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Anticyclotomic diagonal classes and Beilinson--Flach elements

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Pith's one-line read This paper proves that, under a rank-one torsion-free Selmer assumption, the weighted Beilinson–Flach class BF(f,g,h) lies in the balanced Selmer group and equals Ω_{f,γ} L^p_g(f,g,h) times the anticyclotomic diagonal-cycle class κ(f,g,h),

desk verdict A well-built, transparent conditional comparison of diagonal cycles to Beilinson-Flach elements; the advertised equality rests on an unproved rank-one Selmer hypothesis, and the exceptional-zero applications on a conjecture. read the letter →

arxiv 2509.07564 v1 pith:PCEHTT4K submitted 2025-09-09 math.NT

Raúl Alonso , Lois Omil-Pazos , Óscar Rivero This is my paper
classification math.NT MSC 11R2311F85
keywords anticyclotomicEulersystemsBeilinson–FlachclassesdiagonalcyclesHidafamiliesp-adicL-functionsPerrin-RioumapsCMdegenerationirregularweight-oneEisensteinseries
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 claims that two very different p-adic cohomology classes attached to the triple of Hida families (f,g,h)—the anticyclotomic diagonal cycle class and a suitably weighted Beilinson–Flach class—are actually the same class up to an explicit scalar. The scalar is the product of a p-adic period Ω_{f,γ} and the g-unbalanced triple product p-adic L-function L^p_g(f,g,h), and the equality takes place in the balanced Selmer group once a rank-one torsion-free Selmer assumption is granted. The weight-one input f is the irregular Eisenstein series attached to an imaginary quadratic field K, so its Galois representation splits into the cyclotomic and quadratic characters L(1)⊕L(ε_K), and the Beilinson–Flach system correspondingly splits into two Rankin–Selberg pieces. The proof's engine is a factorization of L^p_g(f,g,h)^2 as a product of two Hida–Rankin p-adic L-functions, followed by an injectivity argument through Perrin-Riou maps. If correct, this is the anticyclotomic analogue of the earlier Heegner-point/Beilinson–Kato comparison.

What carries the argument

The central object is the weighted Beilinson–Flach class BF(f,g,h)=v_{f,1}⊗L_p(g,h⊗ε_K)κ_{g,h}+v_{f,ε_K}⊗L_p(g,h)κ_{g,h⊗ε_K}, built from the two Beilinson–Flach classes κ_{g,h} and κ_{g,h⊗ε_K} after the CM family f degenerates to the irregular weight-one Eisenstein series. The identity that carries the argument is the factorization of the triple product p-adic L-function into a product of two Hida–Rankin p-adic L-functions (Proposition 5.2), which is a p-adic Artin-formalism consequence of the decomposition V_f ≅ L(1)⊕L(ε_K). The injectivity of the Perrin-Riou map L^{-+}_{gh} (Proposition 5.10) then forces the two classes to be proportional, and the proportionality constant is read off from

What would settle it

Specialize the equality at a good crystalline point (y,z) with L^p_g(f,g,h)≠0 and compute the Perrin-Riou images of both classes using the interpolation formulas of Theorems 4.8 and 4.15: if the ratio of the images is not Ω_{f,γ}^{-1}λ_{N_g}(g)L^p_g(f,g,h), Theorem 5.18 fails. A direct computation of H^1_{G∪+}(Q,V^†_{fgh}) at one arithmetic point would also settle the rank-one torsion-free premise.

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Extended reading notes

Core claim

Under Assumption 5.3 (the triple product p-adic L-function L^p_g(f,g,h) is not identically zero) and the unproved structural assumption that H^1_{G∪+}(Q,V^†_{fgh}) is a torsion-free Λ_{gh}-module of rank one, the paper proves that the weighted Beilinson–Flach class BF(f,g,h) lies in the balanced Selmer group and satisfies λ_{N_g}(g)·BF(f,g,h)=Ω_{f,γ}·L^p_g(f,g,h)·κ(f,g,h). Here BF(f,g,h) is assembled from the Beilinson–Flach classes of (g,h) and (g,h⊗ε_K), each weighted by the corresponding Hida–Rankin p-adic L-function, after specialization of the CM family f to the irregular Eisenstein series f=Eis_1(ε_K); κ(f,g,h) is the balanced diagonal-cycle class. The identification is made by compari

Load-bearing premise

The comparison rests on an unproved structural premise: the relevant Selmer group is a torsion-free module of rank one over the Iwasawa algebra of the two Hida families, which the authors expect from sign considerations but do not prove; if that fails, the injectivity step identifying the two classes collapses, and in the exceptional-zero section the existence of an improved Beilinson–Flach class is also assumed.

