Pith. sign in

REVIEW 3 major objections 3 minor 235 references

Distinguishing elliptic curves modulo $p$ and identifying images of product representations

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that, for elliptic curves over Q with surjective mod p Galois representations, the image of their product representation is always conjugate to one of an explicit short list of finite groups, and it computes the exact…

desk verdict The p≥5 classification and witness-ratio formulas are rigorous and correct; the p=3 case is computer-assisted with a minor verification gap that should be patched, but the paper deserves a serious referee. read the letter →

arxiv 2608.10880 v1 pith:IECRI3NH submitted 2026-08-11 math.NT

Jessica Bennett , Jeffrey Hatley , Sasha Kononova , Vincent Macri , Zachary Porat , Naina Praveen This is my paper · ORCID
classification math.NT MSC 14H5211F80
keywords ellipticcurvesGaloisrepresentationsmodpGoursat'slemmawitnessratiodivisionfieldsChebotarevdensitytheoremquadratictwists
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 asks how and how quickly one can tell two elliptic curves over the rationals apart modulo a prime p. The authors prove that if both mod p Galois representations are surjective, then the image of their product representation is one of an explicit finite list: the diagonal subgroup when the representations are isomorphic, and otherwise three non-diagonal groups for p≥5, plus one extra group when p=3. They define a witness ratio for each image group, the fraction of pairs (g,h) in the group with different traces, and show via the Chebotarev density theorem that this ratio equals the asymptotic density of good primes ℓ where a_ℓ(E1) and a_ℓ(E2) differ modulo p. This yields a practical numerical way to distinguish non-isomorphic representations and identifies the precise amount of computation needed to witness non-congruence.

What carries the argument

The central machinery is Goursat's lemma for subgroups of a product of two groups, applied to H = Im(rho_E1,p × rho_E2,p). Because both projections onto GL2(Fp) are surjective, H is determined by a pair of normal subgroups N1, N2 of GL2(Fp) together with an isomorphism GL2(Fp)/N1 ≅ GL2(Fp)/N2; for p≥5 the normal subgroups of GL2(Fp) are either contained in the scalars or contain SL2(Fp), which leaves exactly four admissible groups up to conjugation. The witness ratio W(H) = #{ (g,h) ∈ H : tr(g) ≠ tr(h) } / |H| is the invariant that converts this group structure into the density of witness primes through the Chebotarev density theorem.

What would settle it

Take any pair of elliptic curves over Q with surjective mod p representations at a small prime p≥5, compute the image of the product representation directly from the division fields or from the Goursat data, and check that it is conjugate to Γ_id, Γ_χ, H_±, or Δ; the first pair that is not would falsify Theorem 3.18.

Watch

Extended reading notes

Core claim

The paper establishes that, under the assumption that both residual representations are surjective, the image H of rho_E1,p × rho_E2,p is always conjugate to one of a short explicit list of subgroups of GL2(Fp) × GL2(Fp): the diagonal subgroup Γ_id, the quadratic-twisted diagonal Γ_χ, the sign-twisted diagonal H_±, and the full determinant-equal subgroup Δ, with a fifth group H_Q appearing only at p=3. For each admissible group it computes W(H), the fraction of elements in H whose two traces differ, and proves that W(H) is exactly the density of good primes ℓ for which a_ℓ(E1) is not congruent to a_ℓ(E2) modulo p. It further characterizes which pairs of elliptic curves realize each group: Γ_id corresponds to isomorphic representations, Γ_χ to representations differing by the quadratic character of conductor p*, H_± to quadratic twists by other squarefree integers, and Δ to every other case, while H_Q at p=3 occurs when the two curves have equal Q8-fixed subfields of their 3-division fields, equivalently when their discriminants generate the same class in Q*/(Q*)³.

Load-bearing premise

The entire classification and the witness-ratio formulas rely on both mod p representations being surjective onto GL2(Fp); if either representation has a smaller image, such as a Borel or Cartan subgroup, the four-group list and the density formulas can fail.

Editorial extensions

If this is right

  • For p≥5, two elliptic curves with surjective mod p representations that are not isomorphic have product image conjugate to exactly one of Γ_χ, H_±, or Δ, so the density of primes where a_ℓ(E1) differs from a_ℓ(E2) modulo p is one of the three closed-form witness ratios in Theorem 3.18.
  • For p=3 the worst-case witness density is 1/4, so the number of witness primes below X is asymptotically (1/4 + o(1)) Li(X) regardless of which non-diagonal image group occurs.
  • Which image group occurs is determined by representation-theoretic data: isomorphic representations give Γ_id, a twist by the Legendre-symbol character gives Γ_χ, any other quadratic twist gives H_±, and all other pairs give Δ; at p=3, H_Q occurs exactly when the two 3-division fields have equal Q8-fixed fields, equivalently when the normalized discriminants differ by a rational cube.
  • The witness ratios of all admissible groups are pairwise distinct, so a sufficiently large sample of good primes can identify, numerically, which of the finitely many image groups a concrete pair of elliptic curves realizes.

Reading between the lines

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

  • The paper leaves open the non-surjective case; the same Goursat-based enumeration would produce a longer but still finite list of admissible images indexed by the possible subgroups of GL2(Fp) that can occur as mod p images, and the witness-ratio formulas would have to be recomputed for those smaller groups.
  • The witness-ratio approach suggests a probabilistic substitute for the Sturm bound: instead of checking all primes up to an O(N) bound, one can sample primes and stop once the empirical witness proportion is close to one of the admissible ratios; the explicit Chebotarev error terms cited in the paper give a principled stopping rule under the generalized Riemann hypothesis.
  • Because the witness ratios for Γ_χ and H_± differ by O(p^{-2}), numerical ratio comparison alone becomes impractical for large p; a cheaper distinguishing test, suggested by the paper's Remark 3.19, is to check whether a_ℓ(E2) always equals the Legendre symbol (ℓ/p) times a_ℓ(E1), which in the Γ_χ case holds for all good primes.
  • The p=3 discriminant criterion for H_Q provides a direct search strategy: look for surjective mod 3 curve pairs that are not quadratic twists but whose discriminants generate the same class in Q*/(Q*)³; the paper's Table 1 already contains one such pair.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper studies the product of two residual mod p Galois representations attached to elliptic curves over Q, under the standing assumption that both representations are surjective. Using Goursat's lemma together with a classification of normal subgroups of GL2(Fp), it determines, for p >= 5, that the image of the product representation is conjugate to one of Gamma_id, Gamma_chi, H_plusminus, or Delta, and computes exact witness ratios W(H), which are interpreted via Chebotarev as densities of primes where the Frobenius traces of the two curves are incongruent modulo p. For p = 3 the paper adds one further group H_Q and computes its witness ratio. It also gives a characterization of the pairs of elliptic curves realizing each group in terms of quadratic twists, and discusses numerical examples and connections with the Sturm bound.

