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Quadratic Chabauty for Atkin-Lehner quotients of modular curves via weakly holomorphic modular forms: Hodge Filtrations

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper gives an algorithm that computes the Hodge filtration on X_0^+(N) from q-expansions of modular forms instead of a plane model.

desk verdict Useful model-free Hodge filtration algorithm, but the 'always works' claim is proved only for j=d-1 while the algorithm uses minimal j_dR; the worked examples don't cover the proved case. read the letter →

arxiv 2509.02291 v1 pith:XJAMDJRP submitted 2025-09-02 math.NT

Isabel Rendell This is my paper · ORCID
classification math.NT MSC 11F1111F2511G1814G05
keywords quadraticChabautyHodgefiltrationAtkin-Lehnerquotientsweaklyholomorphicmodularformsq-expansionsdeRhamcohomologyp-adicheightsX_0^+(N)
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 aims to make quadratic Chabauty computations of rational points on the modular curves X_0^+(N) independent of an explicit plane model of the curve. It does this by giving an algorithm, Algorithm 3.8, that computes the Hodge filtration on the universal connection AZ using only q-expansions of weight-2 and weight-12 cusp forms in the plus eigenspace of the Atkin-Lehner involution. A key step is a q-expansion basis of de Rham cohomology built by multiplying holomorphic differentials by a quotient of weight-12 cusp forms; the paper proves such a quotient exists and verifies the basis property numerically in the worked examples. This matters because plane models become impractical as the genus grows, and the Hodge filtration is the main input for the p-adic heights that define the finite set of p-adic points containing the rational points.

What carries the argument

The central object is the Hodge filtration matrix λ_Fil, a (2g+2)-block lower-triangular unipotent matrix recording how the Hodge filtration sits inside the universal connection AZ. The innovation is to build the de Rham basis and λ_Fil from weakly holomorphic modular forms: the function f_dR = s_{d−j_dR}/s_d, a quotient of two weight-12 plus-eigen cusp forms, has poles only at ∞ and supplies the missing g differentials; Serre's cup-product formula verifies the basis; the Hecke operator T_p, computed on q-expansions, produces the correspondence Z and the connection Λ; solving the gauge equation G_∞^{-1} dG_∞ = Λ yields ψ_{αβ}, and the universal property determines β_Fil and γ_Fil by the inte

What would settle it

For a prime N not among 67, 97, 193 with genus(X_0^+(N)) ≥ 2, compute the minimal j_dR from (14), form {ω_i, (s_{d−j_dR}/s_d)ω_i}, and evaluate the exact cup product determinant; a zero determinant would refute the claim that the algorithm always produces a basis of H^1_dR.

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Extended reading notes

Core claim

The paper claims that for prime N with genus(X_0^+(N)) ≥ 2, the matrix λ_Fil — the data of the Hodge filtration on the rank-(2g+2) unipotent bundle with connection AZ — can be computed from modular form q-expansions alone. Algorithm 3.8 obtains a basis of H^1_dR(X/Q) as {ω_0,...,ω_{g−1}, f_dR ω_0,..., f_dR ω_{g−1}}, where the ω_i come from S_2^+(Γ0(N)) and f_dR = s_{d−j_dR}/s_d is a quotient of weight-12 plus-eigen cusp forms with poles only at the cusp ∞. It then computes the Hecke operator T_p and the nice correspondence Z exactly, solves for the gauge matrix ψ_{αβ}, and determines β_Fil and γ_Fil, with γ_Fil built from the same family of quotients. The worked examples include genus 2 (N=6

Load-bearing premise

The algorithm chooses the smallest j_dR satisfying the valuation condition (14), but the paper proves the basis property only for j = d−1; for the minimal j_dR the invertibility of the cup product matrix is checked numerically, not proved.

