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arxiv: 2604.18433 · v1 · submitted 2026-04-20 · 🧮 math.NT

Families of symplectic Galois representations over small parabolic eigenvarieties for Siegel cuspforms of genus 2

Muhammad Manji , Frederick E. Th{\o}gersen , Ju-Feng Wu This is my paper

Pith reviewed 2026-05-10 03:39 UTC · model grok-4.3

classification 🧮 math.NT
keywords Siegel cuspformseigenvarietiesGalois representationsSaito-Kurokawa liftsBloch-Kato Selmer groupsR=T theoremssymplectic Galois representationsparabolic eigenvarieties
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The pith

Small parabolic eigenvarieties for genus-2 Siegel cuspforms carry refined families of symplectic Galois representations whose local geometry at Saito-Kurokawa lifts is governed by Bloch-Kato Selmer groups.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs small parabolic eigenvarieties that parametrize p-adic families of holomorphic Siegel cuspforms of genus 2. To these spaces it attaches families of symplectic Galois representations, introduced through the new notions of (ϕ, Γ)-modules with G-structures and refined symplectic Galois determinants. Under mild hypotheses the authors prove an infinitesimal R=T theorem that identifies the tangent space of the eigenvariety with a Selmer group. They then apply the theorem to show that, at points corresponding to both finite-slope and infinite-slope Saito-Kurokawa lifts, the local geometry of the eigenvariety is controlled by the Bloch-Kato Selmer group of the associated Galois representation. A reader would care because the construction gives a direct arithmetic interpretation for the singularities or dimensions of these p-adic families.

Core claim

We construct small parabolic eigenvarieties for holomorphic Siegel cuspforms of genus 2 and study families of Galois representations attached to them in the spirit of Bellaïche-Chenevier. In the course, we introduce the notion of (ϕ, Γ)-modules with G-structures and the notion of refined families of symplectic Galois representations by implementing the theory of symplectic Galois determinant. We then prove an infinitesimal R=T theorem under mild hypotheses. As an application, we study the relationship between the geometry of the small parabolic eigenvarieties at the Saito-Kurokawa lifts for cuspidal eigenforms (both finite-slope and infinite-slope) and the Bloch-Kato Selmer groups of those e

What carries the argument

Small parabolic eigenvarieties equipped with refined families of symplectic Galois representations, constructed via (ϕ, Γ)-modules with G-structures.

Load-bearing premise

The mild hypotheses on weights, slopes or residual representations under which the infinitesimal R=T theorem holds.

What would settle it

A concrete counterexample would be a Saito-Kurokawa lift where the dimension of the tangent space to the eigenvariety at that point fails to equal the dimension of the Bloch-Kato Selmer group of the corresponding Galois representation.

Figures

Figures reproduced from arXiv: 2604.18433 by Frederick E. Th{\o}gersen, Ju-Feng Wu, Muhammad Manji.

Figure 12
Figure 12. Figure 12: ] [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 1
Figure 1. Figure 1: A depiction of Spa(AF , AF ). The point xF (resp., xFq((π))) corresponds to AF → OF → F (resp., AF mod ϖ −−−−−→ Fq[[π]] → Fq((π)), where ϖ is a uniformiser of OF and π is the image of π in C ♭ p ). For any [a, b] ⊂ (0, ∞), we define Y[a,b] := the interior of the preimage ϑ −1 [a, b]. 5Since π is transcendental over OF , we have an isomorphism of topological rings OF [[T]] → OF [[π]] = AF , T 7→ π, where OF… view at source ↗
read the original abstract

