Pith. sign in

REVIEW 3 major objections 4 minor 53 references

The eigencurve at crystalline points with scalar Frobenius and Gross-Stark regulators

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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

Pith's one-line read At p-irregular weight-one points, the eigencurve is locally four transversally crossing branches, and the localized ordinary étale cohomology is not a free Hecke module.

desk verdict A technically strong paper that gives a new conditional description of the eigencurve at p-irregular weight-one points, but the abstract overstates the unconditional content. read the letter →

arxiv 2502.00876 v1 pith:3EV7VVJ4 submitted 2025-02-02 math.NT

Adel Betina , Alexandre Maksoud , Alice Pozzi This is my paper · ORCID
classification math.NT MSC 11F3311G1811F8011R23
keywords p-adiceigencurveweightoneformsp-irregularpointsHidafamiliesGorensteinnessordinaryétalecohomologyGross-Starkregulatorstranscendence
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 pins down the local geometry of the p-adic eigencurve at classical weight-one cusp forms that are p-irregular, meaning the two roots of the p-th Hecke polynomial coincide. There the usual deformation-theoretic machinery breaks down: the ordinary deformation functor is not representable, and several Hida families can pass through the same weight-one point. The authors prove that, under two non-vanishing p-adic regulator hypotheses, the completed local ring is the fourfold power series ring $\overline{\mathbb{Q}}_p[[X_1,X_2,X_3,X_4]]$ modulo the relations $X_iX_j=0$ for $i\neq j$; geometrically, four irreducible components cross transversally at the point. The same hypotheses imply that the ordinary p-adic étale cohomology of the tower of modular curves, localized at the corresponding Hecke prime, is not free over the Hecke algebra, so a common freeness assumption in Iwasawa theory fails precisely in this p-irregular situation.

What carries the argument

The load-bearing object is a degree-at-most-four polynomial $Q(S)$, built from p-adic logarithms of S-units of the number field cut out by the adjoint representation $\mathrm{ad}\,\rho$; its roots are exactly the possible residual slopes $s$ of the ordinary line $V^+$ at $x$ for which the Selmer group $\mathrm{Sel}(\mathrm{ad}\,\rho,V^+)$ is one-dimensional. The hypothesis (sl) is the nonvanishing of the discriminant of $Q(S)$, and (reg) is the nonvanishing of the resultant of $Q(S)$ with a second log-polynomial $P_L(S)$; together they make the four roots simple and distinct and force the trace-zero Selmer group $\mathrm{Sel}(\mathrm{ad}^0\rho,V^+)$ to vanish for every line. Higher infinitesimal deformations of the Artin representation $\rho$ along a candidate component are then analyzed: the first cocycle is forced to be the unique trace-$\lambda$ Selmer class, and all higher cocycles vanish, proving that each component is étale over the weight space.

What would settle it

Take an explicit p-irregular weight-one newform and compute the matrices $L$ and $M$ from the S-units of its Galois field; if $\mathrm{Disc}(Q(S))=0$ for a single such form, the hypotheses (sl) fails and the predicted four-branch geometry cannot hold by this mechanism, while a numerical check of $\mathrm{res}_S(P_L(S),Q(S))\neq 0$ would turn the conditional theorem into an unconditional statement for that point.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem A: if $f_\alpha$ is the p-stabilization of a p-irregular weight one newform $f$, and if the two regulator non-vanishing conditions (sl) and (reg) hold, then the completed local ring $T$ of the eigencurve at $x$ is isomorphic to $\overline{\mathbb{Q}}_p[[X_1,X_2,X_3,X_4]]/(X_iX_j)_{i<j}$, with the weight map sending $X$ to $X_1+X_2+X_3+X_4$. Consequently $T$ is not Gorenstein, and exactly four irreducible components of the eigencurve pass through $x$, each étale over the weight space. The companion Theorem B states that the localized ordinary étale cohomology $H_{\infty,\mathfrak{p}_f}$ of the tower $X_1(Np^r)$ is not free over the localized Hida–Hecke algebra $h_{\mathfrak{p}_f}$, that the standard ordinary cohomology exact sequence does not split as Hecke modules, and that the reduction modulo the maximal ideal is $\rho\oplus\rho$, where $\rho$ is the Artin representation attached to $f$. The regulator hypotheses are shown to follow from the weak p-adic Schanuel conjecture when $\rho$ is exotic and from the p-adic four exponentials conjecture when $\rho$ has real multiplication; in the Klein (RM+CM) case they hold unconditionally.

