{"id":"676343cb-a254-4864-94d0-275c8a41c249","arxiv_id":"2507.15992","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An algorithm based on Ribet's isogeny computes point counts for Shimura curves, yielding 116 record curves, automorphism results for 9288 of 10609 curves, and a tetragonal classification up to 32 exceptions each.","lead":"This paper computes the number of points over finite fields for thousands of Shimura curves and their Atkin-Lehner quotients, using modular form data instead of defining equations. It finds 116 quotient curves with record point counts, proves that most of the studied curves have only their expected automorphisms, and nearly classifies the tetragonal Shimura curves.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.3's claimed scope looks inconsistent: S is defined by star-quotient genus > 2, but 10609 counts full-genus > 2, and low-star pairs like (6,11) are not covered.","rationale":"I read the central claim as the algorithmic point-count computation together with the automorphism theorem in the introduction. The reader's weakest assumption was the non-squarefree extension of Ribet's isogeny; I do not press that as the primary objection because the authors explicitly cite Martin's basis problem theorem and Roberts's thesis for that extension, and the automorphism theorem itself is stated only for squarefree N, where the classical squarefree Ribet isogeny is available. The more immediate and textually verifiable problem is internal: the definition of S in Section 5 does not match the count 10609 reported in Remark 1.4 and the abstract. The example (6,11) shows that full genus > 2 does not imply star-quotient genus > 2, so the set for which Theorem 5.3 and Corollary 5.4 are proved is genuinely smaller than the set named in Theorem 1.3. This is not a complaint about external consensus or a stylistic issue; it is a concrete gap in the proof of a stated theorem. The proposed check settles it by recomputing the exact cardinality of S and listing the omitted low-star pairs. If the check confirms |S| = 10609, then my objection fails and the result can stand; if it instead finds omitted pairs, the theorem statement and the abstract's 9288 count need amendment. I therefore keep the reader's CONDITIONAL verdict, but for a different reason than the one emphasized by the reader.","tokens_in":50494,"tokens_out":33611,"duration_ms":391369,"concrete_test":"Recompute g(X_0^D(N)^*) for every squarefree coprime pair (D,N) with DN <= 10000 using Proposition 2.1 for the genus and Proposition 2.2 for the fixed-point counts of Atkin-Lehner involutions. Compare the resulting cardinality of S with 10609 and with 9288 + 1321. In particular, enumerate all pairs with g(X_0^D(N)) > 2 but g(X_0^D(N)^*) <= 2; if any such pair is absent from Table 7, then Theorem 1.3 lacks a proof for it, and if |S| != 10609, the headline '9288 of 10609' should be corrected.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 5 defines S by the conditions (1) g(X_0^D(N)^*) > 2 and (2) DN <= 10000, and immediately asserts |S| = 10609. Remark 1.4, however, says that 10609 is the total number of coprime pairs with g(X_0^D(N)) > 2 and DN <= 10000. These two genus conditions are not equivalent. A concrete witness is (D,N) = (6,11): Proposition 6.13 places (6,11) among curves with Q-gonality 2 and genus at least 2, and Table 9 lists (6,11) with m = 66 and quotient genus 0, so g(X_0^6(11)^*) = 0 even though g(X_0^6(11)) > 2. Thus (6,11) cannot belong to S. Theorem 5.3 and Corollary 5.4 only establish results for pairs in S; the descent argument from a non-Atkin-Lehner involution to the star quotient relies on the star quotient having genus at least 2 via Proposition 2.5/2.6. Low-star-quotient pairs such as (6,11) are not handled by that argument and do not appear among the twelve exceptions in Table 7. Consequently, Theorem 1.3, as stated for all 10609 full-genus > 2 curves, is not proved unless a separate argument is supplied. Moreover, if |S| is actually smaller than 10609, the reported count 9288, which appears to be |S| - 1321, cannot literally be '9288 of the 10609 curves'.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":50844,"tokens_out":11165,"duration_ms":123040,"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":[{"comment":"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.","section":"Section 5 / Theorem 1.3 / Remark 1.4"},{"comment":"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.","section":"Theorem 5.3, proof"}],"minor_comments":[{"comment":"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.","section":"Theorem 1.3"},{"comment":"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.","section":"Proposition 6.16(2), proof"},{"comment":"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.","section":"Lemma 4.2"},{"comment":"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.","section":"Tables 1 and 2"}],"recommendation":"major_revision","confidential_remarks":"The inconsistency between the definition of S and the count 10609 appears to be an editing slip, but it directly affects the headline Theorem 1.3 and the '9288 of 10609' claim. I would ask the authors to recompute the actual size of S and the corresponding counts, or to restate Theorem 1.3 for the set actually treated. I would also ask for a clear statement of the finite-check termination criterion in the proof of Theorem 5.3, since that is essential for the claimed proof."