Atkin–Lehner Quotients

Galbraith's example, $X_0^*(137)$

Star curves

The star curve of level $N$ is the quotient of $X_0(N)$ by the full group of Atkin–Lehner involutions, $$X_0^*(N) \colonequals X_0(N)/W(N).$$

• Throughout, we assume $N$ is squarefree.
• Since $N$ is squarefree, $X_0^*(N)$ has exactly one cusp, and it is rational.
• Rational points on $X_0^*(N)$ give low degree points on $X_0(N)$.

Moduli interpretation

Elkies: for any number field $K$, the non-cuspidal points of $X_0^*(N)(K)$ are the $K$-curves of degree $d \mid N$, i.e. $E/\overline{K}$ that is $d$-isogenous to all of its Galois conjugates.

Over $\Q$: these are the $\Q$-curves, a mild generalization of elliptic curves over $\Q$, arising as quotients of $X_1(M)$.

So Galbraith's point on $X_0^*(137)$ is exceptional.

Star curves tend to have many rational CM points.

Two questions

Part I. How many exceptional points are there?

A generalization of Serre uniformity to discrete arithmetic subgroups of $\PGL_2^+(\Q)$. Elkies states it for points of any fixed degree $d$; we take $d = 1$.

Part II. Can we explain the existence of exceptional points, when they arise?

The algorithm

steps 1–4: finding exceptional points

1. Construct a canonical model for $X_0^*(N)$ from a basis $f_1, \dots, f_g \in S_2(\Gamma_0(N))$ of the forms with Atkin–Lehner eigenvalue $+1$ for every $p \mid N$.
2. Search for rational points in a box: a lower bound on $X_0^*(N)(\Q)$.
3. Count the rational CM points exactly, by Shimura reciprocity.
4. Compare. More points than CM points plus one $\Rightarrow$ an exceptional point.

Counting the CM points

• $h(R)$ is the class number.
• $N(R)$ is the product of the primes $p \mid N$ that split in $K$ or divide the conductor of $R$.
• $\varepsilon(N,R) \in \{0,1\}$ records whether $\operatorname{disc}(R)$ belongs to an order whose CM points are fixed by some $w_m$, $m > 1$.

Which of them are rational?

Fields of definition

Because $\Gal_\Q$ acts transitively on $\overline{\CM}(R)$, we have:

• Degree $d$ forces $h(R) \leq d \cdot 2^{\omega(N)}$: only finitely many orders contribute.
• Summing over the orders with $d = 1$ gives the rational CM points exactly.
• Taking $d = 2$ gives the quadratic CM points, which we will need in Part II.

Genus 3 and 4: new exceptional points

205 star curves of squarefree level, genus 2 through 7. In genus 2 there are exactly 51 exceptional points, by elliptic and quadratic Chabauty:

Bars–González–Xarles · Adžaga–Chidambaram–Keller–Padurariu · Balakrishnan–Dogra–Müller–Tuitman–Vonk

In genus 3 and 4, one exceptional point per curve. Galbraith found those at 137 and 311; the other twelve are new.

Genus 5, 6 and 7: nothing

99 curves. No exceptional points found.

* Identifying a point as a CM point is numerical, to high precision, not rigorous.

Proving that no exceptional point exists at all would need quadratic Chabauty, where the obstacle is the local heights at $N$.

The coefficients of these models are small: over 75% of the nonzero ones have a single digit, so a much larger search is unlikely to help.

Why do exceptional points appear?

Mazur and Ogg: on $X_1(N)$ and $X_0(N)$, the rational points that exist are the ones geometry accounts for.

We know which cusps and CM points exist. To explain a point, we need to know where they lie.

The algorithm

steps 5–8: explaining exceptional points

5. Identify the CM points on the canonical model, by evaluating $f_i(\tau)$.
6. Identify the cusp, as the limit $\tau \to i\infty$.
7. Find the degree 2 CM points by Shimura reciprocity, and identify them on the model.
8. Explain the exceptional point, using the CM points and the cusp.

Automorphisms

Automorphisms explain 58 of the 65 known exceptional points.

Bars and González determined $\Aut(X_0^*(N))$ for all squarefree $N$ and found every nontrivial case. We checked each one: no automorphism produces an exceptional point beyond those already found.

Collinearity

Write $K_X$ for a canonical divisor on $X = X_0^*(N)$, and $\equiv$ for linear equivalence.

$D$ is supported on closed points, so it is defined over $\Q$; then so is $K_X - D$, and it has degree 1, which is why $x$ is rational.

• $g = 2$: $\deg K_X = 2$, so $D = \iota(x)$ for the hyperelliptic involution $\iota$
• $g = 3$: $X$ is a plane quartic, and three cusp-or-CM points on a line force the fourth to be rational

Introduced by Derickx, H., Najman and Shnidman, for all modular curves $X_G$.

Two points left over

Collinearity explains all 14 exceptional points in genus 3 and 4, and in genus 2 it coincides with the hyperelliptic involution.

Together with automorphisms, that leaves two of the 65 unexplained: a pair on the genus 2 curve $X_0^*(286)$. They form an orbit for $\Aut(X_0^*(286)) \simeq \Z/2\Z$, but both are exceptional, so neither is the image of a cusp or a CM point.

Highlighted as mysterious in the earlier genus 2 work.

Elliptic covers

Degree 2 is bielliptic, already counted, so the new content is degree 3.

Explanations summary

Elliptic triple covers explain the two remaining points on $X_0^*(286)$, and give second explanations for many of the others.

Every known exceptional point on a squarefree star curve arises from cusps and CM points by geometry.

For automorphisms and elliptic covers we also check the mechanism is exhausted: they recover only the exceptional points already found.

Links

• Paper: arXiv:2609.00516, Assaf–Hashimoto–Shnidman
• Code: github.com/sachihashimoto/AtkinLehnerQuotients — equations for $X_0^*(N)$, and coordinates for every CM, cusp and exceptional point
• Upstream machinery: assaferan/ShimuraCurveALQuotients, Assaf–Hashimoto
• Models: sachihashimoto/X0Nstarquotients, Hashimoto–Keller–Le Fourn, and TimoKellerMath/QuadraticPoints, Adzaga–Keller–Michaud-Jacobs–Najman–Ozman–Vukorepa

These slides: sachihashimoto.github.io/talks/opatija