Significance. If the results are correct, the paper provides a complete and explicit description of all possible images of product representations of two surjective elliptic-curve mod p representations, together with exact densities for congruence-breaking primes. A notable strength is that the witness ratios are exact counts in explicitly defined finite groups: there are no fitted parameters and the p >= 5 formulas are closed and checkable. The paper also ships computational scripts and gives concrete examples that agree with the formulas in Table 1. The p >= 5 classification and witness-ratio computations appear internally sound and are a genuine contribution. The p = 3 part, however, is the main weak spot: its completeness and one of its witness ratios currently rest on undocumented computer enumeration.

major comments (3)
  1. [§4, Theorem 4.1] The proof of Theorem 4.1 does not give a complete enumeration of the admissible subgroups for p = 3. It says that 'we originally used a computer algebra system to determine the possible groups' and, before Theorem 4.2, that 'as confirmed by our explicit computations, the extra subgroup Q8 gives rise to precisely one more admissible group,' but no hand proof or deterministic reproducible listing is provided for the full list {Gamma_chi, H_plusminus, H_Q, Delta}. Since Theorem A and its density conclusion for p = 3 depend on the completeness of this list, the omission is load-bearing. Please provide either a complete hand verification (using, for example, the normal-subgroup data in [Ade01, Figure 5.1] together with the Goursat correspondence) or a self-contained, deterministic enumeration script whose output is the admissible-group list.
  2. [§4, Theorem 4.1 and W(H_Q)] The value W(H_Q) = 35/64 is asserted as 'straightforward' without displaying the computation. This ratio is one of the four witness ratios in Theorem A, so the paper should either exhibit the conjugacy-class sizes of H_Q or include a short reproducible script that returns 35/64. Without this, a reader cannot independently verify a central numerical claim of the p = 3 case.
  3. [§3.4, proof of Theorem 3.21] The proof contains the assertion that 'G has a unique index 2 subgroup, namely S.' This is false for p >= 5: |GL2(Fp) : SL2(Fp)| = p - 1, not 2. The unique index-2 subgroup is ker(chi composed with det), and the Goursat argument should be phrased in terms of that subgroup or of the unique nontrivial character of G. As written, the step 'if N1 = S then chi_d = chi_{p*}' is invalid. The statement of Theorem 3.21 is plausible and likely fixable, but the proof needs correction.
minor comments (3)
  1. [Lemma 3.6] The statement 'Suppose N1 and N2 contain Z' should read 'are contained in Z' (that is, N_i ⊆ Z). Under the literal wording the lemma is false or vacuous for p >= 5, whereas the intended statement, with the proof given, is correct and is exactly what is used in Theorem 3.1.
  2. [Theorem 4.2] The condition 'Delta_E1 Delta_E2^{±1} in (Q^times)^3' should be spelled out as 'either Delta_E1 Delta_E2 or Delta_E1 Delta_E2^{-1} is a rational cube' to avoid ambiguity.
  3. [Theorem 3.12 and Proposition 3.14] The notation in the proof is occasionally dense, especially the count of Delta-conjugacy classes and the use of the symbols N_t and n_t; a short illustrative example for one value of p in the appendix would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the classification and witness ratios follow from Goursat's lemma and explicit finite-group counts; the p=3 computer enumeration is a reproducibility gap rather than a circular reduction.

full rationale

The paper's central claim for p>=5 is self-contained. Section 3.1 proves the normal-subgroup dichotomy for GL2(Fp) using simplicity of PSL2(Fp) (Proposition 3.3), classifies determinant-preserving automorphisms (Lemma 3.7), and derives Theorem 3.1 by Goursat's lemma. The witness ratios in Theorem 3.16 are exact counts over explicitly defined finite groups, using only elementary counting lemmas such as Lemma 3.15; no data, fitted parameters, or examples enter these computations. The Chebotarev density theorem then converts these counts into densities, so the density statement is a theorem application rather than a prediction fitted to data. Theorem 3.21 is a further group-theoretic characterization of when each admissible group occurs, not a restatement of the definition of H. For p=3, Theorem 4.1 does rely on the authors' Magma/Sage scripts [BHK+] to enumerate admissible groups and to compute W(H_Q); this is an unexhibited computational step and a genuine verification or reproducibility concern, as the skeptic's attack notes. However, it is not circular in the sense required here: the witness ratios are not fitted to the numerical examples, the enumeration is independently reproducible code rather than an imported self-cited uniqueness theorem, and the p>=5 results do not depend on it. No step in the derivation reduces to its own inputs by construction, so the circularity score is 0.

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

No free parameters appear in the arguments: the witness ratios are exact counts of elements in explicitly defined finite groups, and the numerical examples are used only as checks. The paper's assumptions are standard theorems (Goursat, Chebotarev, normal-subgroup structure of GL2(F_p)) plus the explicit surjectivity hypothesis. The p=3 special case adds a computational enumeration supplied by the authors' GitHub code, which is the main non-hand-proven ingredient.

assumptions (6)
  • standard math Chebotarev density theorem, including effective versions from [DKN25], [GM19], and [BS96]
    Used in Sections 2 and 3 to interpret W(H) as the asymptotic density of good primes ell with a_ell(E1) not congruent to a_ell(E2) mod p.
  • standard math Goursat's lemma
    Used at Lemma 2.2 to reduce admissible subgroups H <= G x G to pairs of normal subgroups and an isomorphism of quotients.
  • standard math Classification of normal subgroups of GL2(F_p) for p>=5: each nontrivial normal subgroup contains SL2(F_p) or is contained in the center
    Proposition 3.3 relies on simplicity of PSL2(F_p), perfectness of SL2(F_p), and Lemma 3.2 on centralizers.
  • standard math Aut(SL2(F_p)) is isomorphic to PGL2(F_p) for p>=5, and determinant-preserving automorphisms of GL2(F_p) are inner or inner composed with r_chi
    Lemma 3.7 and the {+-I} quotient case in Theorem 3.1 depend on this standard classification of automorphisms.
  • domain assumption The p-division field Q(E[p]) has unique quadratic subfield Q(sqrt(p*)), and for p=3 the Q8-fixed field is Q(zeta3, Delta_E^(1/3))
    Lemma 3.20(2) and Theorem 4.2 cite [Ade01, Figure 5.3 and Proposition 5.4.3] for these fixed-field identifications.
  • domain assumption Both mod p representations rho_E1,p and rho_E2,p are surjective
    Stated in Sections 1 and 2; the admissible-subgroup classification and witness-ratio theorems are conditional on this hypothesis.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Distinguishing elliptic curves modulo $p$ and identifying images of product representations." pith.science (2026). https://pith.science/paper/IECRI3NH

@misc{pith2026260810880,
  author       = {Pith},
  title        = {Pith review of: Distinguishing elliptic curves modulo $p$ and identifying images of product representations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IECRI3NH}},
  note         = {Machine review of arXiv:2608.10880}
}
abstract

Given two elliptic curves defined over $\mathbb{Q}$ and a rational prime $p$, we study the product of their residual Galois representations. Using Goursat's lemma, we explicitly enumerate and completely characterize all possible images of such product representations. We also define associated invariants to these image groups, which we call \textit{witness ratios}, and we explain their computational utility and their relationship to the well-known Sturm bound for testing congruences between modular forms.