We construct small parabolic eigenvarieties for holomorphic Siegel cuspforms of genus $2$ and study families of Galois representations attached to them in the spirit of Bella\"iche--Chenevier. In the course, we introduce the notion of $(\varphi, \Gamma)$-modules with $G$-structures and the notion of refined families of symplectic Galois representations by implementing the theory of symplectic Galois determinant d'apr\`es Moakher--Quast. We then prove an infinitesimal $R=\mathbb{T}$ theorem under mild hypotheses. As an application, we study the relationship between the geometry of the small parabolic eigenvarieties at the Saito--Kurokawa lifts for cuspidal eigenforms (both finite-slope and infinite-slope) and the Bloch--Kato Selmer groups of those eigenforms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs small parabolic eigenvarieties parametrizing holomorphic Siegel cuspforms of genus 2. It introduces the notions of (φ, Γ)-modules with G-structures and refined families of symplectic Galois representations, implementing the symplectic Galois determinant theory of Moakher-Quast. An infinitesimal R=T theorem is proved under mild hypotheses. As an application, the local geometry of these eigenvarieties at Saito-Kurokawa lifts (both finite- and infinite-slope) is related to the dimensions of the associated Bloch-Kato Selmer groups.

Significance. If the infinitesimal R=T theorem holds under the mild hypotheses, the work extends the Bellaïche-Chenevier framework to the symplectic setting for genus-2 Siegel forms and supplies a concrete link between eigenvariety tangent spaces and Selmer groups at SK points, including infinite-slope loci. The technical innovations of G-structured (φ, Γ)-modules and refined symplectic families are reusable tools that could support further arithmetic applications, such as Bloch-Kato type conjectures in this context.

major comments (2)
  1. [infinitesimal R=T theorem and application to SK lifts] The statement of the infinitesimal R=T theorem (appearing after the construction of the eigenvariety and the refined families) lists only 'mild hypotheses' without an enumerated list of conditions on the residual representation, the slope, or the (φ, Γ)-module. Because the subsequent application equates the tangent-space dimension of the eigenvariety at infinite-slope SK points with the dimension of the Bloch-Kato Selmer group, the precise hypotheses (irreducibility of the residual representation, validity of the refined symplectic determinant condition, absence of embedded components) must be stated explicitly and verified for the infinite-slope case; otherwise the Selmer-group interpretation does not follow.
  2. [application section on SK lifts] In the application to infinite-slope Saito-Kurokawa lifts, the manuscript invokes the geometry of the small parabolic eigenvariety but does not confirm that the Hecke algebra remains étale or that the refined family satisfies the symplectic determinant condition of Moakher-Quast at those points. If either fails, the claimed equality between the local dimension of the eigenvariety and the Selmer-group dimension is not justified.
minor comments (2)
  1. [preliminaries on (φ, Γ)-modules] Notation for the G-structure on (φ, Γ)-modules is introduced without a self-contained definition or comparison table to the ordinary (φ, Γ)-module case; a short preliminary subsection would improve readability.
  2. [construction of eigenvarieties] The construction of the small parabolic eigenvariety is compared to Bellaïche-Chenevier but lacks an explicit list of differences in the weight space or the Hecke operators used; adding such a comparison would clarify the novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major points below and will revise the text accordingly to improve clarity and rigor.

read point-by-point responses

  1. Referee: The statement of the infinitesimal R=T theorem (appearing after the construction of the eigenvariety and the refined families) lists only 'mild hypotheses' without an enumerated list of conditions on the residual representation, the slope, or the (φ, Γ)-module. Because the subsequent application equates the tangent-space dimension of the eigenvariety at infinite-slope SK points with the dimension of the Bloch-Kato Selmer group, the precise hypotheses (irreducibility of the residual representation, validity of the refined symplectic determinant condition, absence of embedded components) must be stated explicitly and verified for the infinite-slope case; otherwise the Selmer-group interpretation does not follow.

    Authors: We agree that the hypotheses should be stated explicitly rather than described as 'mild.' In the revised manuscript we will insert an enumerated list immediately following the statement of the infinitesimal R=T theorem: (i) the residual representation is irreducible, (ii) the refined symplectic determinant condition of Moakher-Quast holds, (iii) the eigenvariety has no embedded components at the point in question, and (iv) the slope and (φ, Γ)-module satisfy the standard non-degeneracy conditions used in the construction of the small parabolic eigenvariety. We will then verify, in a new paragraph in the application section, that these four conditions are satisfied at the infinite-slope Saito-Kurokawa points by appealing to the G-structure on the (φ, Γ)-modules and the uniform construction of the refined families. revision: yes

  2. Referee: In the application to infinite-slope Saito-Kurokawa lifts, the manuscript invokes the geometry of the small parabolic eigenvariety but does not confirm that the Hecke algebra remains étale or that the refined family satisfies the symplectic determinant condition of Moakher-Quast at those points. If either fails, the claimed equality between the local dimension of the eigenvariety and the Selmer-group dimension is not justified.

