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Modular abelian surfaces of small conductor with nontrivial Tate--Shafarevich groups

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Pith's one-line read This paper proves that geometrically simple abelian surfaces over Q with conductor at most (10000)^2 exist whose Tate-Shafarevich groups contain a subgroup isomorphic to (Z/pZ)^2 for p = 5, 7, 11, 13, including the first known examples for

desk verdict First examples of 11- and 13-torsion in Sha of simple abelian surfaces, carefully proven, with a reproducibility gap around unpublished database dependencies that a referee should close. read the letter →

arxiv 2602.19813 v2 pith:SGQZ7ND5 submitted 2026-02-23 math.NT

Sam Frengley , Dylan Laird This is my paper · ORCID
classification math.NT MSC 11G1011F3311G40
keywords Tate-Shafarevichgroupabeliansurfacemodularformscongruencesvisibilitygenus2Jacobianrealmultiplicationconductor
topics Birch and Swinnerton-Dyer
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 proves that for each prime p = 5, 7, 11, 13 there is a geometrically simple abelian surface A/Q, with conductor at most (10000)^2, whose Tate-Shafarevich group X(A/Q) contains a subgroup isomorphic to (Z/pZ)^2. These are, according to the authors, the first known such examples for p = 11 and 13. The surfaces are given explicitly as Jacobians of genus-2 curves, and the proof works by detecting congruences between the associated weight-2 newforms and those of auxiliary abelian varieties, then using the visibility theorem to turn the congruence into a non-trivial element of Sha. The paper also gives a conjectural example of an order-7 class in Sha that is not visible in any abelian threefold, which would mean its visibility dimension is 4.

What carries the argument

The load-bearing identity is the (p,q)-congruence: an isomorphism A[p] ≅ B[q] of Galois modules for prime ideals p and q of the endomorphism rings of two abelian varieties. On the modular-forms side this is equivalent to congruence of Fourier coefficients modulo a prime above p, and the paper proves a finite criterion (generalising a classical one) so that only coefficients up to a computable bound need be checked. The visibility theorem then identifies a subgroup of X(A/Q) with a quotient of Mordell-Weil groups of an isogenous variety, provided the rank of A is 0, B has positive rank, and the Tamagawa numbers are coprime to p. This converts a congruence into an unconditional statement about

What would settle it

Choose one row of Table B.3, recompute the conductor of the Jacobian of the given Weierstrass equation (it must equal the square of the newform level), verify real multiplication by the listed order, and check the mod-p Fourier-coefficient congruence up to the stated bound; any failure would disprove the corresponding claim of Theorem 1.1.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is a systematic way to produce unconditional examples of non-trivial Tate-Shafarevich groups. A finite enumeration of congruences between weight-2 newforms with coefficient fields of degree at most 4 and levels up to 10000 yields, after passing to the corresponding abelian varieties and applying a visibility criterion, abelian surfaces A/Q such that (Z/pZ)^2 embeds into X(A/Q). The key advance over previous constructions is that the congruences are proved rather than conjectured: a generalised finite-coefficient criterion checks only Fourier coefficients up to a prescribed bound and, once the mod-p representation is shown to be irreducible, upgrades to

Load-bearing premise

The identification of each explicit genus-2 curve with the abelian variety attached to the stated newform depends on unpublished database work, and some Tamagawa-number computations are unpublished; if any of these data are wrong, the corresponding row of Theorem 1.1 would no longer be a proved example.

Editorial extensions

If this is right

  • For each p = 5, 7, 11, 13 there is now an explicit, geometrically simple abelian surface over Q known unconditionally to have (Z/pZ)^2 in its Tate-Shafarevich group; the p = 11 and 13 cases are new.
  • Because the surfaces are Jacobians of explicit genus-2 curves, they can serve directly as test cases for the standard conjecture relating L-functions to ranks and Sha orders.
  • The paper's enumeration of congruences among level-at-most-10000 newforms with coefficient fields of degree at most 4 gives a complete list for this range, so future visibility constructions can be checked against it.
  • The conjectural p = 7 example (label 9603.2.a.o) provides evidence that the visibility dimension of a Sha element can be as large as 4, and reduces the conjecture to determining rational points on two explicit plane quartic curves.