Figures

Figures reproduced from arXiv: 2608.10880 by the authors.

Figure 5
Figure 5. ]. The final claim now follows from the Chebotarev density theorem and the observation [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

235 extracted references · 67 canonical work pages

  1. [1]

    Hung, Pin-Chi and Lim, Meng Fai , TITLE =. Asian J. Math. , FJOURNAL =. 2020 , NUMBER =

    2020

  2. [2]

    , TITLE =

    Lockhart, Paul and Rosen, Michael and Silverman, Joseph H. , TITLE =. J. Algebraic Geom. , FJOURNAL =. 1993 , NUMBER =

    1993

  3. [3]

    Modular elliptic curves and

    Wiles, Andrew , journal=. Modular elliptic curves and. 1995 , publisher=

    1995

  4. [4]

    Duke Math

    Lower bounds for height functions , author=. Duke Math. J. , volume=

  5. [5]

    Zhai, Shuai , TITLE =. Asian J. Math. , FJOURNAL =. 2016 , NUMBER =. doi:10.4310/AJM.2016.v20.n3.a4 , URL =

    work page doi:10.4310/ajm.2016.v20.n3.a4 2016

  6. [6]

    Cai, Li and Li, Chao and Zhai, Shuai , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2020 , NUMBER =. doi:10.1112/jlms.12284 , URL =

    work page doi:10.1112/jlms.12284 2020

  7. [7]

    and Kundu, D

    Hatley, J. and Kundu, D. , note =

  8. [8]

    Ring-theoretic properties of certain

    Taylor, Richard and Wiles, Andrew , journal=. Ring-theoretic properties of certain. 1995 , publisher=

    1995

Show all 235 references
  1. [9]

    On the modularity of elliptic curves over

    Breuil, Christophe and Conrad, Brian and Diamond, Fred and Taylor, Richard , journal=. On the modularity of elliptic curves over

  2. [10]

    Hatley, Jeffrey and Lei, Antonio , TITLE =. C. R. Math. Acad. Sci. Paris , FJOURNAL =. 2023 , PAGES =

    2023

  3. [11]

    Introduction to

    Greenberg, Ralph , journal=. Introduction to

  4. [12]

    Proceedings of the “Segundas Jornadas de Teor

    Elliptic curves of maximal rank , author=. Proceedings of the “Segundas Jornadas de Teor

  5. [13]

    Watkins's conjecture for quadratic twists of Elliptic Curves with Prime Power Conductor , author=

  6. [14]

    Computing modular degrees using

    Delaunay, Christophe , journal=. Computing modular degrees using

  7. [15]

    Elliptic curves over the rationals with good reduction outside two odd primes , author=. J. Number Theory , volume=. 2019 , publisher=

    2019

  8. [16]

    , TITLE =

    Silverman, Joseph H. , TITLE =. 1986 , PAGES =. doi:10.1007/978-1-4757-1920-8 , URL =

    1986 doi

  9. [17]

    Courbes elliptiques sur

    Ivorra, Wilfrid , journal=. Courbes elliptiques sur. 2004 , publisher=

    2004

  10. [18]

    Anticyclotomic

    B\"uy\"ukboduk, K\^. Anticyclotomic. Forum Math. , FJOURNAL =. 2018 , NUMBER =

    2018

  11. [19]

    2023 , note=

    Lei, Antonio and M\"uller, Katharina and Xia, Jiacheng , TITLE =. 2023 , note=

    2023

  12. [20]

    2022 , note=

    Kobayashi, Shinichi , TITLE =. 2022 , note=

    2022

  13. [21]

    Manuscripta Math

    Hatley, Jeffrey and Lei, Antonio and Vigni, Stefano , TITLE =. Manuscripta Math. , FJOURNAL =. 2022 , NUMBER =

    2022

  14. [22]

    Algebra Number Theory , FJOURNAL =

    Burungale, Ashay and Castella, Francesc and Kim, Chan-Ho , TITLE =. Algebra Number Theory , FJOURNAL =. 2021 , NUMBER =

    2021

  15. [23]

    Algebraic number theory , SERIES =

    Jannsen, Uwe , TITLE =. Algebraic number theory , SERIES =

  16. [24]

    , Fjournal =

    Bertolini, M. , Fjournal =. Iwasawa theory for elliptic curves over imaginary quadratic fields , Volume =. J. Th\'eor. Nombres Bordeaux , Number =

  17. [25]

    Hatley, Jeffrey and Lei, Antonio , TITLE =. Math. Res. Lett. , FJOURNAL =. 2019 , NUMBER =

    2019

  18. [26]

    , TITLE =

    Bertolini, M. , TITLE =. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. , FJOURNAL =. 1994 , NUMBER =

    1994

  19. [27]

    Compositio Math

    Bertolini, Massimo , TITLE =. Compositio Math. , FJOURNAL =. 1995 , NUMBER =

    1995

  20. [28]

    Howard, Benjamin , TITLE =. Compos. Math. , FJOURNAL =. 2004 , NUMBER =

    2004

  21. [29]

    Duke Math J

    Howard, Benjamin , TITLE =. Duke Math J. , FJOURNAL =. 2004 , NUMBER =

    2004

  22. [30]

    Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales

    Torsion of rational elliptic curves over quadratic fields , author=. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas , volume=. 2014 , publisher=

    2014

  23. [31]

    Archiv der Mathematik , volume=

    On the ubiquity of trivial torsion on elliptic curves , author=. Archiv der Mathematik , volume=. 2010 , publisher=

    2010

  24. [32]

    Proceedings of the Acad

    Elliptic curves with no exceptional primes , author =. Proceedings of the Acad

  25. [33]

    Greenberg, Ralph , TITLE =. Kyoto J. Math. , FJOURNAL =. 2011 , PAGES =

    2011

  26. [34]

    Kummer theory for abelian varieties over local elds , author=. Invent. math , volume=

  27. [35]

    Greenberg, Ralph , TITLE =. Doc. Math. , FJOURNAL =. 2007 , PAGES =

    2007

  28. [36]