    Authors: We thank the referee for highlighting this gap. Once the hypotheses are listed explicitly, the infinitesimal R=T theorem directly implies that the Hecke algebra is étale at the points under consideration. The refined families are constructed precisely so that the symplectic determinant condition holds everywhere in the family, including at infinite-slope loci; this follows from the definition of G-structured (φ, Γ)-modules and the implementation of Moakher-Quast's theory. In the revision we will add a short paragraph in the application section that records these two facts and therefore justifies the identification of the tangent-space dimension with the dimension of the Bloch-Kato Selmer group at infinite-slope SK lifts. revision: yes

Circularity Check

0 steps flagged

No circularity; constructions, R=T theorem, and application are independently derived

full rationale

The paper constructs small parabolic eigenvarieties for genus-2 Siegel forms, introduces (φ, Γ)-modules with G-structures and refined symplectic Galois representations by implementing cited external theory (Bellaiche-Chenevier, Moakher-Quast), proves an infinitesimal R=T theorem under stated mild hypotheses, and applies the result to relate eigenvariety geometry at Saito-Kurokawa lifts (finite- and infinite-slope) to Bloch-Kato Selmer groups. No quoted step reduces a claimed prediction or uniqueness result to a fitted parameter, self-definition, or self-citation chain; the central claims retain independent mathematical content outside the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The paper relies on standard background from p-adic Hodge theory, eigenvariety theory, and Galois deformation theory; new definitions are introduced but their independence from prior work cannot be verified.

axioms (2)
  • domain assumption Standard properties of Galois representations attached to Siegel cuspforms and the existence of eigenvarieties in the Bellaiche-Chenevier framework.
    Invoked throughout the abstract as the setting for the new constructions.
  • domain assumption The theory of symplectic Galois determinant as developed by Moakher-Quast.
    Used to define refined families of symplectic Galois representations.
invented entities (2)
  • (φ, Γ)-modules with G-structures no independent evidence
    purpose: To encode additional structure on the Galois side for studying families over eigenvarieties.
    New notion introduced in the paper.
  • refined families of symplectic Galois representations no independent evidence
    purpose: To parametrize the Galois representations attached to points on the small parabolic eigenvarieties.
    New notion introduced to implement the symplectic determinant theory.

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Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    ‘p-adic families of Siegel modular cus- pforms’

    [AIP15] Fabrizio Andreatta, Adrian Iovita and Vincent Pilloni. ‘p-adic families of Siegel modular cus- pforms’. In:Annals of Mathematics181.2 (2015), pp. 623–697. [AIS14] Fabrizio Andreatta, Adrian Iovita and Glenn Stevens. ‘Overconvergent modular sheaves and modular forms forGL2/F’. In:Israel J. Math.201.1 (2014), pp. 299–359. [All16] Patrick B. Allen. ‘...

    work page 2015

  2. [2]

    ‘On Siegel eigenvarieties at Saito–Kurokawa points’

    [BB22] Tobias Berger and Adel Betina. ‘On Siegel eigenvarieties at Saito–Kurokawa points’. In:Ann. Inst. Fourier (Grenoble)72.3 (2022), pp. 901–961. [BC09] Joël Bellaïche and Gaëtan Chenevier.Families of Galois representations and Selmer groups. Astérisque

    work page 2022

  3. [3]

    ‘Bernstein eigenvarieties’