Load-bearing premise

The result stands or falls on the nonvanishing of two p-adic regulator expressions, the discriminant of $Q(S)$ and the resultant of $P_L(S)$ and $Q(S)$; in the exotic case neither is proven, and each would follow from the weak p-adic Schanuel conjecture, while in the real-multiplication case one reduces to the p-adic four exponentials conjecture and is unconditional only in the Klein (RM+CM) case.

Editorial extensions

If this is right

  • At any p-irregular weight-one point satisfying (sl) and (reg), the eigencurve is locally a transversal union of four smooth curves, and the completed local ring is not Gorenstein.
  • The generalized overconvergent eigenspace for the Hecke eigensystem of $f_\alpha$ inside weight-one overconvergent forms is two-dimensional and isomorphic to $H^1(\mathbb{Q},\mathrm{ad}^0\rho)$, confirming a conjecture on the dimension of this generalized eigenspace.
  • The localized ordinary étale cohomology $H_{\infty,\mathfrak{p}_f}$ and its $\pm$-parts are not free over the localized Hida–Hecke algebra, so the standard ordinary cohomology exact sequence does not split there; the characteristic-zero fiber is $\rho\oplus\rho$.
  • The triangulation of $\varphi,\Gamma$-modules over the normalization does not descend to any open neighborhood of $x$ in the eigencurve.
  • For exotic $\rho$, the regulator hypotheses are consequences of the weak p-adic Schanuel conjecture, so the whole description is unconditional at all weight-one points modulo that conjecture; in the real-multiplication case the analogous role is played by the p-adic four exponentials conjecture, with the Klein case unconditional.

Reading between the lines

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

  • The polynomial $Q(S)$ and the two regulators give a practical algorithm for deciding, on any concrete eigencurve point, whether four branches meet: compute the S-units and the p-adic logarithms, then test simplicity of the roots of $Q(S)$.
  • The non-freeness of $H_{\infty,\mathfrak{p}_f}$ suggests that around p-irregular weight-one points the canonical-period and reciprocity-law constructions that assume freeness will require a Cohen–Macaulay or derived replacement, rather than a free module argument.
  • One may expect analogous four-branch non-Gorenstein local rings at irregular weight-one points of other eigenvarieties, wherever the same Selmer-group criterion produces four admissible ordinary lines.
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 / 4 minor

Summary. The paper studies the completed local ring T of the p-adic eigencurve at a p-irregular classical weight-one cusp form f whose attached Galois representation has scalar Frobenius at p. Under two explicit regulator nonvanishing hypotheses, (sl) and (reg), introduced in Section 4.2, the authors prove Theorem A: T is isomorphic to Qpbar[[X1,X2,X3,X4]]/(XiXj) for i<j, with the weight map sending X to X1+X2+X3+X4. This implies that four irreducible components pass through the point, meet transversally, and each is étale over the weight space, and that T is not Gorenstein. The proof proceeds by analyzing the possible ordinary lines through the residual representation, encoding them as roots of a quartic polynomial Q(S), and then using S-unit regulators and p-adic logarithms to show that exactly four Hida families occur. As applications, Theorem B shows that the localized ordinary étale cohomology of the modular tower X1(Np^r) is not free over the localized Hecke algebra, that Ohta's exact sequence does not split, and Theorem 6.7 proves a conjecture of Darmon–Lauder–Rotger on the dimension of the generalized overconvergent eigenspace. The paper is long and technically detailed, with several steps relying on prior work of the authors and of Betina–Dimitrov.

Significance. If correct, this is a significant contribution to the arithmetic of eigencurves and Iwasawa theory. It gives the first complete description of the local geometry at p-irregular weight-one non-CM points, where the usual R=T method fails and the ordinary deformation functor is not representable. The explicit structure T = Qpbar[[X1,...,X4]]/(XiXj) is striking and has concrete consequences: non-Gorensteinness, non-freeness of localized étale cohomology, and a resolution of the Darmon–Lauder–Rotger conjecture in the irregular setting. The hypotheses (sl) and (reg) are not fitted parameters: the regulator matrices L and M are defined from S-units and p-adic logarithms independently of the eigencurve, and the conditional statements in the body are honestly labeled. The paper also connects the hypotheses to Gross–Stark regulators and to the weak p-adic Schanuel conjecture, which is a useful and natural framework. However, the advertised scope in the abstract is broader than what is actually proved unconditionally, since for exotic representations and for RM non-CM representations the hypotheses are only known under unproved transcendence conjectures, except for the Klein case.