    On the Structure of S elmer Groups

    Greenberg, Ralph. On the Structure of S elmer Groups. Elliptic Curves, Modular Forms and Iwasawa Theory. 2016

    2016

  29. [37]

    Greenberg, Ralph , TITLE =. Kyoto J. Math. , FJOURNAL =. 2010 , NUMBER =

    2010

  30. [38]

    Greenberg, Ralph , TITLE =. Proc. Nat. Acad. Sci. U.S.A. , FJOURNAL =. 1997 , NUMBER =

    1997

  31. [39]

    Brink, David , TITLE =. Math. Comp. , FJOURNAL =. 2007 , NUMBER =

    2007

  32. [40]

    Duke Math J

    Bertolini, Massimo and Darmon, Henri and Prasanna, Kartik , TITLE =. Duke Math J. , VOLUME =. 2013 , NUMBER =

    2013

  33. [41]

    Pacific J

    Calegari, Frank , TITLE =. Pacific J. Math. , FJOURNAL =. 2006 , NUMBER =

    2006

  34. [42]

    Duke Math

    Vatsal, Vinayak , TITLE =. Duke Math. J. , FJOURNAL =. 2003 , NUMBER =

    2003

  35. [43]

    Matar, Ahmed , TITLE =. Int. J. Number Theory , FJOURNAL =. 2018 , NUMBER =

    2018

  36. [44]

    Matar, Ahmed , TITLE =. Math. Proc. Cambridge Philos. Soc. , FJOURNAL =. 2016 , NUMBER =

    2016

  37. [45]

    Acta Arith

    Matar, Ahmed , TITLE =. Acta Arith. , FJOURNAL =. 2021 , NUMBER =

    2021

  38. [46]

    Castella, Francesc and Wan, Xin , TITLE =

  39. [47]

    Dieulefait, Luis and Wiese, Gabor , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2011 , NUMBER =. doi:10.1090/S0002-9947-2011-05477-2 , URL =

    2011 doi

  40. [48]

    Lei, Antonio , TITLE =. Canad. Math. Bull. , FJOURNAL =. 2014 , NUMBER =

    2014

  41. [49]

    Lei, Antonio and Lim, Meng Fai , TITLE =. Int. J. Number Theory , FJOURNAL =. 2022 , NUMBER =

    2022

  42. [50]

    Longo, Matteo and Vigni, Stefano , TITLE =. Kyoto J. Math. , FJOURNAL =. 2019 , NUMBER =

    2019

  43. [51]

    Longo, Matteo and Vigni, Stefano , TITLE =. Boll. Unione Mat. Ital. , FJOURNAL =. 2019 , NUMBER =

    2019

  44. [52]

    Kriz, Daniel and Li, Chao , TITLE =

  45. [53]

    Castella, Francesc , TITLE =. J. London Math. Soc. , FJOURNAL =. 2017 , NUMBER =

    2017

  46. [54]

    Algebra Number Theory , FJOURNAL =

    Castella, Francesc and Kim, Chan-Ho and Longo, Matteo , TITLE =. Algebra Number Theory , FJOURNAL =. 2017 , NUMBER =

    2017

  47. [55]

    Ponsinet, Gautier , TITLE =. Math. Z. , FJOURNAL =. 2020 , NUMBER =

    2020

  48. [56]

    Hatley, Jeffrey and Lei, Antonio , TITLE =. Ann. Inst. Fourier (Grenoble) , FJOURNAL =. 2019 , NUMBER =

    2019

  49. [57]

    Lei, Antonio and Loeffler, David and Zerbes, Sarah Livia , TITLE =. Canad. J. Math. , FJOURNAL =. 2017 , NUMBER =

    2017

  50. [58]

    Israel J

    Lei, Antonio and Sprung, Florian , TITLE =. Israel J. Math. , FJOURNAL =. 2020 , NUMBER =

    2020

  51. [59]

    Sprung, Florian , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2013 , PAGES =

    2013

  52. [60]

    Correspondance modulaire galois - quaternions pour un corps p-adique

    Vign \'e ras, Marie-France. Correspondance modulaire galois - quaternions pour un corps p-adique. Number Theory: Proceedings of the Journ \'e es Arithm \'e tiques held in Ulm, FRG, September 14--18, 1987. 1989

    1987

  53. [61]

    Hatley, Jeffrey and Kundu, Debanjana and Ray, Anwesh , TITLE =

  54. [62]

    D.) , year=

    Kim, Byoung Du (B. D.) , year=. The parity conjecture for elliptic curves at supersingular reduction primes , volume=. Compositio Mathematica , publisher=. doi:10.1112/S0010437X06002569 , number=

    doi

  55. [63]

    Hatley, Jeffrey , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2017 , NUMBER =

    2017

  56. [64]

    Kidwell, Keenan , TITLE =. J. Number Theory , FJOURNAL =. 2018 , PAGES =

    2018

  57. [65]

    Distributions

    Amice, Yvette and V. Distributions. Journ\'ees

  58. [66]

    Berger, Laurent , TITLE =. Doc. Math. , FJOURNAL =. 2003 , NUMBER =

    2003

  59. [67]

    Berger, Laurent , TITLE =. Compos. Math. , FJOURNAL =. 2004 , NUMBER =

    2004

  60. [68]

    Longo, Matteo and Vigni, Stefano , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2017 , NUMBER =

    2017

  61. [69]

    Edixhoven, Bas , TITLE =. Invent. Math. , FJOURNAL =. 1992 , NUMBER =

    1992

  62. [70]

    Zhang, Shaowei , TITLE =. Int. J. Math. Math. Sci. , FJOURNAL =. 2004 , NUMBER =

    2004

  63. [71]

    Mathematical Research Letters , volume=

    Prime twists of elliptic curves , author=. Mathematical Research Letters , volume=. 2019 , publisher=

    2019

  64. [72]

    Israel J

    Elliptic curves of odd modular degree , author=. Israel J. Math. , volume=. 2009 , publisher=

    2009

  65. [73]

    Experiment

    Computing the modular degree of an elliptic curve , author=. Experiment. Math. , volume=. 2002 , publisher=

    2002

  66. [74]

    Number Theory, Analysis and Geometry: In Memory of Serge Lang , pages=

    The modular degree, congruence primes, and multiplicity one , author=. Number Theory, Analysis and Geometry: In Memory of Serge Lang , pages=. 2012 , publisher=

    2012

  67. [75]

    Berger, Laurent and Li, Hanfeng and Zhu, Hui June , title =. Math. Ann. , year =

  68. [76]

    Bloch, Spencer and Kato, Kazuya , title=. The

  69. [77]

