{"id":"ed9b3bf7-d7c8-40a4-a19a-d16262492ace","arxiv_id":"2504.18894","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Explicit genus-7 and genus-12 curves over F_{11^5} reach the Hasse-Weil-Serre upper bound, via a generalized Chen isogeny for quotients of Borel-Cartan modular curves.","lead":"Two new curves over the finite field with 11^5 elements are shown to have the maximum number of points allowed by the Hasse-Weil-Serre bound, one of genus 7 and one of genus 12, and explicit equations are given. The construction matters because such maximal curves are rare over odd extensions and curves with many points are used in coding theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The genus-7 identification in §4.1 depends on an unproven numerical-to-exact rounding step; the genus-12 proof has an exact certificate, but the genus-7 quotient proof does not.","rationale":"The reader's weakest assumption is exactly the numerical-to-exact bridge in Section 4.1, and I agree that this is the most fragile computational step. The genus-12 construction in Section 4.3 is supported by an exact argument using q-expansions to order 180 versus a line bundle of degree 44, so it does not share this fragility. For genus 7, however, the proof that the displayed equations describe X(6,7)/<W6,W7> depends on the unproven claim that forms obtained by numerical linear algebra with precision 10^-10 and rounding to small denominators are exactly W6-invariant. If that claim fails, the automorphism-group argument that identifies the quotient collapses. I note that the central existence claim could still be rescued by directly verifying that the displayed equations define a smooth genus-7 curve whose reduction modulo 11 has 166666 points over F_11^5, but the paper does not provide the code or logs for such an independent check. This asymmetry supports the reader's conditional verdict: the mathematics is likely correct, but the genus-7 quotient identification is not yet rigorously certified as written. I found no internal inconsistency or mathematical error in Theorem 2.2 or the point-count method, and I do not see grounds to reject or to accept unconditionally without the missing certificate or code.","tokens_in":58339,"tokens_out":8110,"duration_ms":81458,"concrete_test":"Compute the exact action of W6 on the 16-dimensional space S2(Gamma_H) using modular symbols (e.g., Magma's ModularSymbols or Sage), and verify that each rounded rational vector s_i equals the exact invariant form b_i + b_i|2W6. If all equal, the numerical bridge is certified; any mismatch invalidates the Section 4.1 proof of the quotient identification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In §4.1 the authors compute br|2m6 numerically to precision 10^-10 and \"look for denominators less than 40\"; the resulting s_i are claimed to lie exactly in Omega^1(Y) and to be W6-invariant. The subsequent proof that the displayed quadrics describe X(6,7)/<W6,W7> starts from \"the differential forms s1,...,s7 in Omega^1(Y)\" and uses Riemann-Hurwitz to make Y -> X a degree-2 quotient by an automorphism u, then proves u=W6. If the rounding only gives approximate invariance, the map is not W6-equivariant and the automorphism argument has no foundation. Unlike §4.3, where F_j(g)=O(q^180) and degree 4g-4=44 force exact vanishing, §4.1 provides no exact certificate for the s_i; the 5800-coefficient check can prove quadratic relations exact only after s_i are known to be exact forms. Thus the identification of the genus-7 equations with the intended modular quotient rests on an unverified numerical assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies quotient modular curves of Borel-Cartan type and their quotients by Atkin-Lehner type involutions. Its main theoretical contribution is Theorem 2.2, a generalization of Chen's isogeny theorem giving an explicit isogeny decomposition of the Jacobian of each quotient. Using this theorem together with LMFDB eigenvalue data, the authors compute point counts for many such curves over finite fields and identify 36 new records. The central arithmetic claims are that the genus-7 curve X(6,7)/<W6,W7> and the genus-12 curve X(156,1)/<W13> over F_{11^5} attain the Hasse-Weil-Serre bound, with explicit equations for both. The genus-12 identification is backed by an exact degree-bound argument, while the genus-7 identification relies on a numerical-to-exact rounding step in Section 4.1 that is not fully certified.","tokens_in":1269,"tokens_out":1249,"duration_ms":119061,"significance":"If the results are correct, this is a meaningful contribution to the curves-with-many-points literature: the two curves would be the first known genus-7 and genus-12 curves over F_{11^5} reaching the Serre bound, and the paper provides explicit equations for both. The generalization of Chen's isogeny theorem is independently useful, and the large tables in Appendices B and C give many new record lower bounds. A clear strength is that the computations are supported by a public Magma repository, and the genus-12 proof contains a genuine exact certificate. The main caveat is that the genus-7 identification rests on an unverified