major comments (3)
  1. [Abstract and §1; §4.2, Lemmas 4.6–4.8] The abstract presents the main result and its application as unconditional: it says 'A complete description ... is given' and 'as an application, we show that ... is not free', without displaying the hypotheses (sl) and (reg). In the body, however, Theorem A is explicitly conditional on (sl) and (reg), and those hypotheses are not known unconditionally for the main non-CM cases: for exotic ρ, Lemmas 4.6(ii), 4.7(ii) and 4.8(ii) deduce (res), (sl) and (reg) only from Conjecture 2.1 (the weak p-adic Schanuel conjecture); for RM non-CM ρ, Lemma 4.7(iii) reduces (sl) to the inequality (1), which is unconditional only in the Klein (RM+CM) case and otherwise is a consequence of Conjecture 2.4. Consequently, the 'complete description' advertised in the abstract is not yet established for those cases. I recommend that the abstract and the introductory summary be revised to state the conditional nature of the theorems explicitly and to distinguish the unconditional Klein case from the conjecturally conditional RM and exotic cases.
  2. [§4.3.2] The assertion that conditions (sl) and (reg) 'unconditionally hold' in the Klein case is stated without proof; the reader must infer this from the displayed relations in the paragraph. Since this is the only unconditional non-CM case mentioned in the introduction, the verification should be included or at least made explicit enough that the claim can be checked directly from L−(φ), L−(φ), Sφ and the displayed formula for Q(S). As written, the sentence rests on an unshown computation, albeit a short one.
  3. [§6, Proposition 6.6 and Theorem 6.5] The proof that there are exactly four Hida families (Theorem 6.5(i)) uses Proposition 6.6(iii), whose determinant computation is not shown in detail; the text says 'Using explicit expressions provided by Proposition 4.1, we see that' and then gives the final formula. This is acceptable in a research paper provided the formula is correct, but the displayed expression for det(E) has an apparent typo: the denominator in the displayed quotient repeats (s−s''') and omits (s′−s'''). Since the conclusion only needs the slopes to be pairwise distinct, the nonvanishing is unaffected, but the formula should be corrected.
minor comments (4)
  1. [§1, Acknowledgments] The word 'Acknolwedgements' is misspelled; it should be 'Acknowledgements'.
  2. [§4.3.1, Eq. (1)] The sentence 'We note that the above inequality is (1) unconditionally satified if ρ has both RM and CM' contains the typo 'satified' and the phrasing is awkward; it should be 'the inequality (1) is unconditionally satisfied'.
  3. [§5.2] The symbol K is used both for a quadratic field in the earlier dihedral cases and for the quotient field of A0 in Section 5.2; although the context makes the meaning clear, using a different letter for the quotient field would avoid confusion.
  4. [§7.2, Corollary 7.4] The notation M±_{O_tildeC} and M_{O_C} in the statement of Corollary 7.4 is introduced after the statement; a brief indication of the gluing construction before the corollary would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the regulator hypotheses are independent inputs.

full rationale

The paper's central theorem is conditional on two explicit nonvanishing hypotheses, (sl) and (reg), stated in Section 4.2. These are conditions on matrices L and M defined in (25) from p-adic logarithms of S-units of the field H cut out by the adjoint representation. Neither L nor M involves the eigencurve or Hecke algebra, and the nonvanishing is not fitted to any subset of data. The conclusion is not used to establish these hypotheses. The proof derives T from them via polynomials Q(S) and P_L(S); slopes are roots of Q(S) by Proposition 5.4(ii), and (sl) and (reg) force distinctness and nonvanishing. The isomorphism is not the same as Disc(Q) != 0. Prior results [Mak23] and [Mak24] are external published theorems with independent proofs; they are cited as lemmas, not self-supporting premises. The CM comparison with [BD21b] is not the method for non-CM cases. The abstract's omission of the hypotheses is an exposition issue, not circularity.