Editorial extensions

If this is right

  • Algorithm 3.8 computes λ_Fil for any prime N with genus(X_0^+(N)) ≥ 2 using only q-expansions of S_2^+(Γ0(N)) and S_12^+(Γ0(N)), eliminating the plane model as an input.
  • Because T_p is computed as exact rational numbers rather than p-adic approximations, the same Hodge filtration computation can be reused for multiple primes p.
  • The method verifiably handles genus 7: for X_0^+(193), the q-expansion cup product matrix has nonzero determinant, yielding a symplectic basis, T_3, Z, β_Fil, and γ_Fil.
  • The congruence experiments for N=67 and N=97 show a corank difference of g−1 = ρ(J)−1 between matrices built from iterated integrals of plus-eigen cusp forms and single integrals of minus-eigen cusp forms, indicating a systematic relation.
  • This is the first step toward a model-free quadratic Chabauty algorithm; the remaining step is the matching Frobenius-structure (λ_ϕ) computation.

Reading between the lines

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

  • If the basis construction extends to every j satisfying (14), the same algorithmic pattern would likely apply to other modular curves with a single cusp; for curves with several cusps the single valuation condition would become a system of pole conditions.
  • The corank difference g−1 in the congruence experiments suggests that the iterated integrals of plus-eigen cusp forms and the single integrals of minus-eigen cusp forms are related by a quotient of rank g−1, possibly a manifestation of the correspondence Z; this could be tested for further primes.
  • The open gap between the proven choice j = d−1 and the algorithm's minimal j_dR could be closed by proving the cup-product determinant is nonzero for every j satisfying (14), or by a deterministic criterion for choosing j; the algorithm's speed depends on using the minimal choice.
  • Since the Hecke matrix is computed exactly and rationally, the choice of prime p can be optimized after the cohomology is computed, unlike model-based approaches where p is effectively fixed when computing Frobenius.
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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

3 major / 4 minor

Summary. The paper presents an algorithm (Algorithm 3.8) for computing the Hodge filtration on the universal connection AZ for Atkin–Lehner quotients X_0^+(N), N prime, genus at least 2, using only q-expansions of the plus eigenspaces S_2^+(Γ_0(N)) and S_12^+(Γ_0(N)), thereby avoiding an explicit plane model. The first half of the de Rham basis is obtained from holomorphic differentials ω_i attached to S_2^+, and the second half from products s_{d-j_dR}/s_d · ω_i, where s_i are ordered basis elements of S_12^+ and j_dR is the smallest integer satisfying the valuation condition (14). The algorithm then computes T_p, the nice correspondence Z, the gauge matrix ψ_{αβ}, and finally β_Fil and γ_Fil. Worked computations are given for X_0^+(67) (genus 2) and X_0^+(193) (genus 7), together with corank computations for N=67,97,193 suggesting congruences between iterated integrals in the plus eigenspace and single integrals in the minus eigenspace.

Significance. If the correctness gap described below is closed, this is a useful technical advance: it replaces a plane-model input with modular-form data, extends the worked range to genus 7, and gives an explicit, checkable route to a Hodge filtration for quadratic Chabauty. The paper provides many detailed q-expansion matrices and cup-product computations, so the numerical content is concrete and partly verifiable. The empirical congruence observations in §4 are interesting, though they are presented as evidence rather than as a proven theorem. The main theoretical weakness is that the basis-producing step used in the actual algorithm is only verified numerically, not proved.

