{"id":"26c5986c-566f-4966-a75d-e795778479db","arxiv_id":"2501.14060","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For prime level p, the Ceresa cycle of X0(p) is nontrivial if and only if X0(p) is not hyperelliptic; for general N, only finitely many X0(N) can have a vanishing Ceresa cycle.","lead":"For prime levels p, the Ceresa cycle of the modular curve X0(p) is shown to be non-torsion exactly when the curve is not hyperelliptic. The proof uses shadow points on modular Jacobians and shows that there are only finitely many composite levels N with a vanishing Ceresa cycle.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 24's irreducibility claim for C1 modulo p is under-proved; if it fails for some p in the class p ≡ 1 mod 24 and p ≡ 1,2,4 mod 7, the Bezout bound and the complete classification in Theorem 1 collapse.","rationale":"The reader's weakest-assumption identification is accurate. Proposition 24 is the only step where a statement needed for infinitely many primes is justified by a vague monodromy assertion and an unspecified computer analysis. The rest of the proof is a combination of standard results (Mazur, Ogg, Gross-Zagier) and finite computations, which are reproducible from the linked repository. The finite exceptions in Tables 2 and 3, the explicit bounds, and the period-lattice computations give independent support. Thus the central theorem is likely true, but the proof as written has a genuine gap: if the irreducibility assertion in Proposition 24 fails, the derived Bezout bound and hence the class-number bound do not follow, so the claimed complete classification for prime levels is not established. The proposed computational check would settle this. Since the reader already conditioned on this point, the verdict remains conditional.","tokens_in":22264,"tokens_out":24648,"duration_ms":217054,"concrete_test":"Use a computer algebra system to compute the branch points of f1, in particular the five branch points in the degree-5 orbit. For each such branch point β, compute the fiber polynomial f1(x)-β and its discriminant (or the resultant with f1'(x)) to detect collisions of ramification points above β. Take the product of these discriminants over the five β's, together with the discriminant of the degree-5 polynomial defining the β-orbit, and factor the resulting fixed integer. Verify that no prime p ≡ 1 mod 24 with p ≡ 1,2,4 mod 7 divides this integer. If none do, the collision condition in Proposition 24 holds for every relevant prime. As an additional sanity check, for each prime in the class up to the numerical bound (244897), directly compute the number of geometrically irreducible components of C1 over F_p and compare with the number over Q (expected to be 2).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 5.3, Proposition 24 is the load-bearing step for the hardest residue class in Theorem 1. The argument that both 6 Sh(T2) and 6 Sh(T3) cannot reduce to zero for large p uses Proposition 23, whose conclusion #C1∩C2 ≥ g(X0^+(p)) is only useful if #C1∩C2 is bounded by Bezout. That requires C1 and C2 to have no common component modulo p, i.e. that the irreducible components of C1 over Q remain irreducible modulo every p≡1 mod12, p≡1 mod8, p≡1,2,4 mod7. Proposition 24 asserts exactly this. Its proof cites [4, Lemma 2] for a monodromy reduction criterion and states two hypotheses: p does not divide the order of any monodromy element, and no ramification points above a fixed branch point collide modulo p. For the first, the set has size 80, so any monodromy order has prime factors ≤80; since every p in the residue class exceeds 80, this is automatic, although the paper does not say so. For the second, the paper checks collisions above 0 and ∞ by factoring differences of the relevant points, but for the five remaining branch points (a degree-5 orbit) it only says 'if there were a collision ... this would violate the Riemann-Hurwitz formula ... unless two of the branch points also collided. Analysing the polynomial shows that again, this cannot happen for a suitable p.' No polynomial, no resultant, and no list of exceptional primes is provided. 