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Point counts, automorphisms, and gonalities of Shimura curves

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Ribet's isogeny turns Shimura-curve point counting into modular-form eigenvalue sums, and the resulting counts settle most automorphism and gonality questions for these curves.

desk verdict Useful point-count and gonality work, but the automorphism theorem conflates two genus conditions and is false as stated for pairs like (6,11). read the letter →

arxiv 2507.15992 v2 pith:YDW3YUH5 submitted 2025-07-21 math.NT

Pietro Mercuri , Oana Padurariu , Frederick Saia , Claudio Stirpe This is my paper · ORCID
classification math.NT MSC 11G1811G2011G3011G15
keywords ShimuracurvespointcountsoverfinitefieldsAtkin-LehnerinvolutionsautomorphismgroupsgonalityRibetisogenyquaternionicmultiplicationmaximal
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper implements an algorithm that counts the $\mathbb{F}_{p^r}$-rational points on Shimura curves $X_0^D(N)$ over $\mathbb{Q}$ and their Atkin–Lehner quotients, without requiring defining equations. The counts drive two arithmetic classifications: for 9288 of the 10609 curves $X_0^D(N)^*$ of genus $g>2$ with $DN\le 10000$ the automorphism group is trivial, so up to short exception lists every quotient has only Atkin–Lehner automorphisms and $X_0^D(N)$ itself has $W_0(D,N)$ as its full automorphism group; and all tetragonal and geometrically tetragonal Shimura curves $X_0^D(N)$ are determined up to 32 possible exceptions each. Separately, among 783514 tested Atkin–Lehner quotients the algorithm finds 116 whose rational point counts exceed those of any previously known curve of the same genus over the same finite field, and 898 maximal quotients, including at least 4 previously unknown isomorphism classes of maximal curves. The central claim is that Ribet's isogeny reduces Shimura-curve point counting to modular-form eigenvalue computations, making these classifications feasible without equations.

What carries the argument

The load-bearing object is the Ribet isogeny comparing the Jacobian of the Shimura curve with a subquotient of the Jacobian of the classical modular curve $X_0(DN)$: it gives $\mathrm{Jac}(X_0^D(N))$ as the $D$-new part of $\mathrm{Jac}(X_0(DN))$, together with the sign rule $\epsilon_f(D,N)=(-1)^{\omega(\gcd(D,m))}\epsilon_f(1,DN)$ for the Atkin–Lehner involution $w_m$ on each isogeny factor $A_f$, where Hall divisors $m\parallel DN$ index the involutions. Algorithm 1.1 combines this with the multiplicity formula of Lemma 4.2, which expresses the Jacobian of an Atkin–Lehner quotient as $\prod_f A_f^{m_f}$ with multiplicities $m_f$ built from Atkin–Lehner eigenvalues, and then computes $\#X_0^D(N)(\mathbb{F}_q)=q+1-\sum_f m_f\,\operatorname{tr}(\operatorname{Frob}_q\mid A_f)$. This machinery is what lets the paper count points on curves for which no defining equation is used.

What would settle it

A concrete check is to take a non-squarefree level such as $(D,N)=(6,49)$, compute $\#X_0^6(49)(\mathbb{F}_p)$ for a prime $p\nmid DN$ by Algorithm 1.1, and recompute the same count independently from an explicit equation or model of the curve or from a different cohomological method; any disagreement would falsify the generalized Ribet step. Alternatively, one can compute both sides of the sign identity $\epsilon_f(D,N)=(-1)^{\omega(\gcd(D,m))}\epsilon_f(1,DN)$ for a non-squarefree $N$ and look for a violation.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is that Algorithm 1.1 computes $\#(X_0^D(N)/W)(\mathbb{F}_{p^r})$ for every tuple $(D,N,W,p,r)$ in the stated range. The proof mechanism is Ribet's isogeny, which identifies $\mathrm{Jac}(X_0^D(N))$ with the $D$-new part of $\mathrm{Jac}(X_0(DN))$ and relates the Atkin–Lehner signs by $\epsilon_f(D,N)=(-1)^{\omega(\gcd(D,m))}\epsilon_f(1,DN)$; with the multiplicity formula of Lemma 4.2 this turns point counts into sums of weighted Frobenius traces of newforms. Using these counts, the paper proves that $X_0^D(N)^*$ has trivial automorphism group for 9288 of the 10609 pairs $(D,N)$ with $DN\le 10000$, that $\mathrm{Aut}(X_0^D(N))=W_0(D,N)$ for all but 12 such pairs, and that any quotient's automorphism group is purely Atkin–Lehner for all but 1321 pairs. For gonality, the same point counts certify that every geometrically tetragonal $X_0^D(N)$ either appears among the 161 listed geometrically tetragonal curves or among the 32 pairs of Table 1, and every tetragonal-over-$\mathbb{Q}$ curve either appears among the 141 listed curves or among the 32 pairs of Table 2.