    [BD24] Christophe Breuil and Yiwen Ding. ‘Bernstein eigenvarieties’. In:Peking Math. J.7.2 (2024), pp. 471–642. [Bel09] Joël Bellaïche.An introduction to the conjecture of Bloch and Kato. Lecture Notes at the Clay Mathematical Institute Summer School. Available at:https : / / virtualmath1 . stanford . edu/~conrad/BSDseminar/refs/BKintro.pdf

    work page 2024

  4. [4]

    ‘Generic smoothness forG-valued potentially semi-stable deformation rings’

    [Bel16] Rebecca Bellovin. ‘Generic smoothness forG-valued potentially semi-stable deformation rings’. en. In:Annales de l’Institut Fourier66.6 (2016), pp. 2565–2620. [Ben11] Denis Benois. ‘A generalization of Greenberg’sL-invariant’. In:Amer. J. Math.133.6 (2011), pp. 1573–1632. [Ber02] Laurent Berger. ‘Représentationsp-adiques et équations différentielle...

    work page 2016

  5. [5]

    2008, pp

    Re- présentationsp-adiquesdegroupesp-adiques.I.Représentationsgaloisienneset(ϕ,Γ)-modules. 2008, pp. 13–38. [Ber17] John Bergdall. ‘Paraboline variation overp-adic families of(ϕ,Γ)-modules’. In:Compos. Math. 153.1 (2017), pp. 132–174. [BLGGT14] Thomas Barnet-Lamb, Toby Gee, David Geraghty and Richard Taylor. ‘Local-global compat- ibility forl=p, II’. In:A...

    work page 2008

  6. [6]

    ‘Onp-adicL-functions forGL2n in finite slope Shalika families’

    [BSDW26] Daniel Barrera Salazar, Mladen Dimitrov and Chris Williams. ‘Onp-adicL-functions forGL2n in finite slope Shalika families’. In:Adv. Math.486 (2026), Paper No. 110741,

    work page 2026

  7. [7]

    ‘Parabolic eigenvarieties via overconvergent co- homology’

    [BSW21] Daniel Barrera Salazar and Chris Williams. ‘Parabolic eigenvarieties via overconvergent co- homology’. In:Math. Z.299.1-2 (2021), pp. 961–995. [Buz07] Kevin Buzzard. ‘Eigenvarieties’. In:L-Functions and Galois Representations. Ed. by David Burns, Kevin Buzzard and Jan Nekovář. London Mathematical Society Lecture Note Series. Cambridge University P...

    work page 2021

  8. [8]

    ‘Thep-adic local Langlands cor- respondence forGL 2(Qp)’

    [CDP14] Pierre Colmez, Gabriel Dospinescu and Vytautas Pašk¯ unas. ‘Thep-adic local Langlands cor- respondence forGL 2(Qp)’. In:Camb. J. Math.2.1 (2014), pp. 1–47. [Che14] Gaëtan Chenevier. ‘Thep-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations overarbitrary rings’. In:Automorphic forms and Galois representations. Vol

    work page 2014

  9. [9]

    London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge, 2014, pp. 221–285. [CM98] RobertF.ColemanandBarryC.Mazur.‘TheEigencurve’.In:Galois Representations in Arith- metic Algebraic Geometry. Ed. by A. J. Scholl and R. L.Editors Taylor. London Mathematical Society Lecture Note Series. Cambridge University Press, 1998, pp. 1–114. [Col08] Pierr...

    work page 2014

  10. [10]

    Représentationsp- adiques de groupesp-adiques. I. Représentations galoisiennes et(ϕ,Γ)-modules. 2008, pp. 213–

    work page 2008

  11. [11]

    [Col97] Robert F. Coleman. ‘p-adic Banach spaces and families of modular forms’. In:Invent. Math. 127.3 (1997), pp. 417–479. [CS04] Robert F. Coleman and William A. Stein. ‘Approximation of eigenforms of infinite slope by eigenforms of finite slope’. In:Geometric aspects of Dwork theory. Vol. I, II. Walter de Gruyter, Berlin, 2004, pp. 437–449. [DH25] Mla...

    work page 1997

  12. [12]

    ‘On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms’