numerical bridge in Section 4.1; this is a correctness-risk concern rather than a demonstrated error, and it appears repairable by adding an exact verification step.","major_comments":[{"comment":"The identification of the displayed genus-7 canonical model with X(6,7)/<W6,W7> rests on an unproven numerical-to-exact step. The forms s1,...,s7 are obtained by solving a linear system to precision 10^-10 and rounding coefficients to rationals with denominators less than 40; the text then asserts, without proof, that these forms lie exactly in Omega^1(Y) and are exactly W6-invariant. The subsequent Riemann-Hurwitz degree argument and the automorphism argument proving u=W6 use this exactness as a hypothesis, so an approximate result would leave the displayed equations unconnected to the intended modular curve. This contrasts with Section 4.3, where F_j(g)=O(q^180) is forced to vanish exactly by the degree bound 4g-4=44. Please provide an exact certificate for each s_i, for example by verifying the linear relation br + sum_j c_j b_j = 0 on Fourier coefficients through an exact or provably sufficient computation, and state that certificate in the text.","section":"§4.1, paragraphs 'For the computation of S' and 'To turn clues into a proof'"},{"comment":"Even if the s_i were known to lie in Omega^1(Y), the claim that the ten quadratic equations are exactly satisfied by them is not proved in the text; the numerical computation with 5800 Fourier coefficients and precision 10^-10 does not by itself establish exact vanishing, and no degree-bound argument like the one in Section 4.3 is supplied. The same paragraph also asserts smoothness over Z[1/42] without describing the verification. Please add a certificate for the exact vanishing of each quadratic relation and a clear description of the smoothness check, or replace the numerical part of the proof by an exact computation.","section":"§4.1, paragraph after the ten displayed quadrics"}],"minor_comments":[{"comment":"The phrase 'no cusp forms of weight 2 and level 14 or less than 10' is confused; the intended statement is likely 'no cusp forms of weight 2 and level at most 14'.","section":"Example 3.5"},{"comment":"The coefficient '(4x^3+4x^2+2−1)y^8' contains '2−1', which should presumably be '1'; please correct this typo.","section":"§4.2, planar equation"},{"comment":"The operator |2m6 is defined only by reference to [Miy05]; a one-line definition in the text would improve readability.","section":"§4.1"},{"comment":"The injectivity of the reduction map Aut(X_Q) -> Aut(X_F5) is asserted in a single sentence; a short justification would help the reader trust that the Magma bound #Aut(X_F5) <= 2 indeed bounds the rational automorphism group.","section":"§4.1, final paragraph"},{"comment":"The GitHub repository is cited without a commit hash or version; since the paper relies on it for equations and for the proofs of several computational claims, please include a persistent version identifier.","section":"Data availability"}],"recommendation":"major_revision","confidential_remarks":"The central concern is the missing exactness certificate in Section 4.1; if the authors supply an exact verification of the forms and quadratic relations, the paper should be in good shape. I found no evidence of circularity: the point counts come from modular form eigenvalues, and the genus-12 proof has a genuine exact argument. I would not reject on the current evidence, but the numerical-to-exact bridge must be made auditable before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know up front. First, the main existence claims hold up: the quotient modular curves X(6,7)/<W6,W7> and X(156,1)/<W13> do have the stated genera and do meet the Serre bound over F_{11^5}, and the point counts come from modular form eigenvalues, not from the equations. Theorem 2.2 genuinely generalizes Chen-type isogeny decompositions to all quotient groups by Atkin-Lehner type involutions, and the 36 new table records are a concrete payoff. Second, the explicit equations for the genus-7 curve are shakier than the rest of the paper. The rounding from numerical linear algebra (precision 10^-10, denominators <40) to exact differential forms on Y is asserted rather than certified. The genus-12 equations have a real certificate: the q^180 vanishing plus the degree 4g-4=44 bound forces exact zero. The genus-7 section has no comparable Sturm-bound or exact modular-symbol argument, so the identification of those quadrics with X(6,7)/<W6,W7> is genuinely conditional. The stress-test note landed on the right spot here.\n\nThe paper does several things well besides the theorem. The worked examples make the point-count algorithm transparent, and the genus-12 planar model is backed by a rigorous vanishing argument. The automorphism-based strategy for proving the canonical model is a good idea, and the authors are honest about the approximation process. I would also credit the clean separation between existence, which relies on the isogeny theorem and known eigenvalue data, and the explicit-equation claims, which are more fragile.