major comments (3)
  1. [§3.1, Corollary 3.5 and Algorithm 3.8(3)] The central algorithmic claim is not proved for the choice made in Step 3. Proposition 3.2 proves existence of a j satisfying (14) by taking j=d-1, and the argument that the construction 'always works' in §3.1 explicitly uses fdR=s1/sd. Algorithm 3.8 instead chooses jdR to be the smallest j satisfying (14). The paper itself concedes after Corollary 3.5 that the basis property for this minimal choice is checked only numerically in §4. For N=67, jdR=1 while d-1=31, and for N=193, jdR=3 while d-1=89, so the worked examples do not exercise the proved case at all. If a prime N admitted a minimal jdR satisfying (14) but producing a degenerate cup product, Step (7) would not be defined and the claimed λ_Fil would not exist. The algorithm should either be changed to take jdR=d-1, or a proof must be supplied that the minimal admissible j inherits nondegeneracy of the cup product.
  2. [§3.2, Algorithm 3.8(7)] The normalization γ_Fil(b)=0 requires a rational point b on X in the affine open Y=X\{∞}, but Algorithm 3.8 does not list b (or the hypothesis X(Q)≠∅) among its inputs, and the text merely says 'we can take b to be the upper half plane representative of a rational CM point of X'. No proof is given that such a point exists for every N in the stated range, and the worked examples do not identify the point used. Since the uniqueness of γ_Fil depends on this normalization, the algorithm as stated is incomplete. Please state the nonemptiness assumption explicitly and, for the examples, specify the point b or explain how γ_Fil was normalized.
  3. [§3.1, after (20)] Even for the value j=d-1 covered by Proposition 3.2, the paper does not give a complete proof that {ω_0,...,ω_{g-1}, (s1/sd)ω_0,...,(s1/sd)ω_{g-1}} has nonzero cup-product determinant. The valuation condition (14) is necessary for the displayed differentials not to lie in H^0(X,Ω^1), but it does not by itself imply that they span a complementary g-dimensional subspace of H^1_dR. The text derives a sufficient pole-order supply of functions from Corollary 3.5, then states that the method 'will always work' and appeals to numerical cup-product checks. A theorem establishing nondegeneracy for the chosen j, or an explicit counterexample, is needed for the central claim.
minor comments (4)
  1. [Algorithm 3.8(2)] Step (2) says 'd = dim S_2^+(Γ_0(N))' but d is used as the dimension of S_12^+(Γ_0(N)); this is surely a typo, but it is confusing in a pseudocode algorithm.
  2. [Proposition 3.1] The equality w_N(Δ(z)) = Δ(-1/(Nz)) = Δ(Nz) suppresses the automorphy factor and the normalization of the Atkin–Lehner operator; the statement would be cleaner if the slash action and the resulting constant factor were written explicitly. The subsequent quotient construction is unaffected, but the notation should be corrected.
  3. [§3.2, equation (22)] In Step (7) of Algorithm 3.8 the text refers to '(22 holds)' and the displayed condition is stated twice with slightly different notations; please clean up the formatting and the references to equations.
  4. [§4.3] The displayed symplectic basis elements contain denominators written as '22', '32', '33' (e.g. 22·23·101·617). These are clearly powers of 2 and 3 but should be typeset as 2^2, 3^2, 3^3 to avoid misreading the numerical coefficients.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the only flagged issue is an unproven minimal-jdR basis claim, which is a proof gap rather than a circular reduction.

full rationale

The paper's central computation, Algorithm 3.8, derives the Hodge filtration matrix lambda_Fil from q-expansions of S_2^+(Gamma_0(N)) and S_12^+(Gamma_0(N)), Hecke operators, and cup products. No target value of the Hodge filtration is used to set constants or choose parameters, so the 'prediction' is not fitted to its own output. The basis construction is verified by an independent criterion: Serre's cup product formula and a nonzero determinant check. The paper explicitly concedes a limitation regarding the minimal jdR choice: after Corollary 3.5 it states, 'Moreover, by computing the cup product, we have found that taking jdR as in (14) always produces a basis in our examples in §4.' This is an admission that the basis property for the minimal jdR is only numerical, whereas Proposition 3.2 proves existence for j = d-1. That is a correctness/completeness gap, not circularity: the cup product test is independent of the desired Hodge filtration, and the minimal-jdR choice is not fitted to any target value. The recipe for the nice correspondence Z is imported from prior work [BDM+23], [BBB+21], but those are not self-citations of the present author and are not used to force the conclusion. The congruence observations in Section 4 are made after the computation and are not used as inputs. No self-definitional, fitted-input, or self-citation-load-bearing pattern is present.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The algorithm introduces no new mathematical entities: the vector bundle, connection, Hodge filtration, and Hecke correspondence are objects from the prior quadratic Chabauty framework. The main extra assumptions are standard theorems plus two domain assumptions about Hecke action on weakly holomorphic differentials and Hadian's universal property. The unproven minimal-jdR basis statement is the principal algorithmic assumption.