'Suitable p' is not a quantifier: the argument must hold for every prime in the congruence class, including infinitely many beyond the numerical bound. If C1 splits modulo some such p, the Bezout bound fails, the class-number inequality has an extra term, and the finite verification could miss a prime for which both shadows are torsion, invalidating the 'if and only if' classification.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Ceresa cycle of the complete modular curve X0(N), viewed as an element of the rational Chow group CH^1(J0(N)) with the canonical half-degree basepoint. The main result, Theorem 1, asserts a complete dichotomy for prime level p: Cer(p) is non-zero if and only if X0(p) is not hyperelliptic, i.e. for p>71 or p in {43,53,61,67}. Theorem 2 asserts that there are only finitely many N with Cer(N)=0 and gives an explicit numerical bound. The proof follows the Zhang-type route: vanishing of the Ceresa cycle implies vanishing of the Gross-Kudla-Schoen diagonal, which in turn forces certain 'shadow' divisors, obtained by intersecting the diagonal cycle with Hecke correspondences, to be torsion. The bulk of the paper is devoted to proving that carefully chosen shadows from T2, T3, and the Atkin-Lehner involution are non-torsion, using reduction modulo p, supersingular j-invariants, CM orders, Mazur's torsion theorems, class-number bounds, and a substantial amount of Magma computation.","tokens_in":22620,"tokens_out":15736,"duration_ms":144161,"significance":"If the arguments are correct, this is a strong and interesting result: it gives the first complete classification of non-vanishing for a natural infinite family of modular curves, and it reduces an arithmetic cycle question to explicit non-torsion statements in Jacobians. The paper also provides a useful framework connecting Ceresa classes, Chow-Heegner points, and triple-product L-functions, and it includes reproducible code for the computational parts. The main theorems go considerably beyond the recent results of Kerr-Li-Qiu-Yang and are stated as a sharp dichotomy for prime levels. However, the full classification rests on several large computational checks and on an irreducibility statement modulo p whose proof is only sketched; these points need to be made verifiable before the main theorem can be considered established.","major_comments":[{"comment":"The proof of Proposition 24 is not complete. The argument that the irreducible components of C1 remain irreducible modulo every prime in the relevant infinite congruence classes is load-bearing: Proposition 23's conclusion #C1(Fp)∩C2(Fp) ≥ g(X0^+(p)) is only useful when coupled with the Bezout bounds 79^2 and 79×129, which in turn require that no new components appear over Fp. For the first reduction hypothesis, the set has size 80, so p does not divide any monodromy order automatically for the primes in question, but the paper does not say this. For the second hypothesis, collisions above 0 and ∞ are checked, but for the remaining five branch points the text only says that a collision would violate Riemann-Hurwitz unless branch points also collided, and that 'analysing the polynomial' rules this out for a suitable p. No polynomial, resultant, or finite list of exceptional primes is supplied. The quantifier 'suitable p' is not acceptable because the conclusion must hold for every prime in the residue class. This step must be either proved generally or documented with the actual computations and exceptional primes.","section":"§5.3, Proposition 24"},{"comment":"The finite verification that completes the classification is not reported in the paper. Table 3 gives the CM orders and some exceptional primes, but the text does not give the full output of divisor checker.m, primes1T2.m, or primes11AL.m. In particular, the claims 'the exceptions can all be seen to be primes where X0(p) is hyperelliptic' and 'applying Proposition 21 shows that for all of them, 6Sh(T2) is non-zero' are assertions about computations whose results are not listed. Since these checks are used to rule out infinitely many primes by reducing to a finite list, the complete list of primes and the chosen supersingular j-invariant for each should appear in the paper or an appendix. A reader should not have to rerun the scripts to verify the main theorem.","section":"§5.2–5.4, Tables 2–3 and computational scripts"},{"comment":"The step from torsion of Sh(wp) to the