Reading between the lines

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

  • The same sieve could be run on larger level bounds or higher-degree coefficient fields once the modular-forms database grows; the restriction to levels up to 10000 and degree up to 4 is a database artefact, not a limitation of the method.
  • The proof upgrades semisimple isomorphisms to true Galois-module isomorphisms using irreducibility of the mod-p representations, so a variant of the method might handle reducible cases only at the cost of an extra argument.
  • If Conjecture 1.5 is proved, it would supply the first explicit element of Sha of an abelian surface whose minimal visualizing abelian variety is of dimension 4, giving a sharp answer to a natural quantitative question about visibility.
  • One could test the method's reach by looking for congruences of the same type among newforms of conductor just above 10000; a first example there would suggest the phenomenon is not tied to small levels.
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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

2 major / 4 minor

Summary. The paper proves that certain explicitly listed geometrically simple abelian surfaces A/Q of conductor at most (10000)^2 have (Z/pZ)^2 contained in the Tate--Shafarevich group X(A/Q) for p = 5, 7, 11, 13. The method is to find congruences between weight-2 newforms using a Kraus--Oesterlé-style criterion, pass to RM abelian varieties, and then apply a visibility theorem of Agashe--Stein and Fisher (corrected in Appendix A). The paper also gives a conjectural example of 7-torsion in X(A/Q) not visible in any abelian threefold, and reports a computational enumeration of all such congruences in the LMFDB for coefficient fields of degree at most 4.

Significance. If the results are correct, the paper supplies the first geometrically simple abelian surfaces with nontrivial 11- and 13-torsion in X(A/Q). The computational framework is a natural extension of Cremona--Freitas, and the authors are admirably explicit about which rows depend on unpublished work, which Tamagawa numbers could not be computed, and which rows are excluded from the main theorem. The paper ships Magma/Sage code and uses a Sturm-bound certificate for the congruences and an independent visibility argument for the Sha elements, so the proof strategy is transparent and checkable in principle. The central weakness is that the genuinely new p = 11 and p = 13 rows rely on the unpublished database [CEH+a,b] for the curve-to-newform correspondence, and the published text does not contain enough data to certify that correspondence independently.

major comments (2)
  1. [Proposition 2.3 / Table B.2, rows (11, 9025.2.a.r) and (13, 6776.2.a.r)] The new p=11 and p=13 examples in Theorem 1.1 rest on identifying the genus-2 curves in Table B.3 with the newforms 9025.2.a.r/9025.2.a.n and 6776.2.a.b. The proof of Proposition 2.3 says this is checked by a Hilbert modular surface point, conductor computation, and trace/norm matching, but none of the actual data is given; Table B.3 says it 'relies heavily' on the unpublished [CEH+a,b]. Since the other p=11 row (6962.2.a.q) is dagger-marked and excluded, an error in the 9025 row would collapse the p=11 novelty claim. Please include the full certificate: Hilbert point, conductor verification, and the trace/norm values for enough primes.
  2. [Theorem 1.4 / Table B.2] Theorem 1.4 is stated for 'any of the tuples in Table B.2', but Table B.2 contains rows marked * ('unproved, since we could not compute a Tamagawa number'), e.g. 6864.2.a.bf, 7632.2.a.v, and 9802.2.a.j. For those rows the Tamagawa-number hypothesis of Theorem 4.1(i) is not verified, so Theorem 1.4 is false as stated. The statement must be restricted to rows without * and without †, or the unproved/unpublished rows must be proved and the marks removed.
minor comments (4)
  1. [Lemma 3.2, proof] The proof says 'there exists 1 ≤ i ≤ 6', but the sets are indexed L_0,...,L_5. This should be 0 ≤ i ≤ 5.
  2. [Section 3, proof of Theorem 1.8] The statement 'we are left only with the examples given' is a finite computational completeness claim, but the paper does not report the number of hash collisions, the exact version of the code, or a log/certificate. Since Theorem 1.8 is not needed for Theorem 1.1 this is not a blocking issue, but reproducibility would be improved by a frozen repository snapshot or a checksummed output.
  3. [Table B.2 / Table B.3] Several typos: 'computing' should be 'computed', 'assocaited' should be 'associated', and 'yeilds' should be 'yields'. Please proofread the table notes.
  4. [Section 2, Table 2.1] The 'direct calculation' that the local factors of f' and g' agree modulo m is not shown; a reference to the precise case in [KO92, Proposition 4] would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the congruence, visibility, and Sha claims are derived from independent criteria; reliance on unpublished external databases is a verification gap, not a circular step.