Editorial extensions

If this is right

  • For fixed good crystalline specializations g=g_{y0}, h=h_{z0}, the equality descends to a relation between the specialized weighted Beilinson–Flach class and the specialized diagonal cycle class, whenever the balanced Selmer group is one-dimensional and L^p_g(f,g_α,h_α)≠0 (Corollary 5.25).
  • The Perrin-Riou image of BF under Log_{ω_g⊗ω_h} factors as Ω_{f,γ} L^p_g(f,g,h) L^p_f(f,g,h)/λ_{N_g}(g), giving a purely p-adic factorization of a big logarithm by two triple-product L-functions (Corollary 1.4/5.29).
  • In the exceptional-zero case, the improved classes satisfy Ω_{f,γ} cL^p_g(f,g,h) bκ(f,g,h)=λ_{N_g}(g) cBF(f,g,h) (Theorem 1.6/6.17-6.20), so the comparison survives the vanishing of the relevant Euler factors once the predicted improved Beilinson–Flach class is available.
  • The result supplies a new instance of the principle that cyclotomic and anticyclotomic Euler systems can be matched after a degeneration: diagonal cycles here play the role of Heegner points, Beilinson–Flach classes the role of Kato classes.
  • A byproduct is a mixed factorization (Corollary 5.22) relating L^p_g and L^p_h to Hida–Rankin L-functions with and without the quadratic twist ε_K, even though the interpolation regions are disjoint.

Reading between the lines

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

  • If the rank-one torsion-free Selmer assumption can be proved by control theorems—the authors note it is expected from sign considerations—the same argument should give the equality unconditionally for this family of triples, without changing the reciprocity-law core.
  • The mechanism should extend to any CM Hida family specializing to a weight-one Eisenstein series, not only Eis_1(ε_K); the paper's Remark 2.3 indicates the broader class, so the same comparison should hold for those twist families.
  • A natural construction to attempt next is the improved Beilinson–Flach class bκ_{g,h} directly from Rankin–Eisenstein classes, which would upgrade the exceptional-zero theorem from conditional to unconditional.
  • Because the diagonal-cycle construction does not use modular units, the same comparison may be reproducible in settings where Beilinson–Flach classes are not available, such as Shimura curves, giving anticyclotomic comparisons entirely through geometric classes.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper compares the anticyclotomic diagonal-cycle class κ(f,g,h) with a weighted Beilinson–Flach class BF(f,g,h) attached to a triple (f,g,h) of Hida families, where f is a CM family passing through the irregular weight-one Eisenstein series Eis_1(ε_K). The main result, Theorem 1.1 / Corollary 5.19, asserts that, under a rank-one torsion-free Selmer hypothesis on H^1_{G∪+}(Q,V^†_{fgh}) and a non-vanishing hypothesis on the triple product p-adic L-function L^p_g(f,g,h), one has λ_{N_g}(g) BF(f,g,h) = Ω_{f,γ} L^p_g(f,g,h) κ(f,g,h). The proof factors the triple product p-adic L-function into two Hida–Rankin L-functions (Prop. 5.2), constructs BF satisfying the balanced local condition, and compares the images under a Perrin–Riou map L^{+−+}_{fgh} (Thm. 5.18). Injectivity of that map on the Selmer group is then invoked to identify the two global classes. A further section treats the p-exceptional case using improved L-functions, improved diagonal cycles, and a conjectural improved Beilinson–Flach class cBF, giving Theorem 1.6 / Proposition 6.18 under additional hypotheses including Conjecture 6.8.