Assumptions & free parameters 0 free parameters · 14 assumptions · 0 invented entities

The central theorem is verified conditional on the nonvanishing of two p-adic regulators (sl) and (reg); beyond that, the proof relies on standard results in Hida theory, Galois cohomology, and p-adic transcendence theory. The transcendental conjectures (weak p-adic Schanuel, p-adic four exponentials) are used only to show that the regulator hypotheses hold in certain cases, not as assumptions of the main theorem. No fitted parameters or invented entities appear.

assumptions (14)
  • domain assumption rho is odd, irreducible, unramified at p, with Frobenius at p acting by a scalar (p-irregular weight one form).
    Sets the setting of the paper; Section 1 and Section 3, assumptions (i)-(iii).
  • domain assumption Hypothesis (sl): Disc(Q(S)) != 0.
    Assumed in Theorem A and Theorem B; in the exotic case it follows from Conjecture 2.1, in the RM case it reduces to inequality (1).
  • domain assumption Hypothesis (reg): res_S(Q(S), P_L(S)) != 0.
    Assumed in Theorem A, Theorem 6.5, Theorem 7.1; unconditional in the RM case, follows from Conjecture 2.1 in the exotic case.
  • domain assumption Hypothesis (res): res_S(P_L, P_M) != 0.
    Used in Propositions 3.7-3.9, 5.4, 6.6; implied by (reg) and proved unconditionally in the RM case.
  • domain assumption Weak p-adic Schanuel conjecture (Conjecture 2.1).
    Used to prove (res), (sl), (reg) in the exotic case.
  • domain assumption p-adic Four Exponentials conjecture (Conjecture 2.4).
    Used in Lemma 4.7(iii) to show (sl) is a consequence of inequality (36) in the RM case.
  • standard math Baker-Brumer theorem (Prop 2.2).
    Establishes injectivity of the p-adic logarithm on the Q-span of algebraic numbers modulo the p-line; used in Lemma 2.5.
  • standard math Brumer-Waldschmidt-Roy rank bound (Prop 2.3, cited from [Mak23]).
    Used for linear independence of regulator entries and in Corollary 6.2.
  • standard math Hida theory: ordinary Hecke algebra, ordinary part of etale cohomology is finite free over Lambda, exact sequence (63), strong multiplicity one.
    Used throughout Sections 5-7 to relate components of the eigencurve to Hida families.
  • standard math Wiles's ordinarity theorem ([Wil88]) and Nyssen-Rouquier pseudo-representation lifting.
    Provides the ordinary filtration and true representations from pseudo-characters; used in Section 5.2.
  • standard math Poitou-Tate duality and finiteness of class groups.
    Used in Section 3 to compute Selmer group dimensions.
  • standard math Chebotarev density theorem.
    Used in Corollary 6.2 to pick regular primes where the matrices have full rank.
  • standard math Kedlaya-Liu (Phi,Gamma)-modules and Kedlaya-Pottharst-Xiao triangulation.
    Used in Section 7.2 to construct the triangulation and its non-descent.
  • domain assumption p odd for the application in Section 7.
    Theorem B is stated for odd p; Ohta's setup requires odd p.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The eigencurve at crystalline points with scalar Frobenius and Gross-Stark regulators." pith.science (2026). https://pith.science/paper/3EV7VVJ4

@misc{pith2026250200876,
  author       = {Pith},
  title        = {Pith review of: The eigencurve at crystalline points with scalar Frobenius and Gross-Stark regulators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3EV7VVJ4}},
  note         = {Machine review of arXiv:2502.00876}
}
abstract

A complete description of the local geometry of the $p$-adic eigencurve at $p$-irregular classical weight one cusp forms is given in the cases where the usual $R=T$ methods fall short. As an application, we show that the ordinary $p$-adic \'etale cohomology group attached to the tower of elliptic modular curves $X_1(Np^r)$ is not free over the Hecke algebra, when localized at a $p$-irregular weight one point.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

53 extracted references · 51 canonical work pages

  1. [1]

    Arithmetic of critical p -adic l -functions

    Denis Benois and Kâzım Büyükboduk. Arithmetic of critical p -adic l -functions. 2024

    work page 2024

  2. [2]

    e l Bella\

    Jo\" e l Bella\" che and Ga\" e tan Chenevier. Families of G alois representations and S elmer groups. Ast\' e risque , (324):xii+314, 2009

    work page 2009

  3. [3]

    e l Bella \

    Jo \"e l Bella \" che and Mladen Dimitrov. On the eigencurve at classical weight 1 points. Duke Mathematical Journal , 165(2):245--266, 2016

    work page 2016

  4. [4]