full rationale

The paper's central derivation chain is: enumerate candidate congruences by hash-sieving (Theorem 1.8), prove each congruence via the Sturm-bound criterion (Proposition 2.2), promote these to abelian varieties using explicit curves with RM/conductor checks (Proposition 2.3), and then apply the Agashe–Stein/Fisher visibility theorem (Theorem 4.1) with independently checked ranks and Tamagawa numbers. None of these steps feeds the conclusion back into an input: coefficient congruences are verified up to the Sturm bound, the visibility theorem is proved in Appendix A, and the algebraic ranks are obtained by 2-descent and point search rather than assumed from the desired Sha conclusion. The paper itself flags rows depending on the unpublished work [DJ] with daggers and excludes them from Theorem 1.1, and marks uncomputed Tamagawa cases with asterisks; these are honest limitations, not circular reductions. The dependence of Table B.3 on the in-preparation external database [CEH+a,b] is a reproducibility and verification concern, not a self-definitional or fitted-input step: the curves are explicit, the RM and conductor assertions are checked, and the corresponding newform is identified by matching traces and norms. The reuse of the first author's earlier methods in Section 5 is confined to a conjectural non-visibility discussion and to one previously known p=7 example; it is not load-bearing for the new p=11 and p=13 results.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted: the only numerical inputs are LMFDB data and explicit curve equations, with no adjustable constants. The axioms are standard theorems in modular forms, Galois representations, and abelian varieties, plus domain assumptions about the completeness of LMFDB and the correctness of unpublished companion databases and computer algebra outputs. The paper introduces no new physical or mathematical entities such as new forces, fields, or symmetries; the modular curves X_A[p](7) are standard twisted modular curves, not invented objects.

assumptions (7)
  • standard math Shimura–Ribet correspondence between weight-2 newforms with real multiplication and isogeny classes of abelian varieties over Q.
    Used throughout Section 1.1 and Proposition 2.3 to transfer congruences of newforms to congruences of abelian varieties.
  • standard math Serre's modularity conjecture as proved by Khare–Wintenberger, used for the converse direction of the modular correspondence.
    Invoked in Section 1.1 (Ribet [Rib04, Theorem 4.4] and [KW09a, KW09b]) to make the newform-to-abelian-variety assignment bijective.
  • standard math Kraus–Oesterlé / Sturm bound criterion for proving a congruence from finitely many Fourier coefficients.
    Proposition 2.2 states the finite-check criterion; the proof follows [KO92, Proposition 4] and the Sturm bound [Ste07, Corollary 9.20].
  • standard math Agashe–Stein/Fisher visibility theorem (Theorem 4.1) with the corrected use of B' in the Tamagawa condition.
    This is the mechanism that converts a (p,q)-congruence into a visible subgroup of Sha. The corrected proof is given in Appendix A.
  • domain assumption LMFDB completeness and accuracy for newforms of level ≤ 10000 with coefficient field degree ≤ 4.
    Theorem 1.8 claims to list all congruences among such newforms 'contained in the LMFDB'; any missing or erroneous LMFDB entry would break the 'all congruences' enumeration.
  • domain assumption Correctness of the unpublished [CEH+a,b] database identifying genus-2 curves with RM abelian surfaces associated to LMFDB newforms.
    Table B.3 and Proposition 2.3 rely on this in-preparation database; the authors supplement it with Hilbert-modular-surface and conductor checks, but the underlying database is not independently published.
  • domain assumption Reliability of the implemented Tamagawa-number and rank computations (Magma, Sage, PARI, and in some rows the unpublished [DJ]).
    Theorem 1.4 requires the Tamagawa numbers of A and B (or B') to be coprime to p and the ranks to be as recorded. Rows marked with an asterisk or dagger are excluded from Theorem 1.1 when the verification is incomplete.