Load-bearing premise

The load-bearing premise is that Ribet's isogeny, originally proved for squarefree level, remains true for arbitrary level $N$, including the Atkin–Lehner sign rule; if this generalization failed, the Hecke-eigenvalue computation in the algorithm and every point count derived from it would be unsupported.

Editorial extensions

If this is right

  • For 9288 of the 10609 pairs in $S$, the star quotient $X_0^D(N)^*$ has trivial automorphism group; consequently all quotients of those curves have automorphism group $(\mathbb{Z}/2\mathbb{Z})^{\omega(DN)-\mathrm{ord}_2(|W|)}$, and $X_0^D(N)$ itself has only Atkin–Lehner automorphisms except for 12 listed pairs.
  • Every geometrically tetragonal Shimura curve with $DN\le 77416$ that is not among the 161 listed ones must be one of the 32 pairs in Table 1, and every tetragonal-over-$\mathbb{Q}$ curve that is not among the 141 listed ones must be one of the 32 pairs in Table 2.
  • The algorithm produces 116 Atkin–Lehner quotients whose point counts exceed all previously known curves of the same genus over the same finite field, and 898 maximal quotients, including at least 4 previously unknown isomorphism classes of maximal curves.
  • Because the algorithm needs no defining equations, rerunning it with larger Hecke-eigenvalue data would extend the automorphism and gonality classifications beyond the $DN\le 10000$ bound, as the paper notes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The 32 unresolved geometric-tetragonal pairs and the 32 unresolved tetragonal-over-$\mathbb{Q}$ pairs are explicit finite lists; targeted methods such as explicit equations, local solubility, or higher-genus analogues of the paper's tests could settle each remaining case.
  • The 116 record-holding quotients suggest that Atkin–Lehner quotients of Shimura curves form an efficient search family for curves with many rational points; testing larger primes, higher powers $r$, and levels beyond $DN=10000$ could yield further records and new maximal classes.
  • If the generalized Ribet isogeny were ever shown to fail for some non-squarefree level, only the point counts for affected tuples would be at risk; the squarefree-level core of the automorphism and gonality theorems would survive, since the squarefree restriction is used in separate structural lemmas.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper implements Algorithm 1.1 for computing #(X_0^D(N)/W)(F_{p^r}) for Shimura curves and their Atkin–Lehner quotients, using Ribet's isogeny (Theorem 2.3), the multiplicity formula of Lemma 4.2, and LMFDB Hecke data. It reports 116 point-count records among Atkin–Lehner quotients, 898 maximal curves, and 4 new maximal isomorphism classes. As applications, it claims that Aut(X_0^D(N)) equals the Atkin–Lehner group for 9288 of the 10609 curves of full genus >2 with DN≤10000, and it classifies tetragonal and geometrically tetragonal Shimura curves up to 32 possible exceptions each. The computations are supported by Magma code in a GitHub repository.

