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Ceresa Cycles of $X_{0}(N)$

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Pith's one-line read Ceresa cycle vanishes only on hyperelliptic X0(p)

desk verdict 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. read the letter →

arxiv 2501.14060 v2 pith:ZVQARAHM submitted 2025-01-23 math.AG math.NT

Elvira Lupoian , James Rawson This is my paper · ORCID
classification math.AGmath.NT MSC 14C2514G3511G18
keywords CeresacyclemodularcurvesJacobianvarietiesChowgrouphyperellipticHeckeoperatorsmodifieddiagonalChow-Heegnerdivisors
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

4 major / 6 minor

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.

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 (4)
  1. [§5.3, Proposition 24] 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.
  2. [§5.2–5.4, Tables 2–3 and computational scripts] 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.
  3. [§5.4, use of Proposition 9] 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.
  4. [§6.3, Period lattice computations] 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.
minor comments (6)
  1. [Table 2] 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)'.
  2. [§5.3, Proposition 23] 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.
  3. [§5.4] 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.
  4. [Proposition 17] 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.
  5. [§6.3] 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.
  6. [Proposition 11] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the shadow-point contrapositive chain is independent of its conclusions; the only flagged concern (Proposition 24) is an under-specified verification, not a circular reduction.

full rationale

The derivation chain is not circular. The paper proves non-vanishing of the Ceresa cycle through a contrapositive: Zhang's external theorem gives Cer(N)=0 implies the Gross-Kudla-Schoen diagonal class vanishes; Proposition 4, proved from the definition of the shadow and a push-forward calculation, then implies the associated shadow point Sh(phi) is torsion. The paper's actual work is showing that explicitly constructed shadows for Hecke operators T2, T3, and wp are non-torsion. Those shadow divisors are computed from fixed-point data of Hecke correspondences (Propositions 15-18 and Table 2), not from the desired non-vanishing statement. The non-torsion proofs use Mazur-Ogg torsion theorems, reduction to the special fibre, uniform class-number bounds, and finite computations with explicit CM j-invariants; the listed exceptional primes coincide with Ogg's hyperelliptic list rather than being fitted to force the theorem. Theorem 2 is a genuine deduction from Theorem 1 via Proposition 25 together with independent case checks for small prime powers. The one load-bearing step that is under-justified is Proposition 24 in Section 5.3: after checking collisions over 0 and infinity, the proof handles the five remaining branch points by saying 'Analysing the polynomial shows that again, this cannot happen for a suitable p' without exhibiting the polynomial or quantifying the exceptional primes. That is a completeness gap in an important irreducibility claim, and it is a correctness risk, but it is not circular: the monodromy criterion is cited from external work [4], and the irreducibility conclusion is not assumed as an input anywhere in the derivation. No self-citation chain, fitted-parameter-as-prediction, or definitional identification of the target with an input occurs.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central claim rests on well-established results (Zhang, Mazur, Ogg, Gross, Deligne) and on several in-paper computational and geometric assertions: the irreducibility of components of C1 modulo p (Proposition 24), the correctness of the Magma computations, and the period lattice approximations. No free parameters are fitted; the explicit bounds in Theorem 2 are derived.

assumptions (9)
  • standard math Zhang's Theorem 3: vanishing of the Ceresa cycle is equivalent to vanishing of the Gross-Kudla-Schoen diagonal cycle, and the base divisor is the canonical divisor scaled by 1/(2g-2).
    Cited from [44]; this is the external bridge that lets the authors detect Ceresa vanishing via shadow points.
  • standard math Mazur's Theorem 7 and Theorem 8: J0(p)(Q)_tors is generated by c0-c∞, and J0^+(p)(Q)_tors is trivial for positive genus levels.
    Used in Corollary 19 to show that nonzero image in J0^+(p)(Q) implies the shadow point has infinite order.
  • standard math Ogg's Theorem 6: the cuspidal divisor c0-c∞ has order (p-1)/gcd(p-1,12) in J0(p)(Q).
    Used together with Mazur's theorem in the proof of Proposition 11 to pin down the canonical divisor.
  • standard math Manin-Drinfeld theorem: divisors supported only on cusps are torsion in the Jacobian.
    Invoked in Sections 6.1 and 6.3 to compare shadow points with Heegner divisors and to replace cuspidal components by torsion.
  • standard math Gross's Theorem 27: Heegner divisors DP are of infinite order in J0(N)(Q).
    Used in Section 6.1 to conclude that scaled shadows differing from Heegner divisors by cuspidal divisors are non-torsion.
  • standard math Deligne's proof of the Weil conjectures gives the bound |C b_n| ≤ d(n) n^{1/2} for Fourier coefficients.
    Used in Section 6.3 to control the truncation error in the period lattice computations.
  • domain assumption Deligne-Rapoport and Xue describe the special fibers of the minimal regular models of X0(p) and X0^+(p) over Fp.
    Theorems 13 and 14 are cited and used to justify the reduction map and the component group being trivial.
  • ad hoc to paper Proposition 24: the irreducible components of C1 remain irreducible modulo p for all relevant primes in the p≡1 mod 12 classes.
    Asserted with a sketched monodromy/Riemann-Hurwitz argument; load-bearing for the Bezout bound in the p≡1 mod 12 case.
  • ad hoc to paper The Magma computations and period lattice approximations are correct.
    The finite exception checks and non-membership in the period lattice are verified by computer code whose correctness is not formally established.

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Pith. "Pith review of Ceresa Cycles of $X_{0}(N)$." pith.science (2026). https://pith.science/paper/ZVQARAHM

@misc{pith2026250114060,
  author       = {Pith},
  title        = {Pith review of: Ceresa Cycles of $X_0(N)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZVQARAHM}},
  note         = {Machine review of arXiv:2501.14060}
}
abstract

The Ceresa cycle is an algebraic 1-cycle on the Jacobian of an algebraic curve. Although it is homologically trivial, Ceresa famously proved that for a very general complex curve of genus at least 3, it is non-trivial in the Chow group. In this paper we study the Ceresa cycle attached to the complete modular curve $X_{0}(N)$ modulo rational equivalence. For prime level $p$ we give a complete description, namely we prove that if $X_{0}(p)$ is not hyperelliptic, then its Ceresa cycle is non-torsion. For general level $N$, we prove that there are finitely many $X_{0}(N)$ with torsion Ceresa cycle. Our method relies on the relationship between the vanishing of the Ceresa cycle and Chow-Heegner points on the Jacobian. We use the geometry and arithmetic of modular Jacobians to prove that such points are of infinite order and therefore deduce non-vanishing of the Ceresa cycle.

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