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Pith. "Pith review of Modular abelian surfaces of small conductor with nontrivial Tate--Shafarevich groups." pith.science (2026). https://pith.science/paper/SGQZ7ND5

@misc{pith2026260219813,
  author       = {Pith},
  title        = {Pith review of: Modular abelian surfaces of small conductor with nontrivial Tate--Shafarevich groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SGQZ7ND5}},
  note         = {Machine review of arXiv:2602.19813}
}
abstract

We exhibit examples of geometrically simple abelian surfaces $A/\mathbb{Q}$ with conductor bounded by $(10\,000)^2$ whose Tate--Shafarevich groups contain a subgroup isomorphic to $(\mathbb{Z}/p\mathbb{Z})^2$ for each $p = 5, 7, 11, 13$. To find these examples we generalise work of Cremona--Freitas to give a candidate list of all congruences of a certain type between pairs of weight $2$ newforms $f \in S_2^{\mathrm{new}}(\Gamma_0(N))$ and $g \in S_2^{\mathrm{new}}(\Gamma_0(M))$ contained in the LMFDB (i.e., with $N, M \leq 10\,000$) and with coefficient fields of degree $\leq 4$. Passing from the modular forms to the corresponding abelian varieties we use visibility to (unconditionally) prove the existence of non-trivial elements of the Tate--Shafarevich group. Finally we construct an example of an abelian surface with $(\mathbb{Z}/7\mathbb{Z})^2 \subset \mathrm{Sha}(A/\mathbb{Q})$ which is (conjecturally) not visible in any abelian threefold.

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Works this paper leans on

45 extracted references · 23 canonical work pages

  1. [1]

    Agashe and W

    A. Agashe and W. Stein , https://doi.org/10.1006/jnth.2002.2810 Visibility of S hafarevich- T ate groups of abelian varieties , J. Number Theory 97 (2002), no. 1, 171--185. 1939144

    arXiv 2002

  2. [2]

    , https://doi.org/10.1090/S0025-5718-04-01644-8 Visible evidence for the B irch and S winnerton- D yer conjecture for modular abelian varieties of analytic rank zero , Math. Comp. 74 (2005), no. 249, 455--484, With an appendix by J. Cremona and B. Mazur. 2085902

    work page doi:10.1090/s0025-5718-04-01644-8 2005

  3. [3]

    A. J. Best and R. van Bommel, sage-cluster-pictures: C luster pictures in S age M ath , https://alexjbest.github.io/cluster-pictures/index.html, Accessed 05 November 2025

    2025

  4. [4]

    A. J. Best, L. A. Betts, M. Bisatt, R. van Bommel, V. Dokchitser, O. Faraggi, S. Kunzweiler, C. Maistret, A. Morgan, S. Muselli, and S. Nowell, https://doi.org/10.1112/blms.12604 A user's guide to the local arithmetic of hyperelliptic curves , Bull. Lond. Math. Soc. 54 (2022), no. 3, 825--867

    work page doi:10.1112/blms.12604 2022

  5. [5]

    Bosma , J

    W. Bosma , J. Cannon , and C. Playoust , http://dx.doi.org/10.1006/jsco.1996.0125 The M agma algebra system. I . T he user language , J. Symbolic Comput. 24 (1997), no. 3-4, 235--265, Computational algebra and number theory (London, 1993). 1484478

    arXiv 1996

  6. [6]