Significance. If the main results are correct, this is a substantial computational contribution: the point-count algorithm avoids defining equations, is deterministic from modular-form data, and is benchmarked against external tables such as manypoints.org. The automorphism and gonality applications are natural and potentially influential. The paper provides code and cites the relevant literature. However, the headline Theorem 1.3 has a scope inconsistency between the definition of the set S and the claimed 10609-curve count, and the proof of the main automorphism assertion does not state a termination criterion for the finite-field checks. These issues affect the central claims and must be resolved before the results can be accepted as stated.

major comments (2)
  1. [Section 5 / Theorem 1.3 / Remark 1.4] The set S is defined by the condition g(X_0^D(N)^*) > 2 and DN ≤ 10000, and the text states |S| = 10609. Yet Remark 1.4 says that 10609 is the total number of coprime pairs with g(X_0^D(N)) > 2 and DN ≤ 10000. These are not equivalent. A concrete witness is (D,N) = (6,11): Proposition 6.13(3) places it among curves of genus at least 2 with Q-gonality 2, so it satisfies the full-genus condition, while Table 9 lists (6,11) with quotient X_0^6(11)/⟨w_66⟩ of genus 0, so its star quotient has genus 0 and (6,11) cannot lie in S. Thus the statement |S| = 10609 is incompatible with the definition of S. Moreover Theorem 1.3 as stated in the introduction covers all positive N with full genus >2 and DN≤10000, whereas Corollary 5.4 is proved only for pairs in S, and S is further restricted to squarefree N. Consequently the literal claim that automorphisms are Atkin–Lehner for 9288 of the 10609 full-genus curves is not established by the proof given. The authors need to reconcile the definition of S, the count 10609, the squarefree hypothesis, and the statement of Theorem 1.3.
  2. [Theorem 5.3, proof] The proof of Theorem 5.3 invokes Lemma 5.1, whose inequality is a sum over all integers n ≥ 1 with gcd(n, ℓ) = 1. The proof states only that the check uses prime powers q = p^r with p < 100 and r up to 100 'when possible, and more generally as high as the pre-computed trace data we use allows.' No stopping criterion or congruence argument is given to show that this finite range suffices to rule out a non-trivial involution for every one of the 9288 pairs. If the code contains such a criterion, the manuscript should state it; otherwise the triviality assertions are not verifiable from the text. This is a load-bearing step for both parts of Corollary 5.4 and for Theorem 1.3.
minor comments (4)
  1. [Theorem 1.3] The statement of Theorem 1.3 omits the squarefree condition on N that is used in Proposition 2.5, Lemma 2.7, and the definition of S; the abstract includes the squarefree hypothesis. The theorem statement should match the hypotheses actually used.
  2. [Proposition 6.16(2), proof] The proof says 'All 83 pairs handled in this manner ... are listed in Table 12,' while Proposition 6.16(2) states that Table 12 contains 81 pairs. The number 83 appears to be a typo, but it should be corrected for consistency.
  3. [Lemma 4.2] The displayed formula for m_f contains ambiguous fractions: '(v+1)/2' and '(1+(-1)^v)/4' should be parenthesized, and the dependence of ε_{f,ℓ^e} on the Atkin–Lehner eigenvalue should be spelled out in the notation.
  4. [Tables 1 and 2] The bold-format explanation under Table 1 says pairs proved not tetragonal over Q are bold, while Table 2 says pairs proved geometrically tetragonal are bold. Since the tables are meant to list 'unsure' cases, the formatting should be explained in a way that does not confuse the reader: a pair can be bold because one of the two properties has been settled even though the other remains open.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: point counts are deterministic outputs from external Hecke data; self-citations are to independent published results.