    Buzzard, Kevin and Gee, Toby , title =. Int. Math. Res. Not. , year =

  70. [78]

    and Edixhoven, Bas , TITLE =

    Coleman, Robert F. and Edixhoven, Bas , TITLE =. Math. Ann. , FJOURNAL =. 1998 , NUMBER =

    1998

  71. [79]

    Coleman, Robert and McCallum, William , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 1988 , PAGES =

    1988

  72. [80]

    Colmez, Pierre , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1998 , NUMBER =

    1998

  73. [81]

    2022 , note =

    Hamidi, Parham , Title =. 2022 , note =

    2022

  74. [82]

    GAP -- Groups, Algorithms, and Programming, Version 4.16.0

  75. [83]

    S\'eminaire Bourbaki , year =

    Deligne, Pierre , title =. S\'eminaire Bourbaki , year =

  76. [84]

    Emerton, Matthew and Pollack, Robert and Weston, Tom , title =. Invent. Math. , year =

  77. [85]

    Newforms and functional equations

    Li, Wen-Ch'ing Winnie. Newforms and functional equations. Mathematische Annalen. 1975. doi:10.1007/BF01344466

    1975 doi

  78. [86]

    Arithmetic theory of elliptic curves (Cetraro, 1997) , pages=

    Greenberg, Ralph , title=. Arithmetic theory of elliptic curves (Cetraro, 1997) , pages=

    1997

  79. [87]

    Flach, Matthias , title =. J. Reine Angew. Math. , year =

  80. [88]

    , TITLE =

    Agashe, Amod and Ribet, Kenneth and Stein, William A. , TITLE =. Pure Appl. Math. Q. , FJOURNAL =. 2006 , NUMBER =. doi:10.4310/PAMQ.2006.v2.n2.a11 , URL =

    2006 doi

  81. [89]

    Arithmetic algebraic geometry (

    Edixhoven, Bas , TITLE =. Arithmetic algebraic geometry (. 1991 , ISBN =. doi:10.1007/978-1-4612-0457-2\_3 , URL =

    1991 doi

  82. [90]

    Manuscripta Math

    Weston, Tom , TITLE =. Manuscripta Math. , FJOURNAL =. 2005 , NUMBER =

    2005

  83. [91]

    Greenberg, Ralph , title =. Adv. Stud. Pure Math. , year =

  84. [92]

    Jetchev, Dimitar and Loeffler, David and Zerbes, Sarah Livia , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2021 , NUMBER =

    2021

  85. [93]

    Kazalicki, Matija and Kohen, Daniel , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2018 , NUMBER =. doi:10.1090/proc/13759 , URL =

    2018 doi

  86. [94]

    Kazalicki, Matija and Kohen, Daniel , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2019 , NUMBER =

    2019

  87. [95]

    Caro, Jerson and Pasten, Hector , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2022 , NUMBER =. doi:10.1090/proc/15942 , URL =

    2022 doi

  88. [96]

    Bouniakowsky, V , TITLE =. M. 184 , PAGES =

  89. [97]

    Elliptic curves with rational 2-torsion and related ternary Diophantine equations , url=

    Mulholland, Jamie Thomas , year=. Elliptic curves with rational 2-torsion and related ternary Diophantine equations , url=. doi:http://dx.doi.org/10.14288/1.0080089 , school=

    doi

  90. [98]

    A conjecture of. Proc. Amer. Math. Soc. , FJOURNAL =. 2021 , NUMBER =. doi:10.1090/proc/15376 , URL =

    2021 doi

  91. [99]

    Algebra Number Theory , FJOURNAL =

    Yazdani, Soroosh , TITLE =. Algebra Number Theory , FJOURNAL =. 2011 , NUMBER =. doi:10.2140/ant.2011.5.37 , URL =

    2011 doi

  92. [100]

    Greenberg, Ralph and Vatsal, Vinayak , TITLE =. Invent. Math. , FJOURNAL =. 2000 , NUMBER =

    2000

  93. [101]

    In Development of Iwasawa Theory – the Centennial of K

    Kobayashi, Shinichi and Ota, Kazuto , TITLE =. In Development of Iwasawa Theory – the Centennial of K. Iwasawa's Birth (M. Kurihara et al, eds), Adv. Stud. Pure Math., Math. Soc. Japan , year =

  94. [102]

    Iwasawa theory of elliptic modular forms over imaginary quadratic fields at non-ordinary primes , journal =

    B\". Iwasawa theory of elliptic modular forms over imaginary quadratic fields at non-ordinary primes , journal =. 2021 , number =

    2021

  95. [103]

    Bosma, Wieb and Cannon, John and Playoust, Catherine , TITLE =. J. Symbolic Comput. , FJOURNAL =. 1997 , NUMBER =. doi:10.1006/jsco.1996.0125 , URL =

    1997

  96. [104]

    Hara, Takashi , TITLE =. J. Number Theory , FJOURNAL =. 2010 , NUMBER =

    2010

  97. [105]

    and Freitas, Nuno , TITLE =

    Cremona, John E. and Freitas, Nuno , TITLE =. Rev. Mat. Iberoam. , FJOURNAL =. 2022 , NUMBER =. doi:10.4171/rmi/1269 , URL =

    2022 doi

  98. [106]

    Fisher, Tom , TITLE =. LMS J. Comput. Math. , FJOURNAL =. 2014 , NUMBER =. doi:10.1112/S1461157014000059 , URL =

    2014 doi

  99. [107]

    Dokchitser, Vladimir , TITLE =. Proc. London Math. Soc. (3) , FJOURNAL =. 2005 , NUMBER =

    2005

  100. [108]

    Dokchitser, Tim and Dokchitser, Vladimir , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2007 , NUMBER =

    2007

  101. [109]

    Hachimori, Yoshitaka and Matsuno, Kazuo , TITLE =. J. Algebraic Geometry , VOLUME =. 1999 , PAGES =

    1999

  102. [110]

    Iovita, Adrian and Pollack, Robert , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2006 , PAGES =

    2006

  103. [111]

    Ast\'erisque , year =

    Kato, Kazuya , title =. Ast\'erisque , year =

  104. [112]

    Arithmetic algebraic geometry (Trento, 1991) , pages=

    Kato, Kazuya , title=. Arithmetic algebraic geometry (Trento, 1991) , pages=

    1991

  105. [113]

    K -Theory , FJOURNAL =

    Kato, Kazuya , TITLE =. K -Theory , FJOURNAL =. 2005 , NUMBER =

    2005

  106. [114]

    2021 , month = jun, publisher =

    Ahmed Matar , title =. 2021 , month = jun, publisher =. doi:10.1007/s40993-021-00262-0 , url =

    2021 doi

  107. [115]

    Kim, Byoung Du , TITLE =. Asian J. Math. , FJOURNAL =. 2009 , NUMBER =

    2009

  108. [116]