    L. A. Betts, https://doi.org/10.1016/j.jnt.2020.09.021 Variation of Tamagawa numbers of Jacobians of hyperelliptic curves with semistable reduction , J. Number Theory 231 (2022), 158--213 (English)

    work page doi:10.1016/j.jnt.2020.09.021 2020

  7. [7]

    Bruin , E

    N. Bruin , E. V. Flynn , and D. Testa , https://doi.org/10.4064/aa165-3-1 Descent via (3,3) -isogeny on J acobians of genus 2 curves , Acta Arith. 165 (2014), no. 3, 201--223. 3263947

    work page doi:10.4064/aa165-3-1 2014

  8. [8]

    R. van Bommel, https://doi.org/10.1007/978-3-030-80914-0\_6 Efficient computation of BSD invariants in genus 2 , Arithmetic geometry, number theory, and computation, Simons Symp., Springer, Cham, 2021, pp. 237--258. 4427964

    doi:10.1007/978-3-030-80914-0 2021

Show all 45 references
  1. [9]

    Costa, N

    E. Costa, N. D. Elkies, S. Hashimoto, A. Jha, K. Martin, B. Poonen, and J. Voight, A database of curves with modular J acobians , In preparation

  2. [10]

    , Github repository for `` A database of curves with modular J acobians'' , https://github.com/edgarcosta/ModularAbelianSurfaces.git

  3. [11]

    J. W. S. Cassels and E. V. Flynn , https://doi.org/10.1017/CBO9780511526084 Prolegomena to a middlebrow arithmetic of curves of genus 2 , London Mathematical Society Lecture Note Series, vol. 230, Cambridge University Press, Cambridge, 1996. 1406090

    1996 doi

  4. [12]

    J. E. Cremona and N. Freitas , https://doi-org/10.4171/rmi/1269 Global methods for the symplectic type of congruences between elliptic curves , Rev. Mat. Iberoam. 38 (2022), no. 1, 1--32. 4382462

    2022 doi

  5. [13]

    J. E. Cremona and B. Mazur , http://projecteuclid.org/euclid.em/1046889588 Visualizing elements in the S hafarevich- T ate group , Experiment. Math. 9 (2000), no. 1, 13--28. 1758797

    2000

  6. [14]

    Costa, N

    E. Costa, N. Mascot, J. Sijsling, and J. Voight, https://doi.org/10.1090/mcom/3373 Rigorous computation of the endomorphism ring of a Jacobian , Math. Comp. 88 (2019), no. 317, 1303--1339

    2019 doi

  7. [15]

    Conrad and W

    B. Conrad and W. A. Stein, https://doi.org/10.4310/MRL.2001.v8.n6.a5 Component groups of purely toric quotients , Math. Res. Lett. 8 (2001), no. 5-6, 745--766

    2001 doi

  8. [16]

    Dokchitser and C

    T. Dokchitser and C. Doris , https://doi.org/10.1090/mcom/3387 3-torsion and conductor of genus 2 curves , Math. Comp. 88 (2019), no. 318, 1913--1927. 3925491

    2019 doi

  9. [17]

    Dokchitser, V

    T. Dokchitser, V. Dokchitser, C. Maistret, and A. Morgan, https://doi.org/10.1007/s00208-021-02319-y Arithmetic of hyperelliptic curves over local fields , Math. Ann. 385 (2023), no. 3-4, 1213--1322

    2023 doi

  10. [18]

    Dokchitser and G

    T. Dokchitser and G. Jakovac, Regular models of curves, In preparation

  11. [19]

    N. D. Elkies and A. Kumar , https://doi-org/10.2140/ant.2014.8.2297 K3 surfaces and equations for H ilbert modular surfaces , Algebra Number Theory 8 (2014), no. 10, 2297--2411. 3298543