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The genuinely new piece is Algorithm 1.1: a point-count pipeline for X_0^D(N) and its Atkin-Lehner quotients that avoids defining equations, via Ribet's isogeny and Hecke eigenvalue data from LMFDB. That part looks reproducible, and the outputs are concrete: 116 new record curves, 898 maximal curves, and 4 new maximal isomorphism classes. The tetragonal classification (Theorem 1.5) is also substantial, with careful Castelnuovo-Severi arguments and CM-point checks; the 32 undecided pairs are honestly listed.\n\nThe soft spot is the automorphism theorem. Section 5 defines S by g(X_0^D(N)^*) > 2 and asserts |S| = 10609, but Remark 1.4 says 10609 counts pairs with g(X_0^D(N)) > 2. Those are not the same. Concretely, (6,11) has full genus > 2, but Table 9 shows the quotient by w_66 has genus 0, hence the star quotient has genus 0. So (6,11) is not in S as defined, yet Theorem 1.3 covers it and it appears in neither Table 7 nor Table 8. For W = W_0(6,11), the quotient is P^1, whose automorphism group is not a 2-group, so part (2) is false as stated. This is a load-bearing gap, not a cosmetic typo, because the descent to X^* and Proposition 2.5 need star genus at least 2.\n\nThe proof of Theorem 5.3 is also compressed: the text does not state the range of n used in Lemma 5.1 or give enough detail to reproduce the 9288 count from the text alone. The code is on GitHub, which helps, but a referee would need to run it.\n\nOn the Ribet isogeny point: the paper cites Martin and Roberts for the non-squarefree case. I did not find a red flag there.\n\nBottom line: the point-count and gonality work deserves a serious referee; the automorphism claims need a corrected statement and a separate treatment of low-star-genus pairs. I would send this to review, but expect major revision.","headline":"Useful point-count and gonality work, but the automorphism theorem conflates two genus conditions and is false as stated for pairs like (6,11).","tokens_in":51376,"tokens_out":7705,"would_cite":true,"duration_ms":74267,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G18","11G20","11G30","11G15"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Shimura curves","point counts over finite fields","Atkin-Lehner involutions","automorphism groups","gonality","Ribet isogeny","quaternionic multiplication","maximal curves"],"falsifier":"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.","tokens_in":50319,"feed_emoji":"🔢","tokens_out":12041,"duration_ms":116344,"temperature":0.7,"pith_summary":"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.","feed_headline":"9288 of 10609 Shimura curves have trivial automorphism group","feed_subtitle":"An equation-free point-count algorithm also classifies most tetragonal cases and finds 116 record point counts.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the original isogeny identifying Jac(X_0^D(N)) with the D-new part of Jac(X_0(DN)), the foundation of Algorithm 1.1.","marker":"[Rib80]"},{"why":"Gives the Atkin–Lehner action formulation on isogeny factors used in the sign relation of Theorem 2.3.","marker":"[BD96]"},{"why":"Reproves the basis-problem statement that extends Ribet's isogeny to non-squarefree level.","marker":"[Mar20]"},{"why":"Provides an alternate proof of the unrestricted isogeny, cited for the general version of Theorem 2.3.","marker":"[Rob89]"},{"why":"Supplies the point-count method for modular curves that the Shimura-curve algorithm adapts.","marker":"[DLMS23]"},{"why":"Supplies the multiplicity formula of Lemma 4.2 for Jacobians of Atkin–Lehner quotients.","marker":"[DLMS26]"},{"why":"Supplies the stored Hecke and Atkin–Lehner eigenvalue data for levels DN≤10000 used in Steps 2 and 3.","marker":"[LMF25]"},{"why":"Supplies Lemma 5.1, the point-count constraint on automorphisms that proves triviality for most star quotients.","marker":"[Gon17]"},{"why":"Supplies the structure theorems and criteria showing automorphism groups are elementary 2-groups and identifying Atkin–Lehner cases.","marker":"[KR08]"},{"why":"Provides the linear lower bound on geometric gonality that restricts the tetragonal search to genus at most 34.","marker":"[Abr96]"}],"fun_headline_variants":["116 Shimura quotients break point count records","9288 Shimura curves have trivial automorphism group","Point-count algorithm reveals 116 record Shimura curves","Gonality and automorphisms settled for many Shimura curves"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["116 Shimura quotients break point count records","9288 Shimura curves have trivial automorphism group","Point-count algorithm reveals 116 record Shimura curves","Gonality and automorphisms settled for many Shimura curves"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000535,"raw_usage":{"total_tokens":2629,"prompt_tokens":1059,"completion_tokens":1570,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":675,"completion_tokens_details":{"reasoning_tokens":1505}},"tokens_in":675,"tokens_out":1570,"duration_ms":12643,"temperature":1.0,"reasoning_tokens":1505,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:22:23.014480+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}