of our algorithm are always the empty set — except for $4$ exceptions where we know the $j$-invariants are isolated.

Concretely, we know that for any non-CM elliptic curve over $\mathbb{Q}$ with an isolated $j(E) \in \mathbb{Q}$ and with

then $j(E) \in \{ -140625/8, -9317, 351/4, -162677523113838677 \}$. These latter correspond to $\mathbb{P}^1$ isolated points on $X_1(21), X_1(37), X_1(28)$, and $X_1(37)$ (respectively).1 1An appendix to our paper by Derickx and Mark van Hoeij shows that the last $j$-invariant is actually isolated, not merely sporadic.

More generally, we are led to conjecture that these (and the CM $j$-invariants) are all of the isolated points on $X_1(N)$.

Broadly, our algorithm works by first considering the Galois image of elliptic curves (thanks to the code of David Zywina and the efforts of David Roe and others for making this broadly accessible). An earlier result of Bourdon and her collaborators2 2Abbey proposed this problem at the workshop based on the observation that her result might make things tenable. She was right!

allows one to radically narrow the focus of the Galois image to a minimal set of points of interest. We do this narrowing. We also show that no elliptic curve with adelic image of genus $0$ gives an isolated point, which allows us to ignore many potential leafs of computation.

The details are interesting and we try to be as explicit as possible. The code for this project is also available, and can be found on my github at github.com/davidlowryduda/isolated_points.

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Comments (1)
  1. 2024-02-14 DLD

    I'll give a small update as a comment. Our paper has been accepted and will appear in Mathematics of Computation shortly. (And Math Comp was very fast and forthright with their review, which is nice).