\n\nSoft spots, in proportion. The main one is the missing rigorous bridge for the s_i in §4.1. This is fixable, likely by replacing the numerical linear algebra with exact modular-symbol arithmetic and a Sturm-bound check, or by publishing a machine-checked certificate. Reproducibility is a smaller but real concern: the full enumeration code is not supplied as a single commit-hashed artifact, and the LMFDB data dependency is not pinned to a version. Those are workflow issues, not mathematical errors. The paper acknowledges the genus-12 curve already appeared numerically in [DLMS23]; the new content there is the explicit model and the maximality observation, and that is handled honestly.\n\nWho is this for: anyone working on curves with many points or explicit modular curves. The existence results and the isogeny theorem merit attention even if the genus-7 equation display is treated as provisional. With the §4.1 bridge tightened, the paper would be fully convincing. I would send it to a serious referee and ask for that tightening, or for a clear statement that the displayed equations are only conjectural models pending exact verification. The core mathematics deserves the referee time.","headline":"The existence results for maximal curves of genus 7 and 12 over F_{11^5} are solid, but the genus-7 equation identification has an unproven numerical-to-exact rounding step.","tokens_in":59085,"tokens_out":4388,"would_cite":true,"duration_ms":47456,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G20","11G18","11T71","14G35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that quotient modular curves of genus 7 and 12 over $\\mathbb{F}_{11^5}$ attain the Hasse-Weil-Serre bound, giving explicit equations and a generalized Chen isogeny.","keywords":["Hasse-Weil-Serre bound","maximal curves","many points","modular curves","Atkin-Lehner involutions","Chen isogeny","Cartan subgroups","canonical equations"],"falsifier":"Recompute the $W_6$-invariant subspace of $\\Omega^1(X(6,7)/\\langle W_7\\rangle)$ using exact modular symbols instead of $10^{-10}$ numerical evaluation; if its dimension is not 7 or the displayed quadrics are not exactly satisfied by the exact forms, the canonical-model identification fails. Independently, an enumeration of the $\\mathbb{F}_{11^5}$-points on the reduction of the displayed model must return exactly 166666 for genus 7 and 170676 for genus 12.","tokens_in":58128,"feed_emoji":"🔢","tokens_out":8530,"duration_ms":77114,"temperature":0.7,"pith_summary":"The paper proves that two smooth projective curves, of genus 7 and genus 12, exist over the field with $11^5$ elements and have exactly the number of points allowed by the Hasse-Weil-Serre bound: 166666 and 170676 respectively. Both curves are quotients of modular curves of Borel-Cartan type by Atkin-Lehner type involutions, and the paper gives explicit equations for them. The genus-7 curve is described by a canonical model in $\\mathbb{P}^6$ that is smooth over $\\mathbb{Z}[1/42]$, while the genus-12 curve is $X_0(156)/\\langle W_{13}\\rangle$. To compute the point counts, the paper generalizes Chen's isogeny theorem to all such quotients, expressing each Jacobian as a product of newform abelian varieties with explicit multiplicities. The same computation yields 36 improved lower bounds and several new or non-isomorphic maximal curves, and the appendices tabulate all results.","feed_headline":"Two explicit curves attain the Serre bound over F_11^5","feed_subtitle":"New point-count records of 166666 and 170676 points, with explicit equations for both curves.","key_machinery":"The load-bearing object is Theorem 2.2, a Chen-type isogeny for the whole family: for $Y=X(n_0,n_{ns})$ and $K$ a subgroup of its Atkin-Lehner type involutions, $\\operatorname{Jac}(Y/K)$ is isogenous over $\\mathbb{Q}$ to $\\prod_f A_f^{m_f}$, where the $f$ run over weight-2 newforms of levels $d_0 d_{ns}^2$ and $m_f$ is a product of explicit factors involving the Atkin-Lehner eigenvalues $\\varepsilon_{f,p^e}$ and the characters of $K$. This reduces every point count to modular-form data. For the equations, the method is Zywina's computation of a basis of $S_2(\\Gamma_H)$ followed by extracting the $W_6$-invariant subspace of $\\Omega^1$, applying Mercuri's canonical-embedding technique to find quadrics, and completing the identification by proving, through endomorphism rings of the Jacobian factors and absence of small-order automorphisms, that the hidden quotient involution is exactly $W_6$.","core_discovery":"The central claim is that the quotient $X(6,7)/\\langle W_6,W_7\\rangle$ has genus 7 and $166666$ points over $\\mathbb{F}_{11^5}$, and that $X_0(156)/\\langle W_{13}\\rangle$ has genus 12 and $170676$ points, both numbers equal to the Hasse-Weil-Serre upper bound $q+1+g\\lfloor 2\\sqrt{q}\\rfloor$ for $q=11^5$. The paper presents explicit equations for both: a system of quadrics giving the canonical model of the genus-7 curve, smooth over $\\mathbb{Z}[1/42]$, plus singular planar models for both curves. The existence proof is arithmetic: the generalized isogeny determines the Jacobian decomposition, the Hecke eigenvalues at 11 give the Frobenius eigenvalues, and the point counts follow. The equation proof is geometric: the canonically embedded model is computed from invariant differential forms and then certified by showing the candidate quotient automorphism equals $W_6$ on the Jacobian.","pith_inferences":["The numerical-to-exact certification used for the genus-7 model could be reused: any of the 36 record curves in Table 1 whose equations are not yet written down is a candidate for the same pipeline, turning counted quotients into explicit curves.","The appearance of maximality over an odd-degree extension suggests the point-producing mechanism is not simply supersingular elliptic curves rationalizing over even extensions; the Frobenius eigenvalue patterns of the specific newforms are doing the work, so similar searches over other odd-degree fields may yield more records.","The Appendix A lattice identities imply that point counts and genera of all quotients by subgroups of $(\\mathbb{Z}/2\\mathbb{Z})^r$ are determined by a small set of quotient curves; this could be used as a cheap pre-filter before running the isogeny computation on a large family."],"forward_implications":["For both new pairs $(g,q)=(7,11^5)$ and $(12,11^5)$, the Hasse-Weil-Serre bound is attained by curves with explicit equations; the genus-12 curve was previously known only as a point-counted quotient, and the genus-7 curve is new.","The generalized isogeny gives a uniform, equation-free method to compute $\\#X(\\mathbb{F}_q)$ for every quotient by Atkin-Lehner type involutions from Hecke eigenvalues alone.","The systematic search with $n_0 n_{ns}^2\\leq 10^4$ and genus $\\leq 50$ produces 36 new 'nice' lower bounds and several further maximal curves, including examples over odd-degree extensions such as $\\mathbb{F}_{11^5}$.","Some maximal curves with the same point count have different real Weil polynomials, so the family contains non-isomorphic maximal curves of the same genus and field."],"supporting_citations":[{"why":"Supplies the Jacobian isogeny decomposition and point-count algorithm that Theorem 2.2 generalizes; the genus-12 curve was already found there without equations.","marker":"[DLMS23]"},{"why":"Original Chen isogeny for non-split Cartan modular curves that the present result extends to Borel-Cartan quotients.","marker":"[Che98]"},{"why":"Earlier reformulation and proof of Chen's isogeny used as a step in the generalization.","marker":"[SE00]"},{"why":"Extends Chen's isogeny to normalizers of Cartan subgroups of level $p^n$, another precursor of Theorem 2.2.","marker":"[Che04]"},{"why":"Prior work on automorphisms and Chen-type isogenies for Cartan modular curves that the paper generalizes.","marker":"[DLM22]"},{"why":"Algorithm computing a basis of $S_2(\\Gamma_H)$, from which $\\Omega^1$ of the modular curves is obtained.","marker":"[Zyw21]"},{"why":"Method for passing from a basis of differential forms to quadratic equations of a canonical embedding.","marker":"[Mer18]"},{"why":"Endomorphism ring theorem for modular Jacobians, used to show the candidate automorphism commutes with $W_2$ and $W_6$ and equals $W_6$.","marker":"[Kan08]"},{"why":"Test establishing that the intermediate curve has no automorphisms of order 3, 4, or 5, a key step in the certification.","marker":"[Gon17]"},{"why":"Source of the newform q-expansions and Hecke eigenvalues needed for every point count in the search.","marker":"[LMFDB]"}],"fun_headline_variants":["Explicit genus-7 and genus-12 curves hit Serre bound over F_11^5","Two explicit curves reach Serre bound with 166666 and 170676 points","Explicit maximal curves of genus 7 and 12 over F_11^5","Genus 7 and 12 curves over F_11^5 set new point-count records"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the numerically computed invariant differential forms are exact and exactly satisfy the displayed quadrics; if that numerical-to-exact bridge failed, the equations would not necessarily describe the intended quotient curve.","fun_headline_variants_meta":{"raw":{"variants":["Explicit genus-7 and genus-12 curves hit Serre bound over F_11^5","Two explicit curves reach Serre bound with 166666 and 170676 points","Explicit maximal curves of genus 7 and 12 over F_11^5","Genus 7 and 12 curves over F_11^5 set new point-count records"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001115,"raw_usage":{"total_tokens":4601,"prompt_tokens":862,"completion_tokens":3739,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":478,"completion_tokens_details":{"reasoning_tokens":3645}},"tokens_in":478,"tokens_out":3739,"duration_ms":23481,"temperature":1.0,"reasoning_tokens":3645,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:07:32.540919+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the $W_6$-invariant subspace of $\\Omega^1(X(6,7)/\\langle W_7\\rangle)$ using exact modular symbols instead of $10^{-10}$ numerical evaluation; if its dimension is not 7 or the displayed quadrics are not exactly satisfied by the exact forms, the canonical-model identification fails. Independently, an enumeration of the $\\mathbb{F}_{11^5}$-points on the reduction of the displayed model must return exactly 166666 for genus 7 and 170676 for genus 12.","supporting_citations":[],"review_version":1}