free parameters (3)
  • jdR = smallest j in {1,...,d-1} satisfying valuation condition (14); 1 for N=67 and N=97, 3 for N=193
    This algorithmic choice selects fdR = s_{d-jdR}/s_d. Existence of some admissible j is proven, but the stronger property that the minimal choice gives a full basis of H^1_dR is only checked numerically in the examples.
  • p = 3 in all examples
    Auxiliary prime used in the Hecke operator computation. The algorithm only requires T_p not to lie in Z inside End(J); p=3 is a convenient choice, not a fitted constant.
  • n_N = 60 for N=67, 90 for N=97
    Truncation length for the congruence corank matrices in Section 4. The numerical congruence observation depends on this choice and stability in n_N is not discussed.
assumptions (6)
  • standard math Serre's cup product formula computes cup products as sums of residues.
    Used throughout Sections 3.1 and 3.2 to verify bases and to compute the matrix M for the Hecke operator.
  • standard math Riemann-Roch and dimension formulas for modular forms give d = dim S_12^+(Gamma_0(N)) = (N+1)/2 - g and d > 5g-5 for the relevant primes.
    Used in Section 3.1 to show that quotients of weight 12 plus cusp forms can produce functions with sufficiently large pole order at infinity.
  • domain assumption The Hecke operator T_p on H^1_dR is computed by the Laurent q-expansion formula (21), including for differentials of the second kind with poles at infinity.
    Assumed in Section 3.2 without proof. The formula is standard for modular forms, but its validity for the weakly holomorphic differentials used here is load-bearing.
  • domain assumption Hadian's universal property, as formulated in [BDM+19] Theorem 4.4, uniquely determines beta_Fil and gamma_Fil through the integrality condition (12).
    The Hodge filtration computation in Sections 2.3 and 3.2 depends on this prior theorem and on the description in [BM23].
  • domain assumption For X_0^+(N) with N prime and genus at least 2, one has r = g and rho(J) = g > 1, so quadratic Chabauty and the finiteness condition apply.
    Background quoted from [DLF21] and [BDM+23] in Section 2 and Remark 2.10; it is what makes the Hodge filtration computation relevant.
  • ad hoc to paper The smallest jdR satisfying (14) yields a basis of H^1_dR, i.e. the cup product matrix has nonzero determinant.
    Algorithm 3.8 depends on this. The paper proves existence of an admissible j and proves the basis statement for j = d-1, but for the minimal jdR it cites verification in examples rather than a proof.

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Pith. "Pith review of Quadratic Chabauty for Atkin-Lehner quotients of modular curves via weakly holomorphic modular forms: Hodge Filtrations." pith.science (2026). https://pith.science/paper/XJAMDJRP

@misc{pith2026250902291,
  author       = {Pith},
  title        = {Pith review of: Quadratic Chabauty for Atkin-Lehner quotients of modular curves via weakly holomorphic modular forms: Hodge Filtrations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XJAMDJRP}},
  note         = {Machine review of arXiv:2509.02291}
}
abstract

For Atkin-Lehner quotients $X_0^+(N)$, of prime level and of genus at least 2, we provide an algorithm for computing one of the main objects in the quadratic Chabauty algorithm in terms of weakly holomorphic modular forms associated to the curve. In particular, the algorithm computes a Hodge filtration on a certain unipotent vector bundle with connection related to $X_0^+(N)$, which is crucial in computing the $p$-adic height which is used to define the finite set of $p$-adic points containing the rational points on $X_0^+(N)$. This improves the current Hodge filtration algorithm by replacing the input of an explicit plane model of the curve with weakly holomorphic modular forms to produce a faster computation. We implement our algorithm on the genus 7 modular curve $X_0^+(193)$, and discover congruences between iterated integrals of weight 2 cusp forms in the plus eigenspace for the Atkin-Lehner involution and single integrals of weight 2 cusp forms in the minus eigenspace for the Atkin-Lehner involution.

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Works this paper leans on

34 extracted references · 33 canonical work pages · cited by 1 Pith paper

  1. [1]

    Quadratic Chabauty for Atkin-Lehner Quotients of Modular Curves of Prime Level and Genus 4, 5, 6

    Nikola Ad z aga, Vishal Arul, Lea Beneish, Mingjie Chen, Shiva Chidambaram, Timo Keller, and Boya Wen. Quadratic chabauty for atkin-lehner quotients of modular curves of prime level and genus 4, 5, 6. arXiv preprint arXiv:2105.04811 , 2021

    work page Pith review arXiv 2021

  2. [2]