inequality h(-4p)+h(-p) ≥ p/96 is too quick. If a divisor class is only torsion, Proposition 9, which concerns non-trivial sections of O(d c∞), yields a bound on a multiple of the cusp coefficient, not directly on the coefficient itself. The argument must explicitly use Theorem 8: J^+(p)(Q)tors is trivial under the stated hypotheses, so the projected class is zero rather than merely torsion. This is likely harmless, but as written the logic is incomplete and should be stated.","section":"§5.4, use of Proposition 9"},{"comment":"The numerical verification for the eight levels in §6.3 is load-bearing for Theorem 2, but the quantitative details are only sketched. The inequality '|Cb_n| ≤ d(n)n^{1/2}' contains an unspecified constant C, the geometric series used to bound the tail of the Fourier expansion is not written down, and the tolerance used in the Magma Periods() computation and in the linear-algebra check that B(f)D̃∉Λ_f is not specified. To make the computation rigorous, the paper should state the exact error bound, the chosen truncation m for each level, and the precision to which the period lattice is computed.","section":"§6.3, Period lattice computations"}],"minor_comments":[{"comment":"In the row for p≡5 mod 12, p≡1 mod 8, p≡1,2,4 mod 7, the entry '6δT2' appears to be a typo for '6Sh(T2)'.","section":"Table 2"},{"comment":"The proof states that there are g(X0^+(p)) such pairs of supersingular j-invariants, but it does not justify why the number of non-rational supersingular pairs equals the genus of the quotient. This equality should be explained or referenced.","section":"§5.3, Proposition 23"},{"comment":"The sentence 'A suitable j-invariant cannot be found for the bi-elliptic X0(p)' refers to a class of curves without giving the list or a citation. The bielliptic primes should be identified, for example from [2], so the reader can see which cases are being handled individually.","section":"§5.4"},{"comment":"In the proof for ℓ≥5, the text refers to the regular model of X0(p) modulo ℓ and to CM by Z[√−p], which should presumably be X0(ℓ) and Z[√−ℓ]. As written this is confusing and appears to reduce the wrong curve.","section":"Proposition 17"},{"comment":"There are several typographical slips: 'N∈{35,172,192,292,372,412}' is followed by '292,312412' with a missing comma, and the phrase 'a positive-rank newform f : of dimension 1' contains a stray colon. These should be corrected.","section":"§6.3"},{"comment":"In the proof of the canonical divisor formula, the symmetric form is written as p−11/2(c∞+c0)−8D3−3D4, but the preceding computation gives −4D3−3D4; the coefficient 8 appears to be a typo.","section":"Proposition 11"}],"recommendation":"major_revision","confidential_remarks":"This is a substantial paper and the main theorem is likely correct, but the refereeing bottleneck is verifiability rather than plausibility. In particular, Proposition 24 in §5.3 is the weakest link: the hard residue class in Theorem 1 collapses if the irreducibility claim fails for any prime in the class. The gap seems repairable by providing the missing polynomials/resultants or a genuine monodromy argument, and by documenting the finite computational outputs, so I would not recommend rejection. I would ask the authors to supply full data for the finite checks before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, this is a strong paper with a real new result. The authors completely settle the Ceresa cycle for X0(p) for prime level: non-vanishing exactly when the curve is not hyperelliptic. They also prove finiteness for all levels N, recovering and extending the recent Kerr-Li-Qiu-Yang theorem on Γ_N. The novelty claim holds up; prior work treated Γ_N, not the full X0(N). The shadow point method is clean, the reduction arguments are mostly convincing, and the proof of Theorem 2 via covers plus explicit computations for small primes looks sound. Section 7 is a nice bonus connecting shadows to triple product L-functions with explicit examples.