    2014 doi

  12. [20]

    T. A. Fisher , https://doi.org/10.1112/S1461157016000243 Visualizing elements of order 7 in the T ate- S hafarevich group of an elliptic curve , LMS J. Comput. Math. 19 (2016), no. suppl. A, 100--114. 3540949

    2016 doi

  13. [21]

    E. V. Flynn , F. Lepr\' e vost , E. F. Schaefer , W. A. Stein , M. Stoll , and J. L. Wetherell , https://doi.org/10.1090/S0025-5718-01-01320-5 Empirical evidence for the B irch and S winnerton- D yer conjectures for modular J acobians of genus 2 curves , Math. Comp. 70 (2001),...

    2001 doi

  14. [22]

    E. V. Flynn , https://doi.org/10.1016/j.jnt.2015.01.018 Descent via (5,5) -isogeny on J acobians of genus 2 curves , J. Number Theory 153 (2015), 270--282. 3327574

    2015 doi

  15. [23]

    Frengley , https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1340/ Explicit 7 -torsion in the T ate-- S hafarevich groups of genus 2 J acobians , J

    S. Frengley , https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1340/ Explicit 7 -torsion in the T ate-- S hafarevich groups of genus 2 J acobians , J. Th\'eor. Nombres Bordx. 37 (2025), no. 2, 727--746

    2025 doi

  16. [24]

    Gehrunger , Worksheets for `` T he classification of the stable marked reduction of genus 2 curves in residue characteristic 2'' , https://doi.org/10.3929/ethz-b-000743349

    T. Gehrunger , Worksheets for `` T he classification of the stable marked reduction of genus 2 curves in residue characteristic 2'' , https://doi.org/10.3929/ethz-b-000743349

    doi

  17. [25]

    , https://doi.org/10.48550/arXiv.2507.02426 The classification of the stable marked reduction of genus 2 curves in residue characteristic 2 , arXiv e-prints (2025), arXiv:2507.02426

    2025 doi

  18. [26]

    Frengley and D

    S. Frengley and D. Laird , Github repository, https://github.com/dlaird-ens/p-tors-sha

  19. [27]

    Kraus and J

    A. Kraus and J. Oesterl\' e , https://doi.org/10.1007/BF01444715 Sur une question de B . M azur , Math. Ann. 293 (1992), no. 2, 259--275. 1166121

    1992 doi

  20. [28]

    D. R. Kohel and W. A. Stein, https://doi.org/10.1007/10722028_25 Component groups of quotients of J_0(N) , Algorithmic Number Theory, Springer Berlin Heidelberg, Berlin, Heidelberg, 2000, pp. 405--412

    doi

  21. [29]

    Keller and M

    T. Keller and M. Stoll, https://doi.org/10.1017/fms.2024.133 Complete verification of strong BSD for many modular abelian surfaces over \(Q\) , Forum Math. Sigma 13 (2025), 82, Id/No e20

    2024 doi

  22. [30]

    Khare and J.-P

    C. Khare and J.-P. Wintenberger , https://doi.org/10.1007/s00222-009-0205-7 Serre's modularity conjecture. I , Invent. Math. 178 (2009), no. 3, 485--504. 2551763

    2009 doi

  23. [31]

    II , Invent

    , https://doi.org/10.1007/s00222-009-0206-6 Serre's modularity conjecture. II , Invent. Math. 178 (2009), no. 3, 505--586. 2551764

    2009 doi

  24. [32]

    Liu, https://doi.org/10.1093/oso/9780198502845.001.0001 Algebraic geometry and arithmetic curves , Oxford Graduate Texts in Mathematics, vol

    Q. Liu, https://doi.org/10.1093/oso/9780198502845.001.0001 Algebraic geometry and arithmetic curves , Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002, Translated from the French by Reinie Ern\'e, Oxford Science Publications. 1917232

    2002

  25. [33]

    Liu , Desingularization of double covers of regular surfaces , arXiv e-prints (2025), arXiv:2504.16808