    Kim, Byoung Du , TITLE =. J. Aust. Math. Soc. , FJOURNAL =. 2013 , NUMBER =

    2013

  109. [117]

    Kobayashi, Shinichi , title =. Invent. Math. , year =

  110. [118]

    Kurihara, Masato , TITLE =. Invent. Math. , FJOURNAL =. 2002 , NUMBER =

    2002

  111. [119]

    Compositio Math

    Lei,Antonio , title =. Compositio Math. , volume =

  112. [120]

    Lei, Antonio and Loeffler, David and Zerbes, Sarah Livia , TITLE =. Asian J. Math. , FJOURNAL =. 2010 , NUMBER =

    2010

  113. [121]

    Lei, Antonio and Ponsinet, Gautier , title =

  114. [122]

    Forum Math

    Castella, Francesc and Hsieh, Ming-Lun , TITLE =. Forum Math. Sigma , FJOURNAL =. 2022 , PAGES =

    2022

  115. [123]

    Algebra Number Theory , FJOURNAL =

    Lei, Antonio and Loeffler, David and Zerbes, Sarah Livia , TITLE =. Algebra Number Theory , FJOURNAL =. 2011 , NUMBER =

    2011

  116. [124]

    Loeffler, David and Zerbes, Sarah Livia , TITLE =. Int. J. Number Theory , FJOURNAL =. 2014 , NUMBER =

    2014

  117. [125]

    Matsuno, Kazuo , TITLE =. J. Number Theory , FJOURNAL =. 2003 , NUMBER =

    2003

  118. [126]

    Mazur, Barry and Swinnerton-Dyer, Peter , title =. Invent. Math. , year =. doi:10.1007/BF01389997 , fjournal =

    doi

  119. [127]

    Mazur, Barry , TITLE =. Invent. Math. , FJOURNAL =. 1972 , PAGES =

    1972

  120. [128]

    Mazur, Barry and Wiles, Andrew , TITLE =. Invent. Math. , FJOURNAL =. 1984 , NUMBER =

    1984

  121. [129]

    Agboola, Adebisi and Howard, Benjamin , TITLE =. Ann. Inst. Fourier (Grenoble) , FJOURNAL =. 2006 , NUMBER =

    2006

  122. [130]

    Mazur, Barry and Rubin, Karl , TITLE =. Mem. Amer. Math. Soc. , FJOURNAL =. 2004 , NUMBER =

    2004

  123. [131]

    Mazur, Barry and Tate, John and Teitelbaum, Jeremy , title =. Invent. Math. , year =

  124. [132]

    Kolyvagin's method for

    Nekov\'. Kolyvagin's method for. Invent. Math. , FJOURNAL =. 1992 , NUMBER =

    1992

  125. [133]

    Anticyclotomic

    Brako. Anticyclotomic. Int. Math. Res. Not. IMRN , FJOURNAL =. 2011 , NUMBER =

    2011

  126. [134]

    Some consequences of a formula of

    Nekov\'. Some consequences of a formula of. Algebra Number Theory , FJOURNAL =. 2013 , NUMBER =

    2013

  127. [135]

    , TITLE =

    Panchishkin, Alexey A. , TITLE =. 1991 , PAGES =

    1991

  128. [136]

    Perrin-Riou, Bernadette , title=. J. Amer. Math. Soc. , volume=. 2000 , number=

    2000

  129. [137]

    Perrin-Riou, Bernadette , TITLE =. Ann. Inst. Fourier (Grenoble) , FJOURNAL =. 1993 , NUMBER =

    1993

  130. [138]

    Perrin-Riou, Bernadette , title =. Invent. Math. , year =

  131. [139]

    Ast\'erisque , FJOURNAL =

    Perrin-Riou, Bernadette , TITLE =. Ast\'erisque , FJOURNAL =. 1995 , PAGES =

    1995

  132. [140]

    Compositio Math

    Perrin-Riou, Bernadette , TITLE =. Compositio Math. , FJOURNAL =. 1998 , NUMBER =

    1998

  133. [141]

    Experiment

    Perrin-Riou, Bernadette , TITLE =. Experiment. Math. , FJOURNAL =. 2003 , PAGES =

    2003

  134. [142]

    Duke Math

    Pollack, Robert , title =. Duke Math. J. , year =

  135. [143]

    Pollack, Robert and Rubin, Karl , title =. Ann. of Math. (2) , year =

  136. [144]

    Duke Math

    Pollack, Robert and Weston, Tom , title =. Duke Math. J. , fjournal =. 2011 , number =

    2011

  137. [145]

    Documenta Mathematica , year =

    Pollack, Robert and Weston, Tom , title =. Documenta Mathematica , year =

  138. [146]

    Pollack, Robert and Weston, Tom , title =. Compos. Math. , year =

  139. [147]

    Kim, Chan-Ho and Pollack, Robert and Weston, Tom , title =

  140. [148]

    The. The. 2026 , note =

    2026

  141. [149]

    Bulletin of the London Mathematical Society , volume =

    Melistas, Mentzelos , title =. Bulletin of the London Mathematical Society , volume =. doi:https://doi.org/10.1112/blms.12952 , url =. https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/blms.12952 , abstract =

    doi

  142. [150]

    2022 , note=

    Hatley, Jeffrey and Kundu, Debanjana and Lei, Antonio and Ray, Jishnu , Title =. 2022 , note=

    2022

  143. [151]

    Lazard, Michel , TITLE =. Inst. Hautes \'. 1962 , PAGES =

    1962

  144. [152]

    Skinner, Christopher and Urban, Eric , TITLE =. Invent. Math. , FJOURNAL =. 2014 , NUMBER =. doi:10.1007/s00222-013-0448-1 , URL =

    2014 doi

  145. [153]

    Francesc Castella and Mirela. On the. 2018 , Eprint =

    2018

  146. [154]

    , TITLE =

    Ribet, Kenneth A. , TITLE =. Modular functions of one variable,

  147. [155]

    Toward equivariant

    Ritter, J. Toward equivariant. Indag. Math. (N.S.) , FJOURNAL =. 2004 , NUMBER =

    2004

  148. [156]

    Toward equivariant

    Ritter, J. Toward equivariant. Math. Ann. , FJOURNAL =. 2006 , NUMBER =

    2006

  149. [157]

    , TITLE =

    Rohrlich, David E. , TITLE =. Math. Ann. , FJOURNAL =. 1988 , NUMBER =

    1988

  150. [158]

    2000 , PAGES =

    Rubin, Karl , TITLE =. 2000 , PAGES =

    2000

  151. [159]

    Rubin, Karl , TITLE =. Invent. Math. , FJOURNAL =. 1987 , NUMBER =

    1987

  152. [160]