Significance. If the conditional statements are accepted, the paper provides a new explicit bridge between two a priori unrelated types of Euler systems, extending the Bertolini–Darmon–Venerucci comparison of Heegner points and Beilinson–Kato classes to the setting of anticyclotomic diagonal cycles and Beilinson–Flach elements. The paper is careful and transparent about its hypotheses: Assumption 5.3, the rank-one/torsion-free Selmer assumption, and Conjecture 6.8 are all stated explicitly. The use of p-adic Artin formalism in Proposition 5.2 and the appeal to independent reciprocity laws of [KLZ17], [BSV22b], and [DR22] are coherent. The main caveat is that the central equality of global cohomology classes rests on an unproved Selmer rank hypothesis, and the exceptional-zero theorem additionally rests on an unproved conjecture; this limits the unconditional scope of the paper rather than invalidating its internal logic.

major comments (3)
  1. [§5.4, Prop. 5.10] The injectivity of L^{-+}_{gh} is asserted using the fact that H^1(Q_p, V_g^-⊗V_h^+(2-t_1)) is a 'torsion-free Λ_gh-module of rank 1'. However, Lemma 5.9, which is the immediately preceding lemma, proves only torsion-freeness; no proof or reference is given for the rank-one assertion. This rank statement is load-bearing: Prop. 5.10 is used in Prop. 5.14 to show that BF(f,g,h) satisfies the balanced local condition, and it is therefore needed for the main comparison. The rank-one fact may be standard from local Euler characteristic computations, but it needs to be stated and proved or explicitly cited.
  2. [§5.5, Cor. 5.19 and Remark 5.20] The central injectivity step in Cor. 5.19 is exactly the unproved hypothesis that H^1_{G∪+}(Q,V^†_{fgh}) is a torsion-free Λ_gh-module of rank 1. If the rank is greater than one, the kernel of L^{+−+}_{fgh}∘pr^{+−+}∘res_p can contain the difference of the two classes, and if there is torsion the same conclusion can fail. Remark 5.20 only says the rank statement is 'expected by sign considerations' and notes that H^0(Q,ρ^†)=0 would give torsion-freeness; it does not prove rank one. Since this same hypothesis is reused in Cor. 5.22 and Prop. 5.29, the paper's main theorem is conditional on a substantial Selmer-rank assertion that is not established. The authors should either prove this assertion under acceptable hypotheses or present the main theorem explicitly as a conditional result with a more thorough discussion of evidence.
  3. [§6.4, Theorem 1.6 vs. Prop. 6.18] The introduction's Theorem 1.6 states the exceptional-zero comparison as an implication of the Selmer rank/torsion-freeness and non-vanishing assumptions, but it omits Conjecture 6.8. The class cBF(f,g,h) appearing in the theorem is defined in Definition 6.16 only under Assumption 6.10, which is exactly Conjecture 6.8. Thus the statement in the introduction is incomplete: without Conjecture 6.8 the object cBF need not exist. The abstract also advertises 'some arithmetic applications' without mentioning that the main results are conditional on an unproved conjecture in the exceptional-zero case. Please state all hypotheses in Theorem 1.6 and add appropriate caveats to the abstract.
minor comments (4)
  1. [§1, p. 1; references] There are several spelling inconsistencies in author names: 'B¨ uy¨ ukkboduk' and 'B¨ uy¨ yukboduk' appear in the introduction and references. These should be normalized.
  2. [Assumption 5.1(iii)] The statement 'h is a newform of level N_h' is ambiguous because h is a Hida family. It presumably means the specialization h_{z0} is a newform of prime-to-p conductor N_h, or that the family has tame conductor N_h. Please clarify.
  3. [Prop. 5.11 proof] The proof says 'This follows as in [BSV22b, Proposition 7.3], working with V_f instead of V_f.' The last displayed object appears to be a typo; it should probably be 'V_f' or 'V_{f}' as appropriate.
  4. [§6.3] In the sentence beginning 'As it occurs with the p-adic L-function, the factor ... appears in the interpolation property when considering its variation in families', the wording is awkward and the reference to the appearing factor is unclear. Please rephrase and make the precise factor and its role explicit.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central comparison is conditional on a declared Selmer-rank injectivity hypothesis that is not derived, and the main inputs are independent reciprocity laws; the exceptional-zero conjecture is explicitly assumed, and self-citations are not load-bearing.