    A geometric view on I wasawa theory

    Adel Betina and Mladen Dimitrov. A geometric view on I wasawa theory. J. Th\'eor. Nombres Bordeaux , 33(3):703--731, 2021

    work page 2021

  5. [5]

    Geometry of the eigencurve at CM points and trivial zeros of K atz p-adic L -functions

    Adel Betina and Mladen Dimitrov. Geometry of the eigencurve at CM points and trivial zeros of K atz p-adic L -functions. Advances in Mathematics , 384:107724, 2021

    work page 2021

  6. [6]

    On the failure of G orensteinness at weight 1 E isenstein points of the eigencurve

    Adel Betina, Mladen Dimitrov, and Alice Pozzi. On the failure of G orensteinness at weight 1 E isenstein points of the eigencurve. Amer. J. Math. , 144(1):227--265, 2022

    work page 2022

  7. [7]

    Beilinson- F lach elements and E uler systems I : S yntomic regulators and p -adic R ankin L -series

    Massimo Bertolini, Henri Darmon, and Victor Rotger. Beilinson- F lach elements and E uler systems I : S yntomic regulators and p -adic R ankin L -series. J. Algebraic Geom. , 24(2):355--378, 2015

    work page 2015

  8. [8]

    Beilinson- F lach elements and E uler systems II : the B irch- S winnerton- D yer conjecture for H asse- W eil- A rtin L -series

    Massimo Bertolini, Henri Darmon, and Victor Rotger. Beilinson- F lach elements and E uler systems II : the B irch- S winnerton- D yer conjecture for H asse- W eil- A rtin L -series. J. Algebraic Geom. , 24(3):569--604, 2015

    work page 2015

Show all 53 references
  1. [9]

    Heegner points and B eilinson- K ato elements: a conjecture of P errin- R iou

    Massimo Bertolini, Henri Darmon, and Rodolfo Venerucci. Heegner points and B eilinson- K ato elements: a conjecture of P errin- R iou. Adv. Math. , 398:Paper No. 108172, 50, 2022

    2022

  2. [10]

    Critical p -adic l -functions

    Joel Bellaiche. Critical p -adic l -functions. Inventiones mathematicae , 189(1):1--60, 2012

    2012

  3. [11]

    e l Bella \

    Jo \"e l Bella \" che. The Eigenbook: Eigenvarieties, families of Galois representations, p-adic L-functions . Pathways in Mathematics. Springer International Publishing, 2021

    2021

  4. [12]

    e l Bella \

    Jo \"e l Bella \" che. On certain cohen--macaulay modules. 2023

    2023

  5. [13]

    On extra zeros of p -adic L -functions: the crystalline case

    Denis Benois. On extra zeros of p -adic L -functions: the crystalline case. In Iwasawa theory 2012 , volume 7 of Contrib. Math. Comput. Sci. , pages 65--133. Springer, Heidelberg, 2014

    2012

  6. [14]

    Boston, HW

    N. Boston, HW. Lenstra, and KA. Ribet. Quotients of group-rings arising from 2-dimensional representations. Comptes rendus de l'Académie des sciences. Série I. Mathématique , 312(4):323--328, 1991

    1991

  7. [15]

    Zeta elements for elliptic curves and applications, 2024

    Ashay Burungale, Christopher Skinner, Ye Tian, and Xin Wan. Zeta elements for elliptic curves and applications, 2024. preprint available at https://arxiv.org/pdf/2409.01350.pdf

    2024 arXiv

  8. [16]

    Balanced diagonal classes and rational points on elliptic curves

    Massimo Bertolini, Marco Adamo Seveso, and Rodolfo Venerucci. Balanced diagonal classes and rational points on elliptic curves. Ast\'erisque , (434):175--201, 2022. Heegner points, Stark-Heegner points, and diagonal classes

    2022

  9. [17]

    Reciprocity laws for balanced diagonal classes

    Massimo Bertolini, Marco Adamo Seveso, and Rodolfo Venerucci. Reciprocity laws for balanced diagonal classes. Ast\'erisque , (434):77--174, 2022. Heegner points, Stark-Heegner points, and diagonal classes

    2022

  10. [18]

    The geometry of hida families I : -adic de rham cohomology

    Bryden Cais. The geometry of hida families I : -adic de rham cohomology. Mathematische annalen , 372(1-2):781--844, 2018