    Coleman--gross height pairings and the p-adic sigma function

    Jennifer S Balakrishnan and Amnon Besser. Coleman--gross height pairings and the p-adic sigma function. Journal f \"u r die reine und angewandte Mathematik (Crelles Journal) , 2015(698):89--104, 2015

    work page 2015

  3. [3]

    Two recent p-adic approaches towards the (effective) mordell conjecture

    Jennifer S Balakrishnan, Alex J Best, Francesca Bianchi, Brian Lawrence, J Steffen M \"u ller, Nicholas Triantafillou, and Jan Vonk. Two recent p-adic approaches towards the (effective) mordell conjecture. In Arithmetic L-Functions and Differential Geometric Methods: Regulators IV, May 2016, Paris , pages 31--74. Springer, 2021

    work page 2016

  4. [4]

    Balakrishnan, L

    Jennifer S. Balakrishnan, L. Alexander Betts, Daniel Rayor Hast, Aashraya Jha, and J. Steffen Müller. Rational points on the non-split cartan modular curve of level 27 and quadratic chabauty over number fields, 2025

    work page 2025

  5. [5]

    u ller. Quadratic chabauty: p-adic heights and integral points on hyperelliptic curves. Journal f \

    Jennifer S Balakrishnan, Amnon Besser, and J Steffen M \"u ller. Quadratic chabauty: p-adic heights and integral points on hyperelliptic curves. Journal f \"u r die reine und angewandte Mathematik (Crelles Journal) , 2016(720):51--79, 2016

    work page 2016

  6. [6]

    The M agma algebra system

    Wieb Bosma, John Cannon, and Catherine Playoust. The M agma algebra system. I . T he user language. J. Symbolic Comput. , 24(3-4):235--265, 1997. Computational algebra and number theory (London, 1993)

    1997

  7. [7]

    Balakrishnan and Netan Dogra

    Jennifer S. Balakrishnan and Netan Dogra. Quadratic chabauty and rational points, i: p-adic heights. Duke Mathematical Journal , 167(11), August 2018

    work page 2018

  8. [8]

    Quadratic chabauty and rational points ii: Generalised height functions on selmer varieties

    Jennifer S Balakrishnan and Netan Dogra. Quadratic chabauty and rational points ii: Generalised height functions on selmer varieties. International Mathematics Research Notices , 2021(15):11923--12008, 2021

    work page 2021

Show all 34 references
  1. [9]

    A non-abelian conjecture of tate--shafarevich type for hyperbolic curves

    Jennifer S Balakrishnan, Ishai Dan-Cohen, Minhyong Kim, and Stefan Wewers. A non-abelian conjecture of tate--shafarevich type for hyperbolic curves. Mathematische Annalen , 372:369--428, 2018

    2018

  2. [10]

    Explicit chabauty--kim for the split cartan modular curve of level 13

    Jennifer Balakrishnan, Netan Dogra, J Steffen M \"u ller, Jan Tuitman, and Jan Vonk. Explicit chabauty--kim for the split cartan modular curve of level 13. Annals of mathematics , 189(3):885--944, 2019

    2019

  3. [11]

    Quadratic chabauty for modular curves: algorithms and examples

    Jennifer S Balakrishnan, Netan Dogra, J Steffen M \"u ller, Jan Tuitman, and Jan Vonk. Quadratic chabauty for modular curves: algorithms and examples. Compositio mathematica , 159(6):1111--1152, 2023

    2023

  4. [12]

    Computational tools for quadratic chabauty, 2023

    Jennifer S Balakrishnan and Steffen M \"u ller. Computational tools for quadratic chabauty, 2023

    2023

  5. [13]

    Sur les points rationnels des courbes alg \'e briques de genre sup \'e rieura l’unit \'e

    Claude Chabauty. Sur les points rationnels des courbes alg \'e briques de genre sup \'e rieura l’unit \'e . CR Acad. Sci. Paris , 212(882-885):1, 1941

    1941

  6. [14]

    Coleman integration on modular curves, 2024

    Mingjie Chen, Kiran Kedlaya, and Jun Bo Lau. Coleman integration on modular curves, 2024

    2024

  7. [15]

    Effective chabauty

    Robert F Coleman. Effective chabauty. Duke Mathematical Journal , 1985

    1985

  8. [16]