full rationale

The central result (Theorem 1.1 / Corollary 5.19) is not circular. Its proof has two ingredients: Theorem 5.18, an equality of Perrin-Riou images, and an injectivity step that converts image equality into cohomology-class equality. The injectivity is exactly the stated assumption that H^1_{G∪+}(Q,V†_{fgh}) is a torsion-free Λ_gh-module of rank 1; this is declared, not derived. Remark 5.20 explicitly says this is 'expected by sign considerations' and that torsion-freeness can be ensured by imposing H^0(Q,ρ†)=0. Thus the theorem is conditional on an unproved Selmer-rank hypothesis, which is a correctness/coverage risk and not circularity. The factorization in Proposition 5.2 is proved from Artin formalism and interpolation, not assumed. Definition 5.5 deliberately forms BF using Hida--Rankin L-functions as coefficients, so the local image equality in Theorem 5.18 is partly engineered; however, the final global equality is a genuine comparison with the independent diagonal-cycle class from [BSV22b]/[DR22], and the injectivity step is substantive. The reciprocity laws are cited from KLZ17 and BSV22b/DR22, which are external to this paper. The exceptional-zero results assume Conjecture 6.8 explicitly and flag it as conjectural. Self-citations such as [ACR23b] appear only in Corollary 5.25 and Remark 5.26 and are not load-bearing for Theorem 1.1; no uniqueness theorem from the authors is invoked to forbid alternatives, and no fitted parameter is renamed as a prediction. The score of 2 reflects minor, non-load-bearing self-citation and the conditional nature of the main result, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The central claim rests on domain assumptions about the Hida families, two explicit unproved hypotheses (a rank/torsion-free Selmer statement and a non-vanishing of a p-adic L-function), and, in the exceptional-zero half, an unproved conjecture. No numbers are fitted to data; the fixed choices (isomorphism gamma, auxiliary integer c, Hsieh test vectors) are normalizations taken from the cited constructions, not free parameters.

assumptions (6)
  • domain assumption K is imaginary quadratic with p split; f = Eis_1(epsilon_K) is an irregular weight-one Eisenstein series and a CM Hida family f passes through its p-stabilization.
    Section 2.4 and Proposition 2.2 (from BDP22). This sets the CM degeneration framework and is a deep theorem, but it is not re-derived here.
  • domain assumption g and h are residually irreducible and p-distinguished, with gcd(N_g,N_h)=1, chi_g*chi_h = epsilon_K*omega^{2r}, and p does not divide cond(h).
    Assumption 5.1 and running hypotheses. Needed for freeness of big Galois representations and for applicability of the reciprocity laws from KLZ17 and BSV22b/DR22.
  • ad hoc to paper H^1_{G union +}(Q, V^dag_{fgh}) is a torsion-free Lambda_gh-module of rank 1.
    Assumed in Theorem 1.1 and Corollary 5.19, not proved. Used to obtain injectivity of the Perrin-Riou comparison map and hence the equality of classes. Remark 5.20 says it is expected by sign considerations and can be ensured by H^0(Q, rho^dag)=0.
  • ad hoc to paper The triple product p-adic L-function L^p_g(f,g,h) is not identically zero.
    Assumption 5.3. Used to derive nonvanishing of the Hida-Rankin L-functions and injectivity of the local Perrin-Riou map L^{-+}_{gh}.
  • standard math Ramanujan-Petersson bounds for the relevant weight at least 2 modular forms.
    Used in Lemma 5.9 and Propositions 6.5, 6.14, 6.15 to control H^0(Q_p, V/QV) and to assert nonvanishing of certain Euler factors. For weight at least 2 this is Deligne's theorem, a standard proven result.
  • ad hoc to paper Conjecture 6.8: existence of an improved Beilinson-Flach class bkappa_{g,h} (and hence cBF).
    Assumption 6.10. The exceptional-zero results in Section 6 (Theorem 1.6, Corollary 6.20) hold only conditionally on this unproved conjecture.
invented entities (1)
  • bkappa_{g,h} (improved Beilinson-Flach class, and the derived cBF)
    purpose: To remove the vanishing Euler factor in the exceptional-zero setting so the comparison survives specialization at (g,h) with alpha_g*beta_h = p^{(l0+m0-3)/2}.
    Existence is Conjecture 6.8, adopted as Assumption 6.10. The results of Section 6, including Theorem 6.17 and Theorem 1.6, depend on it. No independent construction or verification is provided in this paper.