    89 [Eme06] Matthew Emerton. ‘On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms’. In:Invent. Math.164.1 (2006), pp. 1–84. [FC90] Gerd Faltings and Ching-Li Chai.Degeneration of Abelian Varieties. Ergebnisse der Mathem- atik und ihrer Grenzgebiete. Springer-Verlag Berlin Heidelberg,

    work page 2006

  13. [13]

    Progr. Math. Birkhäuser Boston, Boston, MA, 1990, pp. 249–309. [Gri95] Valeri Gritsenko. ‘Arithmetical lifting and its applications’. In:Number theory (Paris, 1992– 1993). Vol

    work page 1990

  14. [14]

    London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge, 1995, pp. 103–126. [GT05] Alain Genestier and Jacques Tilouine. ‘Systèmes de Taylor-Wiles pourGSp 4’. In:Formes automorphes (II) - Le cas du groupeGSp(4). Ed. by Jacques Tilouine, Henri Carayol, Michael Harris and Marie-France Vignéras. Astérisque

    work page 1995

  15. [15]

    Société mathématique de France, 2005, pp. 177–290. [GT11] Wee Teck Gan and Shuichiro Takeda. ‘The local Langlands conjecture forGSp(4)’. In:Ann. of Math. (2)173.3 (2011), pp. 1841–1882. [Han17] David Hansen. ‘Universal eigenvarieties, trianguline Galois representations, andp-adic Lang- lands functoriality’. In:Journal für die reine und angewandte Mathemat...

    work page 2005

  16. [16]

    ‘Automorphic forms of ∂-cohomology type as coherent cohomology classes’

    [Har90] Michael Harris. ‘Automorphic forms of ∂-cohomology type as coherent cohomology classes’. In:J. Differential Geom.32.1 (1990), pp. 1–63. [Hid86a] Haruzo Hida. ‘Galois representations intoGL 2(Zp[[X]])attached to ordinary cusp forms’. In: Invent. Math.85.3 (1986), pp. 545–613. [Hid86b] Haruzo Hida. ‘Iwasawa modules attached to congruences of cusp fo...

    work page 1990

  17. [17]

    Société mathématique de France, 2002, pp. 271–322. [JN19a] Christian Johansson and James Newton. ‘Extended eigenvarieties for overconvergent cohomo- logy’. In:Algebra and Number Theory13.1 (Feb. 2019), pp. 93–158. [JN19b] Christian Johansson and James Newton. ‘Irreducible components of extended eigenvarieties and interpolating Langlands functoriality’. In...

    work page 2002

  18. [18]

    Cohomologiesp-adiques et applications arithmétiques. III. 2004, pp. ix, 117–290. [Kha06] Chandrashekhar Khare. ‘Serre’s modularity conjecture: the level one case’. In:Duke Math. J. 134.3 (2006), pp. 557–589. [KPX14] Kiran S. Kedlaya, Jonathan Pottharst and Liang Xiao. ‘Cohomology of arithmetic families of (φ,Γ)-modules’. In:J. Amer. Math. Soc.27.4 (2014),...

    work page 2004

  19. [19]

    Princeton University Press, Princeton, NJ, 2013, pp

    London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, 2013, pp. xxvi+561. [Lan16] Kai-Wen Lan. ‘Vanishing theorems for coherent automorphic cohomology’. In:Res. Math. Sci. 3 (2016), Paper No. 39,

    work page 2013

  20. [20]

    Formes automorphes. II. Le cas du groupeGSp(4). 2005, pp. 1–66. [Liu15] Ruochuan Liu. ‘Triangulation of refined families’. In:Comment. Math. Helv.90.4 (2015), pp. 831–904. [Loe23] David Loeffler. ‘p-adicL-functions in universal deformation families’. In:Ann. Math. Qué.47.1 (2023), pp. 117–137. [LSTX17] Aaron Landesman, Ashvin A. Swaminathan, James Tao and...

    work page 2005

  21. [21]

    ‘On the computation of local components of a newform’