full rationale

I find no circular derivation in this paper. Algorithm 1.1 computes point counts of X_0^D(N)/W by combining external Hecke eigenvalue data from LMFDB (Remark 1.2, Step 2, Step 3) with the multiplicity formula of Lemma 4.2 and Ribet's isogeny (Theorem 2.3). These inputs are not fitted to the point counts being output; the algorithm is a deterministic computation, and the resulting counts are checked against the independent tables at manypoints.org (Section 7). There is no fitted parameter that is later renamed as a prediction. The self-citations that occur are not load-bearing in a circular way. Lemma 4.2 is proved by citing [DLMS26, Theorem 2.2], a published companion paper with an independent proof; the same holds for the algorithm framework inherited from [DLMS23]. The tetragonal application uses results of the same authors, [PS25a], [PS25b], and [Sai24], as black boxes, but those results are externally stated theorems with their own proofs and are not derived from the present paper's conclusions. They do not smuggle in the target classification by definition. The only concern I found is an internal consistency issue, not a circularity: Section 5 defines S by g(X_0^D(N)^*) > 2 and asserts |S| = 10609, while Remark 1.4 says 10609 is the total number of coprime pairs with g(X_0^D(N)) > 2 and DN <= 10000. Since the two genus conditions are not equivalent in general, Theorem 1.3's stated scope and the number 9288 may require a correction. This is a correctness or bookkeeping issue and does not indicate that the derivation reduces to its own inputs. Similarly, the reliance on Ribet's isogeny for non-squarefree N is an assumption cited to [Mar20] and [Rob89], not a self-referential step.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central algorithm rests on the Ribet isogeny in the non-squarefree case, the correctness of LMFDB Hecke data for DN<=10000, and the sufficiency of the finite point-count checks in the automorphism sieve. No new entities are postulated and no constants are fitted; the paper's numbers are computed, not tuned.

assumptions (7)
  • domain assumption Ribet isogeny identifying the D-new part of Jac(X0(DN)) with Jac(X0^D(N)), with the Atkin-Lehner sign relation (Eq. 1), holds for arbitrary N and not only squarefree N.
    Invoked in Theorem 2.3 and Step 2 of Algorithm 1.1. All point counts reduce to modular form data only under this identification; the paper cites Martin and Roberts for the non-squarefree extension without proving it.
  • domain assumption Hecke eigenvalue data and Atkin-Lehner signs for newforms of levels dividing DN with DN<=10000 are correct and complete in LMFDB.
    Steps 2 and 3 of Algorithm 1.1 consume these data. No independent verification or frozen snapshot is provided.
  • domain assumption The point-count checks in Theorem 5.3 cover a sufficient range of prime powers to apply Lemma 5.1 to each of the 9288 curves.
    The text says powers up to 100 when possible, but the lemma's finite sum can in principle require n up to 2g+2; the paper does not state the actual range used per curve.
  • standard math Automorphism groups of the curves and quotients studied are elementary abelian 2-groups, so only involutions need to be excluded (Proposition 2.5).
    Kontogeorgis-Rotger's proposition is used in Section 5 to reduce the automorphism problem to ruling out a single non-Atkin-Lehner involution.
  • standard math Castelnuovo-Severi inequality and Lemmas 6.5-6.9 apply over the perfect fields considered.
    These are the main tools in Section 6 for proving non-tetragonality.
  • standard math Abramovich's gonality lower bound with the improved constant 975/8192 (Theorem 6.14) and Saia's genus lower bound (Lemma 6.15) are valid.
    They restrict tetragonal candidates to genus at most 34 and DN<=77416 in Section 6.
  • standard math Good reduction models and the injection of rational automorphisms into mod p automorphisms hold at primes p not dividing DN.
    Used in Lemma 6.11 and Section 5 to pass from Q to finite fields.

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Pith. "Pith review of Point counts, automorphisms, and gonalities of Shimura curves." pith.science (2026). https://pith.science/paper/YDW3YUH5

@misc{pith2026250715992,
  author       = {Pith},
  title        = {Pith review of: Point counts, automorphisms, and gonalities of Shimura curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YDW3YUH5}},
  note         = {Machine review of arXiv:2507.15992}
}
abstract

We implement an algorithm to compute the number of points over finite fields for the Shimura curves $X_0^D(N)$ over $\mathbb{Q}$ and their Atkin--Lehner quotients. Our computations identify $116$ such quotients over finite fields (out of $783514$ tested) that attain a number of rational points exceeding that of any previously known curve of the same genus over the same finite field. To illustrate the utility of our point counts algorithm in addressing arithmetic questions, we prove that all automorphisms are Atkin--Lehner for $9288$ of the $10609$ curves $X_0^D(N)$ of genus $g > 2$ with $D$ the discriminant of an indefinite quaternion algebra over $\mathbb{Q}$, $N$ a squarefree positive integer coprime to $D$, and $DN\leq 10000$, and we determine all tetragonal and geometrically tetragonal curves $X_0^D(N)$ up to a small number of possible exceptions.

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