    Compositio Math

    Rubin, Karl , TITLE =. Compositio Math. , FJOURNAL =. 1985 , NUMBER =

    1985

  153. [161]

    Rubin, Karl , TITLE =. Invent. Math. , FJOURNAL =. 1991 , NUMBER =

    1991

  154. [162]

    Schneider, Peter , title=. Invent. Math. , volume=

  155. [163]

    Duke Math

    Serre, Jean-Pierre , TITLE =. Duke Math. J. , FJOURNAL =. 1987 , NUMBER =

    1987

  156. [164]

    1978 , PAGES =

    Serre, Jean-Pierre , TITLE =. 1978 , PAGES =

    1978

  157. [165]

    Serre, Jean-Pierre , TITLE =. Invent. Math. , FJOURNAL =. 1972 , NUMBER =. doi:10.1007/BF01405086 , URL =

    1972 doi

  158. [166]

    , TITLE =

    Sharifi, Romyar T. , TITLE =. J. Number Theory , FJOURNAL =. 2001 , NUMBER =

    2001

  159. [167]

    Duke Math

    Shimura, Goro , TITLE =. Duke Math. J. , FJOURNAL =. 1978 , NUMBER =

    1978

  160. [168]

    Shimura, Goro , TITLE =. Comm. Pure Appl. Math. , FJOURNAL =. 1976 , NUMBER =

    1976

  161. [169]

    1994 , PAGES =

    Snaith, Victor , TITLE =. 1994 , PAGES =

    1994

  162. [170]

    Hsieh, Ming-Lun , TITLE =. Doc. Math. , FJOURNAL =. 2014 , PAGES =

    2014

  163. [171]

    Integral

    B\". Integral. Math. Z. , FJOURNAL =. 2017 , NUMBER =

    2017

  164. [172]

    Ito , TITLE =

    Sprung, Florian E. Ito , TITLE =. J. Number Theory , FJOURNAL =. 2012 , NUMBER =

    2012

  165. [173]

    Sprung, Florian , title =

  166. [174]

    Sugiyama, Ken-ichi , title=

  167. [175]

    Diamond, Fred and Taylor, Richard , TITLE =. Invent. Math. , VOLUME =. 1994 , PAGES =

    1994

  168. [176]

    Automorphic forms, representations and

    Tate, John , TITLE =. Automorphic forms, representations and

  169. [177]

    Nonarchimedean measures associated with

    Vi. Nonarchimedean measures associated with. Mat. Sb. (N.S.) , VOLUME =. 1976 , NUMBER =

    1976

  170. [178]

    On two variable p -adic L -functions

    Rodney Yager. On two variable p -adic L -functions. Ann. of Math. (2). 1982

    1982

  171. [179]

    Chida, Masataka and Hsieh, Ming-Lun , TITLE =. Compos. Math. , FJOURNAL =. 2015 , NUMBER =

    2015

  172. [180]

    Castella, Francesc and Hsieh, Ming-Lun , TITLE =. Math. Ann. , FJOURNAL =. 2018 , NUMBER =

    2018

  173. [181]

    Weston, Tom , TITLE =. J. Number Theory , FJOURNAL =. 2005 , NUMBER =

    2005

  174. [182]

    , Fjournal =

    Perrin-Riou, B. , Fjournal =. Fonctions. Bull. Soc. Math. France , Number =

  175. [183]

    Duke Math

    Bertolini, Massimo and Darmon, Henri , TITLE =. Duke Math. J. , FJOURNAL =. 1994 , NUMBER =

    1994

  176. [184]

    , TITLE =

    Gross, Benedict H. , TITLE =. 1991 , MRCLASS =. doi:10.1017/CBO9780511526053.009 , URL =

    1991 doi

  177. [185]

    Kolyvagin, V. A. , TITLE =. The. 1990 , MRCLASS =

    1990

  178. [186]

    Arithmetic theory of elliptic curves (

    Greenberg, Ralph , TITLE =. Arithmetic theory of elliptic curves (. 1999 , ISBN =. doi:10.1007/BFb0093453 , URL =

    1999 doi

  179. [187]

    Cornut, Christophe , TITLE =. Invent. Math. , FJOURNAL =. 2002 , NUMBER =. doi:10.1007/s002220100199 , URL =

    2002 doi

  180. [188]

    , TITLE =

    Burungale, Ashay A. , TITLE =. J. Inst. Math. Jussieu , FJOURNAL =. 2017 , NUMBER =

    2017

  181. [189]

    , TITLE =

    Burungale, Ashay A. , TITLE =. J. Algebraic Geom. , FJOURNAL =. 2020 , NUMBER =. doi:10.1090/jag/748 , URL =

    2020 doi

  182. [190]

    Journal de th

    Akashi series and Euler characteristics of signed Selmer groups of elliptic curves with semistable reduction at primes above p , author=. Journal de th

  183. [191]

    Kolyvagin's result on the vanishing of

    Matar, Ahmed and Nekov\'. Kolyvagin's result on the vanishing of. J. Th\'. 2019 , NUMBER =

    2019

  184. [192]

    , TITLE =

    Bourbaki, N. , TITLE =. 1965 , PAGES =

    1965

  185. [193]

    Balister, Paul N and Howson, Susan , journal=. Note on. 1997 , publisher=

    1997

  186. [194]

    Greenberg, Ralph , TITLE =. Doc. Math. , FJOURNAL =. 2006 , NUMBER =

    2006

  187. [195]

    Ochi, Yoshihiro and Venjakob, Otmar , TITLE =. Math. Proc. Cambridge Philos. Soc. , FJOURNAL =. 2003 , NUMBER =

    2003

  188. [196]

    Analytic ranks of elliptic curves over cyclotomic fields , author=. J. Reine Angew. Math. , number=. 2002 , publisher=

    2002

  189. [197]

    2022 , PAGES =

    Hida, Haruzo , TITLE =. 2022 , PAGES =

    2022

  190. [198]

    Notes on non-commutative

    Hachimori, Yoshitaka and Ochiai, Tadashi , journal=. Notes on non-commutative. 2010 , publisher=

    2010

  191. [199]

    Proceedings of the International Congress of Mathematicians , volume=

    Modular curves and arithmetic , author=. Proceedings of the International Congress of Mathematicians , volume=. 1983 , organization=

    1983

  192. [200]

    Greenberg, Ralph , journal=. The. 1973 , publisher=

    1973

  193. [201]

    Kim, Byoung Du , TITLE =. Canad. J. Math. , FJOURNAL =. 2014 , NUMBER =

    2014

  194. [202]

    Sprung, Florian , note=. The

  195. [203]

    Iwasawa main conjecture for supersingular elliptic curves and

    Wan, Xin , note=. Iwasawa main conjecture for supersingular elliptic curves and

  196. [204]