    2018

  11. [19]

    Robert F. Coleman. Classical and overconvergent modular forms. Invent. Math. , 124(1-3):215--241, 1996

    1996

  12. [20]

    Cho and V

    S. Cho and V. Vatsal. Deformations of induced G alois representations. J. Reine Angew. Math. , 556:79--98, 2003

    2003

  13. [21]

    Factorization of p -adic R ankin L -series

    Samit Dasgupta. Factorization of p -adic R ankin L -series. Invent. Math. , 205(1):221--268, 2016

    2016

  14. [22]

    On the G ross- S tark conjecture

    Samit Dasgupta, Mahesh Kakde, and Kevin Ventullo. On the G ross- S tark conjecture. Ann. of Math. (2) , 188(3):833--870, 2018

    2018

  15. [23]

    First order p -adic deformations of weight one newforms

    Henri Darmon, Alan Lauder, and Victor Rotger. First order p -adic deformations of weight one newforms. In L-functions and automorphic forms , volume 10 of Contrib. Math. Comput. Sci. , pages 39--80. Springer, Cham, 2017

    2017

  16. [24]

    L -invariants of A rtin motives

    Mladen Dimitrov and Alexandre Maksoud. L -invariants of A rtin motives. Ann. Math. Qu\' e . , 47(1):49--71, 2023

    2023

  17. [25]

    Formes modulaires de poids 1

    Pierre Deligne and Jean-Pierre Serre. Formes modulaires de poids 1 . In Annales scientifiques de l' \'E cole Normale Sup \'e rieure , volume 7, pages 507--530. Elsevier, 1974

    1974

  18. [26]

    The weight in S erre's conjectures on modular forms

    Bas Edixhoven. The weight in S erre's conjectures on modular forms. Invent. Math. , 109(3):563--594, 1992

    1992

  19. [27]

    Variation of I wasawa invariants in H ida families

    Matthew Emerton, Robert Pollack, and Tom Weston. Variation of I wasawa invariants in H ida families. Inventiones mathematicae , 163(3):523--580, 2006

    2006

  20. [28]

    On conjectures of S harifi

    Takako Fukaya and Kazuya Kato. On conjectures of S harifi. Kyoto J. Math. , 64(2):277--420, 2024

    2024

  21. [29]

    Benedict H. Gross. A tameness criterion for G alois representations associated to modular forms (mod p ). Duke Math. J. , 61(2):445--517, 1990

    1990

  22. [30]

    Galois representations into GL_2 ( Z _p [[X]]) attached to ordinary cusp forms

    Haruzo Hida. Galois representations into GL_2 ( Z _p [[X]]) attached to ordinary cusp forms. Invent. Math. , 85(3):545--613, 1986

    1986

  23. [31]

    Relative p-adic Hodge theory : foundations

    Kiran Sridhara Kedlaya. Relative p-adic Hodge theory : foundations . Astérisque ; 371. Soc. Math. de France, Paris, 2015

    2015

  24. [32]

    Kedlaya, Jonathan Pottharst, and Liang Xiao

    Kiran S. Kedlaya, Jonathan Pottharst, and Liang Xiao. Cohomology of arithmetic families of ( , ) -modules. J. Amer. Math. Soc. , 27(4):1043--1115, 2014

    2014

  25. [33]

    L. J. P. Kilford and Gabor Wiese. On the failure of the G orenstein property for H ecke algebras of prime weight. Experiment. Math. , 17(1):37--52, 2008

    2008

  26. [34]

    Rankin- E isenstein classes in C oleman families

    David Loeffler and Sarah Livia Zerbes. Rankin- E isenstein classes in C oleman families. Res. Math. Sci. , 3:Paper No. 29, 53, 2016

    2016

  27. [35]

    On the birch-swinnerton-dyer conjecture for modular abelian surfaces, 2023

    David Loeffler and Sarah Livia Zerbes. On the birch-swinnerton-dyer conjecture for modular abelian surfaces, 2023. preprint available at https://arxiv.org/pdf/2110.13102.pdf

    2023 arXiv

  28. [36]

    On the rank of L eopoldt's and G ross's regulator maps

    Alexandre Maksoud. On the rank of L eopoldt's and G ross's regulator maps. Doc. Math. , 28(6):1441--1471, 2023

    2023

  29. [37]