\n\nThe stress-test concern about Proposition 24 is legitimate. That proposition is load-bearing: it asserts that the irreducible components of C1 stay irreducible modulo every prime in p ≡ 1 mod 12, p ≡ 1 mod 8, p ≡ 1,2,4 mod 7, without which the Bezout bound on #C1∩C2 fails and the class number inequality could miss primes. The proof checks the branch points 0 and ∞ by showing differences of roots are not divisible by primes in the relevant classes, but for the degree-5 orbit it only says 'analysing the polynomial shows that again, this cannot happen for a suitable p.' That is not a proof for every prime in the class. The missing resultant computation has to be written down, or the exceptional primes explicitly listed and excluded. If a component did split modulo some larger prime, the whole classification argument would collapse.\n\nThe other concerns are minor. The computational checks are only identified by file names, with no versioned scripts or hashes; a referee will want those pinned down. The paper's own admission that the method needs positive rank Jacobians is honest, and they go out of their way to handle exceptions like X0(34).\n\nNet: the arithmetic core looks solid, the main theorem is significant, and the gap is isolated and likely fixable. Anyone working on Ceresa cycles or Chow-Heegner points on modular curves will want to read it. I would send this to a serious referee, with the explicit request to get Proposition 24 fully proved and to specify the computational artifact.","headline":"Substantive and likely correct; the iff classification for X0(p) is new and important, but Proposition 24's irreducibility claim needs a complete proof for all primes in the class.","tokens_in":23161,"tokens_out":3947,"would_cite":true,"duration_ms":33211,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C25","14G35","11G18"],"pacs":[],"model":"deepseek-v4-flash","headline":"Ceresa cycle vanishes only on hyperelliptic X0(p)","keywords":["Ceresa cycle","modular curves","Jacobian varieties","Chow group","hyperelliptic curves","Hecke operators","modified diagonal cycle","Chow-Heegner divisors"],"falsifier":"For a prime $p\\equiv 1\\pmod{12}$ with $p\\equiv 1\\pmod 8$ and $p\\equiv 1,2,4\\pmod 7$, compute the reduction of the auxiliary curve $C_1$ modulo $p$ (equivalently, the reduction of the Galois group of $f_1$) and check whether any irreducible component splits. If some component splits, the bound $\\#C_1\\cap C_2\\le 79^2$ or $79\\times 129$ could fail, and there might be a prime in this class for which both $6\\,\\mathrm{Sh}(T_2)$ and $6\\,\\mathrm{Sh}(T_3)$ are torsion while $\\mathrm{Cer}(p)=0$, directly contradicting Theorem 1. The paper's computations already implicitly rule this out for all primes up to 150889 or 244897 in the relevant classes; the test is to extend that check or to search for a non-reduced fibre in the listed congruence classes.","tokens_in":2142,"feed_emoji":"🔺","tokens_out":2669,"duration_ms":94847,"temperature":0.7,"pith_summary":"This paper aims to settle when the Ceresa cycle, an algebraic 1-cycle on the Jacobian that is homologically trivial but often non-trivial in the Chow group, vanishes for the modular curves $X_0(N)$. For prime levels the answer is complete: the cycle is zero precisely when $X_0(p)$ is hyperelliptic, and non-torsion for every other prime level. For composite levels, the paper proves that only finitely many $X_0(N)$ can have a torsion Ceresa cycle, and it gives an explicit numerical bound beyond which the cycle is guaranteed non-zero. This matters because it converts a delicate Chow-group question into a check that carefully chosen rational points on the Jacobian are of infinite order, and because it supplies new infinite-order points and evidence for conjectures connecting such cycles to triple-product $L$-functions.","feed_headline":"Ceresa cycle vanishes only on hyperelliptic X0(p)","feed_subtitle":"Completes the prime-level classification and leaves only finitely many composite-level cases to check.","key_machinery":"The central object is the shadow of an endomorphism $\\varphi$ of the Jacobian of a curve $C$, defined as $$\\mathrm{Sh}(\\varphi) = (2g-2)F_\\varphi - \\deg(F_\\varphi)K_C - \\varphi(K_C) - \\varphi^\\vee(K_C) + (\\deg\\varphi+\\deg\\varphi^\\vee)K_C,$$ where $F_\\varphi$ is the fixed-point divisor and $K_C$ the canonical divisor; this is a scaled Chow-Heegner divisor. The key mechanism is Proposition 4: if the Ceresa cycle is zero, then $[\\mathrm{Sh}(\\varphi)]$ is torsion in the Jacobian. For $X_0(p)$ the paper chooses $\\varphi$ to be $T_2$, $T_3$, or the Atkin-Lehner involution $w_p$ depending on the congruence class of $p$, pushes the shadow to the quotient $X_0^+(p)$, and shows its reduction modulo $p$ is non-zero in $\\mathrm{Pic}^0$ of the special fibre. A covering argument then extends prime-level non-vanishing to composite levels.","core_discovery":"The paper proves that for every prime $p$, the Ceresa cycle $\\mathrm{Cer}(p)\\in \\mathrm{CH}^1(J_0(p))$ is non-zero if and only if $X_0(p)$ is not hyperelliptic; equivalently, it is non-trivial for all $p>71$ and for $p\\in\\{43,53,61,67\\}$. For general level $N$, it proves that $\\mathrm{Cer}(N)$ is non-zero whenever $N>25\\times 3^4\\times 5^2\\times 7^2\\times \\prod_{11\\le p\\le 71,\\,p\\notin\\{43,53,61,67\\}} p$, so only finitely many levels can have torsion Ceresa cycle. The engine is a contrapositive: if the Ceresa cycle vanishes, then every shadow point attached to an endomorphism of the Jacobian must be torsion; the paper constructs a shadow point from a Hecke operator or the Atkin-Lehner involution, computes it explicitly, and uses reduction modulo $p$, the known structure of rational torsion, class-number bounds, and finite computations to show the point has infinite order.","pith_inferences":["A natural next step is to compute the Ceresa cycle for the finitely many composite levels below the bound of Theorem 2; the paper already handles several low-genus and rank-one cases, and a complete computation would turn the finiteness statement into an exact classification.","The shadow mechanism suggests a general criterion: any curve whose Jacobian admits a correspondence with a non-torsion shadow has non-vanishing Ceresa cycle. Curves with Hecke correspondences, Shimura curves, or more general arithmetic correspondences are immediate candidates for this method.","Because the proof requires positive Mordell-Weil rank, rank-zero cases such as $X_0(64)$ are outside its scope; the framework indicates that separate invariants, such as algebraic equivalence or algorithmic certification, are the natural complement for those curves.","The congruence-class branching in Section 5.3 could be tested directly: checking the monodromy irreducibility of $C_1$ modulo all primes in the listed classes would either confirm the Bezout bound or expose a prime where the classification could miss a case."],"forward_implications":["For every prime $p$, hyperellipticity is an exact obstruction: $\\mathrm{Cer}(p)\\neq 0$ for $p>71$ and for $p=43,53,61,67$, while all other prime levels have vanishing Ceresa cycle.","Only finitely many levels $N$ can have vanishing Ceresa cycle; every $N$ larger than the explicit product in Theorem 2 has $\\mathrm{Cer}(N)\\neq 0$ in $\\mathrm{CH}^1(J_0(N))$.","The proof constructs new rational points of infinite order on $J_0(N)$—the shadow points of Hecke operators—whenever the Ceresa cycle is non-vanishing.","For bielliptic prime levels, non-vanishing of specific Hecke components of the modified diagonal cycle implies non-vanishing of the associated triple-product $L$-function derivatives, giving evidence for the Gross-Kudla conjecture on heights and triple-product $L$-values.","Together with the covering statement, the prime-level theorem recovers and extends previously known non-vanishing results for certain families of modular curves."],"supporting_citations":[{"why":"Introduces the Ceresa cycle and proves that very general curves of genus at least 3 have non-trivial Ceresa cycle, establishing the phenomenon studied here.","marker":"[9]"},{"why":"Proves the equivalence between vanishing of the Ceresa cycle and vanishing of the modified diagonal cycle, the bridge on which the whole argument rests.","marker":"[44]"},{"why":"Provides the rational torsion subgroup of $J_0(p)$ and $J_0^+(p)$, used to upgrade non-zero reduction to infinite order.","marker":"[33]"},{"why":"Computes the order of the difference of the two cusps, needed for the canonical divisor formulas and torsion comparisons.","marker":"[34]"},{"why":"Lists the hyperelliptic modular curves, identifying exactly which prime levels are excluded in Theorem 1.","marker":"[35]"},{"why":"Describes the special fibre of the minimal regular model of $X_0(p)$, used for reduction modulo $p$.","marker":"[14]"},{"why":"Describes