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Pith. "Pith review of Anticyclotomic diagonal classes and Beilinson--Flach elements." pith.science (2026). https://pith.science/paper/PCEHTT4K

@misc{pith2026250907564,
  author       = {Pith},
  title        = {Pith review of: Anticyclotomic diagonal classes and Beilinson--Flach elements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PCEHTT4K}},
  note         = {Machine review of arXiv:2509.07564}
}
abstract

We present a comparison between the anticyclotomic Euler system of diagonal cycles associated with the convolution of two modular forms and the cyclotomic Beilinson--Flach Euler system. This extends the seminal work of Bertolini, Darmon, and Venerucci, who established a link between (anticyclotomic) Heegner points and the Beilinson--Kato system. Our approach hinges on a detailed analysis of $p$-adic $L$-functions and Perrin-Riou maps and exploits the Eisenstein degeneration of diagonal cycles along Hida families, working with a CM family which specializes to an irregular Eisenstein series in weight one. We use these results to derive some arithmetic applications.

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Works this paper leans on

30 extracted references · 28 canonical work pages

  1. [1]

    Polyakov-Alvarez Formula for Curvilinear Polygonal Domains with Slits

    @article LZ, AUTHOR = Loeffler, David and Zerbes, Sarah Livia , TITLE = Iwasawa theory for the symmetric square , JOURNAL = J. Reine Angew. Math. , FJOURNAL = Journal f\"ur die Reine und Angewandte Mathematik. [Crelle's Journal] , VOLUME = 752 , YEAR = 2019 , PAGES = 179--210 , MRCLASS = 11F67 (11F80 11G18 11R23) , URL = https://www-degruyter-com.ezproxy....

    work page Pith review arXiv 2019

  2. [2]

    Ra\'ul Alonso, Francesc Castella, and \'Oscar Rivero, An anticyclotomic euler system for for adjoint modular galois representations, Ann. Inst. Fourier (Grenoble), to appear (2023), available at arXiv:2204.07658 https://arxiv.org/abs/2204.07658

    work page Pith review arXiv 2023

  3. [3]

    , The diagonal cycle euler system for _2 _2 , J. Inst. Math. Jussieu, to appear (2023), available at arXiv:2106.05322 https://arxiv.org/abs/2106.05322

    work page Pith review arXiv 2023

  4. [4]

    Fabrizio Andreatta and Adrian Iovita, Triple product p -adic L -functions associated to finite slope p -adic families of modular forms , Duke Math. J. 170 (2021), no. 9, 1989--2083. 4278669

    work page 2021

  5. [5]

    K\^azim B\"uy\"yukboduk and Daniele Casazza, On the A rtin formalism for triple product p -adic L -functions: Super-factorization , preprint, arXiv:2301.08383 https://arxiv.org/abs/2301.08383

    arXiv

  6. [6]

    el Bella\

    Jo\"el Bella\"iche and Mladen Dimitrov, On the eigencurve at classical weight 1 points, Duke Math. J. 165 (2016), no. 2, 245--266. 3457673

    work page 2016

  7. [7]

    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 (2022), no. 1, 227--265. 4367418

    work page 2022

  8. [8]

    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 (2022), Paper No. 108172, 50. 4379203

    work page 2022

Show all 30 references
  1. [9]

    e l Bella \

    Jo \"e l Bella \"i che, Critical p -adic L -functions , Invent. Math. 189 (2012), no. 1, 1--60. 2929082

    2012

  2. [10]

    K\^azim B\"uy\"yukboduk and Ryotaro Sakamoto, On the A rtin formalism for triple product p -adic L -functions , preprint, arXiv:2501.06541 https://arxiv.org/abs/2501.06541

    arXiv

  3. [11]

    Ashay Burungale, Christopher Skinner, Ye Tian, and Xin Wan, Zeta elements for elliptic curves and applications, preprint, arXiv:2409.01350 (2024)

    2024 arXiv

  4. [12]