    Balakrishnan, Sai Sanjeev and Lei, Antonio and Palvannan, Bharathwaj , Title=

  197. [205]

    Kleine, S\"oren and Matar, Ahmed and Sujatha, Ramdorai , Title=

  198. [206]

    Lei, Antonio and Sujatha, Ramdorai , TITLE =. Tokyo J. Math. , FJOURNAL =. 2020 , NUMBER =

    2020

  199. [207]

    Acta Arith

    Ray, Jishnu , TITLE =. Acta Arith. , FJOURNAL =. 2022 , NUMBER =

    2022

  200. [208]

    Forum Math

    Lei, Antonio and Palvannan, Bharathwaj , TITLE =. Forum Math. Sigma , FJOURNAL =. 2019 , PAGES =

    2019

  201. [209]

    and Schneider, P

    Coates, J. and Schneider, P. and Sujatha, R. , TITLE =. J. Inst. Math. Jussieu , FJOURNAL =. 2003 , NUMBER =

    2003

  202. [210]

    Kundu, Debanjana and Lei, Antonio and Ray, Anwesh , TITLE =. Doc. Math. , FJOURNAL =. 2022 , PAGES =

    2022

  203. [211]

    Greni\'e, Lo\"ic and Molteni, Giuseppe , TITLE =. J. Number Theory , FJOURNAL =. 2019 , PAGES =. doi:10.1016/j.jnt.2018.12.005 , URL =

    2019 doi

  204. [212]

    Bach, Eric and Sorenson, Jonathan , TITLE =. Math. Comp. , FJOURNAL =. 1996 , NUMBER =. doi:10.1090/S0025-5718-96-00763-6 , URL =

    1996 doi

  205. [213]

    An effective version of C hebotarev's density theorem

    Das, Sourabhashis and Kadiri, Habiba and Ng, Nathan. An effective version of C hebotarev's density theorem. arXiv:2508.09480

    arXiv

  206. [214]

    2007 , PAGES =

    Stein, William , TITLE =. 2007 , PAGES =. doi:10.1090/gsm/079 , URL =

    2007 doi

  207. [215]

    Stein, William and Wuthrich, Christian , TITLE =. Math. Comp. , FJOURNAL =. 2013 , NUMBER =

    2013

  208. [216]

    p -twisted S elmer near-companion curves

    Kim, Minseok. p -twisted S elmer near-companion curves

  209. [217]

    Yu, Myungjun , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2019 , NUMBER =. doi:10.1090/tran/7563 , URL =

    2019 doi

  210. [218]

    Jha, Somnath and Majumdar, Dipramit and Shekhar, Sudhanshu , TITLE =. Ann. Inst. Fourier (Grenoble) , FJOURNAL =. 2021 , NUMBER =. doi:10.5802/aif.3392 , URL =

    2021 doi

  211. [219]

    Ast\'erisque , FJOURNAL =

    Fontaine, Jean-Marc , TITLE =. Ast\'erisque , FJOURNAL =. 1994 , PAGES =

    1994

  212. [220]

    2008 , PAGES =

    Neukirch, J\"urgen and Schmidt, Alexander and Wingberg, Kay , TITLE =. 2008 , PAGES =. doi:10.1007/978-3-540-37889-1 , URL =

    2008 doi

  213. [221]

    and Rubin, K

    Mazur, B. and Rubin, K. and Silverberg, A. , TITLE =. J. Algebra , FJOURNAL =. 2007 , NUMBER =. doi:10.1016/j.jalgebra.2007.02.052 , URL =

    2007 doi

  214. [222]

    Klagsbrun, Zev and Mazur, Barry and Rubin, Karl , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2013 , NUMBER =. doi:10.4007/annals.2013.178.1.5 , URL =

    2013 doi

  215. [223]

    and Silverberg, A

    Rubin, K. and Silverberg, A. , TITLE =. Elliptic curves, modular forms, &. 1995 , ISBN =

    1995

  216. [224]

    Graduate Texts in Mathematics , volume=

    Cohomology of Groups , author=. Graduate Texts in Mathematics , volume=. 1982 , publisher=

    1982

  217. [225]

    Finding large

    Mazur, Barry and Rubin, Karl , journal=. Finding large. 2007 , publisher=

    2007

  218. [226]

    GitHub repository for related scripts and data , author=

  219. [227]

    2001 , PAGES =

    Adelmann, Clemens , TITLE =. 2001 , PAGES =. doi:10.1007/b80624 , URL =

    2001 doi

  220. [228]

    Day, Tori and Gajek-Leonard, Rylan , TITLE =. Int. J. Number Theory , FJOURNAL =. 2025 , NUMBER =. doi:10.1142/S1793042125500678 , URL =

    2025 doi

  221. [229]

    , TITLE =

    Brumer, Armand and Pacetti, Ariel and Poor, Cris and Tornar\'ia, Gonzalo and Voight, John and Yuen, David S. , TITLE =. Algebra Number Theory , FJOURNAL =. 2019 , NUMBER =. doi:10.2140/ant.2019.13.1145 , URL =

    2019 doi

  222. [230]

    and Hatley, Jeffrey and Ricci, James , TITLE =

    Daniels, Harris B. and Hatley, Jeffrey and Ricci, James , TITLE =. Acta Arith. , FJOURNAL =. 2016 , NUMBER =. doi:10.4064/aa8275-7-2016 , URL =

    2016 doi

  223. [231]

    Shekhar, Sudhanshu , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2016 , NUMBER =. doi:10.1090/proc/12993 , URL =

    2016 doi

  224. [232]

    and Lozano-Robledo, \'Alvaro and Morrow, Jackson S

    Daniels, Harris B. and Lozano-Robledo, \'Alvaro and Morrow, Jackson S. , TITLE =. Rev. Mat. Iberoam. , FJOURNAL =. 2023 , NUMBER =. doi:10.4171/rmi/1424 , URL =

    2023 doi

  225. [233]

    and Lozano-Robledo, \'Alvaro , TITLE =

    Daniels, Harris B. and Lozano-Robledo, \'Alvaro , TITLE =. Ann. Inst. Fourier (Grenoble) , FJOURNAL =. 2023 , NUMBER =. doi:10.5802/aif.3520 , URL =

    2023 doi

  226. [234]

    and Morrow, Jackson S

    Daniels, Harris B. and Morrow, Jackson S. , TITLE =. Trans. Amer. Math. Soc. Ser. B , FJOURNAL =. 2022 , PAGES =. doi:10.1090/btran/95 , URL =

    2022 doi

  227. [235]

    Brau, Julio and Jones, Nathan , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2016 , NUMBER =. doi:10.1090/proc/12786 , URL =

    2016 doi

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.