the minimal resolution of the Atkin-Lehner quotient, giving the special fibre used for the reduction map to $\\mathrm{Pic}^0$.","marker":"[42]"},{"why":"Establishes finiteness of cuspidal divisor classes, used to ignore cusp-supported differences in composite-level arguments.","marker":"[32]"},{"why":"Gives the companion cuspidal torsion result, also used to reduce shadow points modulo cusp-supported divisors.","marker":"[17]"},{"why":"Supplies the algorithmic framework and shadow terminology used to certify non-vanishing in low-genus cases.","marker":"[18]"}],"fun_headline_variants":["Non-torsion Ceresa for all non-hyperelliptic X0(p)","Prime-level Ceresa: torsion iff hyperelliptic","Ceresa cycle torsion only for hyperelliptic X0(p)","Full prime-level classification: non-torsion unless hyperelliptic"],"cache_read_input_tokens":25216,"weakest_assumption_plain":"The fragile step is the claim that the two auxiliary curves $C_1$ and $C_2$ used to bound the genus do not break into more components when reduced modulo certain primes; if that failed, the degree-based Bezout intersection bound would not hold and some prime could escape the classification.","fun_headline_variants_meta":{"raw":{"variants":["Non-torsion Ceresa for all non-hyperelliptic X0(p)","Prime-level Ceresa: torsion iff hyperelliptic","Ceresa cycle torsion only for hyperelliptic X0(p)","Full prime-level classification: non-torsion unless hyperelliptic"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001422,"raw_usage":{"total_tokens":5749,"prompt_tokens":961,"completion_tokens":4788,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":577,"completion_tokens_details":{"reasoning_tokens":4713}},"tokens_in":577,"tokens_out":4788,"duration_ms":33773,"temperature":1.0,"reasoning_tokens":4713,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T15:24:54.415566+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a prime $p\\equiv 1\\pmod{12}$ with $p\\equiv 1\\pmod 8$ and $p\\equiv 1,2,4\\pmod 7$, compute the reduction of the auxiliary curve $C_1$ modulo $p$ (equivalently, the reduction of the Galois group of $f_1$) and check whether any irreducible component splits. If some component splits, the bound $\\#C_1\\cap C_2\\le 79^2$ or $79\\times 129$ could fail, and there might be a prime in this class for which both $6\\,\\mathrm{Sh}(T_2)$ and $6\\,\\mathrm{Sh}(T_3)$ are torsion while $\\mathrm{Cer}(p)=0$, directly contradicting Theorem 1. The paper's computations already implicitly rule this out for all primes up to 150889 or 244897 in the relevant classes; the test is to extend that check or to search for a non-reduced fibre in the listed congruence classes.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Ceresa cycle and proves that very general curves of genus at least 3 have non-trivial Ceresa cycle, establishing the phenomenon studied here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves the equivalence between vanishing of the Ceresa cycle and vanishing of the modified diagonal cycle, the bridge on which the whole argument rests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the rational torsion subgroup of $J_0(p)$ and $J_0^+(p)$, used to upgrade non-zero reduction to infinite order."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Computes the order of the difference of the two cusps, needed for the canonical divisor formulas and torsion comparisons."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Lists the hyperelliptic modular curves, identifying exactly which prime levels are excluded in Theorem 1."},{"cited_title":"Deligne and M","cited_arxiv_id":null,"evidence_quote":"Describes the special fibre of the minimal regular model of $X_0(p)$, used for reduction modulo $p$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Describes the minimal resolution of the Atkin-Lehner quotient, giving the special fibre used for the reduction map to $\\mathrm{Pic}^0$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes finiteness of cuspidal divisor classes, used to ignore cusp-supported differences in composite-level arguments."},{"cited_title":"Ellenberg, A","cited_arxiv_id":null,"evidence_quote":"Supplies the algorithmic framework and shadow terminology used to certify non-vanishing in low-genus cases."}],"review_version":1}