    434, 2022, Heegner points, Stark-Heegner points, and diagonal classes, pp

    Massimo Bertolini, Marco Adamo Seveso, and Rodolfo Venerucci, Balanced diagonal classes and rational points on elliptic curves, no. 434, 2022, Heegner points, Stark-Heegner points, and diagonal classes, pp. 175--201. 4489474

    2022

  5. [13]

    434, 2022, Heegner points, Stark-Heegner points, and diagonal classes, pp

    , Reciprocity laws for balanced diagonal classes, no. 434, 2022, Heegner points, Stark-Heegner points, and diagonal classes, pp. 77--174. 4489473

    2022

  6. [14]

    K\^ a z m Buy\"ukboduk, On N ekov\'a r's heights, exceptional zeros and a conjecture of M azur- T ate- T eitelbaum , Int. Math. Res. Not. IMRN (2016), no. 7, 2197--2237. 3509952

    2016

  7. [15]

    Francesc Castella and Kim Tuan Do, Diagonal cycles and anticyclotomic iwasawa theory of modular forms, preprint, arXiv:2303.06751 https://arxiv.org/abs/2303.06751

  8. [16]

    Henri Darmon and Victor Rotger, Diagonal cycles and E uler systems I : A p -adic G ross- Z agier formula , Ann. Sci. \'Ec. Norm. Sup\'er. (4) 47 (2014), no. 4, 779--832

    2014

  9. [17]

    , Diagonal cycles and E uler systems II : T he B irch and S winnerton- D yer conjecture for H asse- W eil- A rtin L -functions , J. Amer. Math. Soc. 30 (2017), no. 3, 601--672

    2017

  10. [18]

    434, 2022, Heegner points, Stark-Heegner points, and diagonal classes, pp

    , p -adic families of diagonal cycles , no. 434, 2022, Heegner points, Stark-Heegner points, and diagonal classes, pp. 29--75. 4489472

    2022

  11. [19]

    I , Invent

    Haruzo Hida, A p -adic measure attached to the zeta functions associated with two elliptic modular forms. I , Invent. Math. 79 (1985), no. 1, 159--195. 774534

    1985

  12. [20]

    II , Ann

    , A p -adic measure attached to the zeta functions associated with two elliptic modular forms. II , Ann. Inst. Fourier (Grenoble) 38 (1988), no. 3, 1--83. 976685

    1988

  13. [21]

    2, 411--532

    Ming-Lun Hsieh, Hida families and p -adic triple product L -functions , American Journal of Mathematics 143 (2021), no. 2, 411--532

    2021

  14. [22]

    Guido Kings, David Loeffler, and Sarah Livia Zerbes, Rankin- E isenstein classes and explicit reciprocity laws , Camb. J. Math. 5 (2017), no. 1, 1--122

    2017

  15. [23]

    , Rankin- E isenstein classes for modular forms , Amer. J. Math. 142 (2020), no. 1, 79--138

    2020

  16. [24]

    Antonio Lei, David Loeffler, and Sarah Livia Zerbes, Euler systems for R ankin- S elberg convolutions of modular forms , Ann. of Math. (2) 180 (2014), no. 2, 653--771

    2014

  17. [25]

    Reine Angew

    David Loeffler and \'Oscar Rivero, Eisenstein degeneration of E uler systems , J. Reine Angew. Math. 814 (2024), 241--282. 4793345

    2024

  18. [26]

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

    2016

  19. [27]

    thesis, McGill University, Montr\'eal, 2019

    Alice Pozzi, The E igencurve at W eight O ne E isenstein P oints , Ph.D. thesis, McGill University, Montr\'eal, 2019. 4197785

    2019

  20. [28]

    Javier Polo and \'Oscar Rivero, Eisenstein degeneration of beilinson--kato classes and circular units, preprint, 2025

    2025

  21. [29]

    Eric Urban, Nearly overconvergent modular forms, Iwasawa Theory 2012: State of the Art and Recent Advances (Berlin), Contrib. Math. Comput. Sci., vol. 7, Springer, 2014, pp. 401--441

    2012

  22. [30]

    Rodolfo Venerucci, Exceptional zero formulae and a conjecture of P errin- R iou , Invent. Math. 203 (2016), no. 3, 923--972. 3461369

    2016

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