    Q. Liu , Desingularization of double covers of regular surfaces , arXiv e-prints (2025), arXiv:2504.16808

    2025

  26. [34]

    H. W. Lenstra and F. Oort, https://doi.org/10.1016/0022-4049(85)90079-9 Abelian varieties having purely additive reduction , J. Pure Appl. Algebra 36 (1985), 281--298

    1985 doi

  27. [35]

    Miyake, https://doi.org/10.1007/3-540-29593-3 Modular forms , corrected 2nd printing ed., Springer Monogr

    T. Miyake, https://doi.org/10.1007/3-540-29593-3 Modular forms , corrected 2nd printing ed., Springer Monogr. Math., Springer, Berlin, 2006, Translated from the Japanese by Joshitaku Maeda

    2006 doi

  28. [36]

    J. R. Merriman and N. P. Smart , https://doi.org/10.1017/S030500410007153X Curves of genus 2 with good reduction away from 2 with a rational W eierstrass point , Math. Proc. Cambridge Philos. Soc. 114 (1993), no. 2, 203--214. 1230127

    1993 doi

  29. [37]

    Poonen , E

    B. Poonen , E. F. Schaefer , and M. Stoll , https://doi.org/10.1215/S0012-7094-07-13714-1 Twists of X(7) and primitive solutions to x^2+y^3=z^7 , Duke Math. J. 137 (2007), no. 1, 103--158. 2309145

    2007 doi

  30. [38]

    K. A. Ribet, https://doi.org/10.1007/978-3-0348-7919-4_15 Abelian varieties over Q and modular forms , Modular curves and A belian varieties (J. E. Cremona, J.-C. Lario, J. Quer, and K. A. Ribet, eds.), Birkh \"a user Basel, 2004, pp. 241--261

    2004 doi

  31. [39]

    Shimura, https://doi.org/10.2969/jmsj/02530523 On the factors of the J acobian variety of a modular function field , J

    G. Shimura, https://doi.org/10.2969/jmsj/02530523 On the factors of the J acobian variety of a modular function field , J. Math. Soc. Japan 25 (1973), 523--544. 318162

    1973

  32. [40]

    J. H. Silverman, https://eudml.org/doc/163765 The Neron fiber of Abelian varieties with potential good reduction , Math. Ann. 264 (1983), 1--3

    1983

  33. [41]

    Stein , https://doi.org/10.1090/gsm/079 Modular forms, a computational approach , Graduate Studies in Mathematics, vol

    W. Stein , https://doi.org/10.1090/gsm/079 Modular forms, a computational approach , Graduate Studies in Mathematics, vol. 79, American Mathematical Society, Providence, RI, 2007, With an appendix by P. E. Gunnells. 2289048

    2007 doi

  34. [42]

    Stoll , Two simple 2 -dimensional abelian varieties defined over Q with M ordell- W eil group of rank at least 19 , C

    M. Stoll , Two simple 2 -dimensional abelian varieties defined over Q with M ordell- W eil group of rank at least 19 , C. R. Acad. Sci. Paris S\' e r. I Math. 321 (1995), no. 10, 1341--1345. 1363577

    1995

  35. [43]

    The LMFDB Collaboration , https://www.lmfdb.org The L -functions and modular forms database , https://www.lmfdb.org, [Online; accessed December 2025]

    2025

  36. [44]

    Bordeaux, http://pari.math.u-bordeaux.fr/ PARI/GP version 2.15.4 , 2023, available from http://pari.math.u-bordeaux.fr/

    The PARI G roup, Univ. Bordeaux, http://pari.math.u-bordeaux.fr/ PARI/GP version 2.15.4 , 2023, available from http://pari.math.u-bordeaux.fr/

    2023

  37. [45]

    The Sage Developers , https://www.sagemath.org S agemath, the S age M athematics S oftware S ystem ( V ersion 9.2) , 2023, https://www.sagemath.org

    2023

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