Point counts, automorphisms, and gonalities of Shimura curves

Technical Summary: Point Counts, Automorphisms, and Gonalities of Shimura Curves

Problem Statement
This work addresses three interconnected arithmetic problems concerning Shimura curves $X^D_0(N)$ over $\mathbb{Q}$ and their Atkin–Lehner quotients, where $D > 1$ is the discriminant of an indefinite quaternion algebra and $N$ is a positive integer coprime to $D$.

1. Point Counts: Determining the number of rational points over finite fields $\mathbb{F}_{p^r}$ for these curves and their quotients, specifically to identify curves that achieve record-breaking point counts for a given genus.
2. Automorphism Groups: Determining the full automorphism groups $\text{Aut}(X^D_0(N))$ and $\text{Aut}(X^D_0(N)/W)$, specifically testing the conjecture that these groups consist solely of Atkin–Lehner involutions for all but finitely many cases.
3. Gonalities: Determining which curves are tetragonal (admit a degree 4 map to $\mathbb{P}^1$) or geometrically tetragonal over $\mathbb{Q}$.

A primary challenge in addressing these problems is the scarcity of explicit defining equations for Shimura curves when $D > 1$, largely due to the absence of cuspidal points and real points, which limits the application of standard algebraic geometry techniques used for modular curves ($D=1$).

Methodology
The authors implement an algorithm (Algorithm 1.1) to compute point counts without requiring explicit defining equations. The method relies on the following components:

• Ribet's Isogeny: Utilizing a theorem by Ribet (Theorem 2.3) which establishes an isogeny over $\mathbb{Q}$ between the Jacobian of the Shimura curve $X^D_0(N)$ and a specific "D-new" sub-quotient of the Jacobian of the modular curve $X_0(DN)$.
• Modular Forms Data: The algorithm computes the number of points by analyzing the action of Hecke operators and Atkin–Lehner involutions on the space of weight 2 cusp forms associated with the modular curve $X_0(DN)$. Specifically, it utilizes the eigenvalues of these operators (sourced from the LMFDB [LMF25]) to determine the Frobenius traces acting on the isogeny factors of the Jacobian.
• Automorphism Checks: To determine automorphism groups, the authors combine the computed point counts with a restriction lemma by González (Lemma 5.1). This lemma provides a bound on the number of points in specific extensions if a non-trivial automorphism of prime order exists. By verifying that the point counts violate this bound for various prime powers, they rule out the existence of non-Atkin–Lehner automorphisms.
• Gonality Analysis: To determine gonalities, the authors combine the lower bounds on gonality derived from point counts (via Lemma 3.3) with the Castelnuovo–Severi inequality (Theorem 6.4) and known results on low-genus curves. They systematically filter candidates for tetragonal curves by checking genus bounds, quotient structures, and the existence of rational points on specific quotients.

Key Contributions and Results

• Point Count Records: The authors computed point counts for 783,514 Atkin–Lehner quotients of Shimura curves. They identified 116 curves that attain a number of rational points exceeding any previously known curve of the same genus over the same finite field (based on data from [vdGHLR09]). Additionally, they found 898 curves that are maximal (attaining the theoretical maximum point count) for their specific genus and field parameters. Among these, they identified at least four previously unknown isomorphism classes of maximal curves.
• Automorphism Groups: For the set of 10,609 pairs $(D, N)$ with $D > 1$, $DN \le 10,000$, and genus $g > 2$, the authors proved that the automorphism group consists entirely of Atkin–Lehner involutions for 9,288 of these cases. Specifically:
  • For 9,288 pairs, $\text{Aut}(X^D_0(N)) = W_0(D, N)$.
  • For the quotients, they established that $\text{Aut}(X^D_0(N)/W)$ consists only of Atkin–Lehner involutions for all but 1,321 pairs (which are listed in Table 8).
• Tetragonal Curves: The authors narrowed the classification of tetragonal and geometrically tetragonal Shimura curves.
  • They identified 161 geometrically tetragonal curves and 141 curves that are tetragonal over $\mathbb{Q}$ (excluding a small number of exceptions).
  • They reduced the set of unknowns to 32 pairs $(D, N)$ for which it remains unclear if the curve is geometrically tetragonal (Table 1) and 32 pairs for which it remains unclear if the curve is tetragonal over $\mathbb{Q}$ (Table 2).
  • They proved that for 14 specific triples $(D, N, W)$, the curve is tetragonal over $\mathbb{Q}$ if and only if the quotient $X^D_0(N)/W$ has a rational point.

Significance
The paper claims significance primarily through the implementation of a point-counting algorithm that bypasses the need for explicit defining equations, a major bottleneck in the study of Shimura curves with $D > 1$.

• Arithmetic Utility: The authors demonstrate that point counts serve as a powerful tool for resolving arithmetic questions regarding automorphism groups and gonalities, areas where techniques from the modular curve setting (relying on cusps) are not directly applicable.
• Data Expansion: The identification of 116 new record-holding point counts and 898 maximal curves expands the known landscape of curves with high point counts, providing new examples for the study of the Hasse–Weil bound and related arithmetic geometry problems.
• Classification Progress: The work significantly advances the classification of low-gonality Shimura curves, moving from a state of limited knowledge to a near-complete classification for curves with $DN \le 10,000$, leaving only a finite, manageable list of cases for future investigation.

The authors note that their results are conditional on the availability of eigenvalue data from the LMFDB and the specific bound $DN \le 10,000$, suggesting that further computational resources could extend these results to higher levels. All code and data are made available in a GitHub repository.