    On generalized main conjectures and p -adic S tark conjectures for A rtin motives

    Alexandre Maksoud. On generalized main conjectures and p -adic S tark conjectures for A rtin motives. Trans. Amer. Math. Soc. , 377(6):3845--3904, 2024

    2024

  30. [38]

    B. Mazur. Modular curves and the E isenstein ideal. Inst. Hautes \'Etudes Sci. Publ. Math. , (47):33--186, 1977. With an appendix by Mazur and M. Rapoport

    1977

  31. [39]

    Class fields of abelian extensions of Q

    Barry Mazur and Andrew Wiles. Class fields of abelian extensions of Q . Invent. Math. , 76(2):179--330, 1984

    1984

  32. [40]

    Cohomology of number fields , volume 323 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

    J\" u rgen Neukirch, Alexander Schmidt, and Kay Wingberg. Cohomology of number fields , volume 323 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, Berlin, second edition, 2008

    2008

  33. [41]

    Pseudo-repr \'e sentations

    Louise Nyssen. Pseudo-repr \'e sentations. Mathematische Annalen , 306(1):257--283, 1996

    1996

  34. [42]

    On the p-adic E ichler- S himura isomorphism for -adic cusp forms

    M Ohta. On the p-adic E ichler- S himura isomorphism for -adic cusp forms. Journal für die reine und angewandte Mathematik , 463:49--98, 1995

    1995

  35. [43]

    Ordinary p -adic \'etale cohomology groups attached to towers of elliptic modular curves

    Masami Ohta. Ordinary p -adic \'etale cohomology groups attached to towers of elliptic modular curves. II . Math. Ann. , 318(3):557--583, 2000

    2000

  36. [44]

    Overconvergent modular forms

    Vincent Pilloni. Overconvergent modular forms. Ann. Inst. Fourier (Grenoble) , 63(1):219--239, 2013

    2013

  37. [45]

    p -adic L -functions and p -adic representations , volume 3 of SMF/AMS Texts and Monographs

    Bernadette Perrin-Riou. p -adic L -functions and p -adic representations , volume 3 of SMF/AMS Texts and Monographs . American Mathematical Society, Providence, RI; Soci\'et\'e Math\'ematique de France, Paris, 2000. Translated from the 1995 French original by Leila Schneps and...

    2000

  38. [46]

    K. A. Ribet. On modular representations of Gal ( Q / Q ) arising from modular forms. Invent. Math. , 100(2):431--476, 1990

    1990

  39. [47]

    Caract \'e risation des caract \`e res et pseudo-caract \`e res

    Rapha \"e l Rouquier. Caract \'e risation des caract \`e res et pseudo-caract \`e res. Journal of algebra , 180(2):571--586, 1996

    1996

  40. [48]

    Ribet and William A

    Kenneth A. Ribet and William A. Stein. Lectures on S erre's conjectures. In Arithmetic algebraic geometry ( P ark C ity, UT , 1999) , volume 9 of IAS/Park City Math. Ser. , pages 143--232. Amer. Math. Soc., Providence, RI, 2001

    1999

  41. [49]

    A reciprocity map and the two-variable p -adic L -function

    Romyar Sharifi. A reciprocity map and the two-variable p -adic L -function. Ann. of Math. (2) , 173(1):251--300, 2011

    2011

  42. [50]

    Sur les d\' e formations p -adiques de certaines repr\' e sentations automorphes

    Christopher Skinner and Eric Urban. Sur les d\' e formations p -adiques de certaines repr\' e sentations automorphes. J. Inst. Math. Jussieu , 5(4):629--698, 2006

    2006

  43. [51]

    Un sous-groupe p -divisible de la jacobienne de x_1(n p_r) comme module sur l'algèbre de hecke

    Jacques Tilouine. Un sous-groupe p -divisible de la jacobienne de x_1(n p_r) comme module sur l'algèbre de hecke. Bulletin de la Société mathématique de France , 79:329--360, 1987

    1987

  44. [52]

    Multiplicities of galois representations of weight one

    Gabor Wiese. Multiplicities of galois representations of weight one. Algebra & Number Theory , 1(1):67--85, 2007

    2007

  45. [53]

    On ordinary -adic representations associated to modular forms

    Andrew Wiles. On ordinary -adic representations associated to modular forms. Invent. Math. , 94(3):529--573, 1988

    1988

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.