    Quadratic chabauty for modular curves and modular forms of rank one

    Netan Dogra and Samuel Le Fourn. Quadratic chabauty for modular curves and modular forms of rank one. Mathematische Annalen , 380:393--448, 2021

    2021

  9. [17]

    A first course in modular forms , volume 228

    Fred Diamond and Jerry Michael Shurman. A first course in modular forms , volume 228. Springer, 2005

    2005

  10. [18]

    a tze f \

    Gerd Faltings. Endlichkeitss \"a tze f \"u r abelsche variet \"a ten \"u ber zahlk \"o rpern. Inventiones mathematicae , 73:349--366, 1983

    1983

  11. [19]

    Rational points on x+ 0 (n) and quadratic q-curves (english, with english and french summaries)

    Steven D Galbraith. Rational points on x+ 0 (n) and quadratic q-curves (english, with english and french summaries). J. Th \'e or. Nombres Bordeaux , 14:205--219, 2002

    2002

  12. [20]

    Brief an g

    Alexandre Grothendieck. Brief an g. faltings. Geometric Galois actions , pages 49--58, 1997

    1997

  13. [21]

    Motivic fundamental groups and integral points

    Majid Hadian. Motivic fundamental groups and integral points. Duke Math. J. , 2011

    2011

  14. [22]

    Quadratic chabauty and p-adic gross–zagier

    Sachi Hashimoto. Quadratic chabauty and p-adic gross–zagier. Transactions of the American Mathematical Society , 376(5):3725–3760, February 2023

    2023

  15. [23]

    The motivic fundamental group of the projective line minus three points and the theorem of siegel

    Minhyong Kim. The motivic fundamental group of the projective line minus three points and the theorem of siegel. Inventiones mathematicae , 161(3), 2005

    2005

  16. [24]

    The unipotent albanese map and selmer varieties for curves

    Minhyong Kim. The unipotent albanese map and selmer varieties for curves. Publications of the Research Institute for Mathematical Sciences , 45(1):89--133, 2009

    2009

  17. [25]

    The l-component of the unipotent albanese map

    Minhyong Kim and Akio Tamagawa. The l-component of the unipotent albanese map. Mathematische Annalen , 340(1):223--235, 2008

    2008

  18. [26]

    Refined dimensions of cusp forms, and equidistribution and bias of signs

    Kimball Martin. Refined dimensions of cusp forms, and equidistribution and bias of signs. Journal of Number Theory , 188:1--17, 2018

    2018

  19. [27]

    Modular forms (ma4h9), 2015

    Marc Masdeu. Modular forms (ma4h9), 2015

    2015

  20. [28]

    Rational points on modular curves

    Barry Mazur. Rational points on modular curves. In Modular Functions of one Variable V: Proceedings International Conference, University of Bonn, Sonderforschungsbereich Theoretische Mathematik July 2--14, 1976 , pages 107--148. Springer, 2006

    1976

  21. [29]

    On p-adic height pairings

    Jan Nekov \'a r. On p-adic height pairings. S \'e minaire de th \'e orie des nombres, Paris , 91:127--202, 1990

    1990

  22. [30]

    Towards non--abelian p--adic Hodge theory in the good reduction case

    Martin Olsson. Towards non--abelian p--adic Hodge theory in the good reduction case . American Mathematical Society, 2011

    2011

  23. [31]

    On spaces of modular forms spanned by eta-quotients

    Jeremy Rouse and John J Webb. On spaces of modular forms spanned by eta-quotients. Advances in Mathematics , 272:200--224, 2015

    2015

  24. [32]

    Quadratic chabauty for modular curves

    Samir Siksek. Quadratic chabauty for modular curves. arXiv preprint arXiv:1704.00473 , 2017

    2017 arXiv

  25. [33]

    Counting points on curves using a map to P ^1

    Jan Tuitman. Counting points on curves using a map to P ^1 . Mathematics of Computation , 85(298):961--981, 2016

    2016

  26. [34]

    Counting points on curves using a map to P1, ii

    Jan Tuitman. Counting points on curves using a map to P1, ii. Finite Fields and Their Applications , 45:301--322, 2017

    2017

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