    [LW12] David Loeffler and Jared Weinstein. ‘On the computation of local components of a newform’. In:Math. Comp.81.278 (2012), pp. 1179–1200. [LW25] Pak-Hin Lee and Ju-Feng Wu. ‘Onp-adic adjointL-functions for Bianchi cuspforms: thep-split case’. In:Trans. Amer. Math. Soc.378.8 (2025), pp. 5579–5640. [Maa79] Hans Maaß. ‘Über eine Spezialschar von Modulfor...

    work page 2012

  22. [22]

    The theory of group schemes of finite type over a field

    Cambridge Studies in Advanced Mathematics. The theory of group schemes of finite type over a field. Cambridge University Press, Cambridge, 2017, pp. xvi+644. [Mil25] James S. Milne.Tannakian Categories.https://jmilne.org/math/Books/tcdraft.pdf

    work page 2017

  23. [23]

    ‘Galois representations attached to automorphic forms onGL 2 over CM fields’

    [Mok14] Chung Pang Mok. ‘Galois representations attached to automorphic forms onGL 2 over CM fields’. In:Compos. Math.150.4 (2014), pp. 523–567. [MQ23] Mohamed Moakher and Julian Quast.Symplectic determinant laws and invariant theory. Pre- print. Available at:https://arxiv.org/abs/2310.15822

    work page arXiv 2014

  24. [24]

    ‘Prolongement analytique sur les variétés de Siegel’

    91 [Pil11] Vincent Pilloni. ‘Prolongement analytique sur les variétés de Siegel’. In:Duke Math. J.157.1 (2011), pp. 167–222. [Pil13] Vincent Pilloni. ‘Overconvergent modular forms’. In:Annales de l’Institut Fourier63.1 (2013), pp. 219–239. [Pil20] Vincent Pilloni. ‘Higher coherent cohomology andp-adic modular forms of singular weights’. In:Duke Math. J.16...

    work page 2011

  25. [25]

    Springer, Berlin, 1973, pp. 191–268. [SGA1] Alexander Grothendieck.Revêtements étales et groupe fondamental. Fasc. I: Exposés 1 à

    work page 1973

  26. [26]

    Institut des Hautes Études Scientifiques, Paris, 1963, pp

    Troisième édition, corrigée, Séminaire de Géométrie Algébrique, 1960/61. Institut des Hautes Études Scientifiques, Paris, 1963, pp. iv+143. [Sor10] Claus M. Sorensen. ‘Galois representations attached to Hilbert-Siegel modular forms’. In:Doc. Math.15 (2010), pp. 623–670. [SR72] Neantro Saavedra Rivano.Catégories Tannakiennes. Vol. Vol

    work page 1960

  27. [27]

    Springer-Verlag, Berlin-New York, 1972, pp

    Lecture Notes in Mathem- atics. Springer-Verlag, Berlin-New York, 1972, pp. ii+418. [SS13] Abhishek Saha and Ralf Schmidt. ‘Yoshida lifts and simultaneous non-vanishing of dihedral twists of modularL-functions’. In:J. Lond. Math. Soc. (2)88.1 (2013), pp. 251–270. [Sta25] The Stacks project authors.The Stacks project.https://stacks.math.columbia.edu

    work page 1972

  28. [28]

    ‘Sur les déformationsp-adiques de certaines représenta- tions automorphes’

    [SU06] Christopher Skinner and Eric Urban. ‘Sur les déformationsp-adiques de certaines représenta- tions automorphes’. In:J. Inst. Math. Jussieu5.4 (2006), pp. 629–698. [SW20] Peter Scholze and Jared Weinstein.Berkeley lectures onp-adic geometry. Annals of Mathem- atics Studies. Princeton University Press,

    work page 2006

  29. [29]

    ‘On thel-adic cohomology of Siegel threefolds’

    [Tay93] Richard Taylor. ‘On thel-adic cohomology of Siegel threefolds’. In:Invent. Math.114.2 (1993), pp. 289–310. [Urb05] Eric Urban. ‘Sur les représentationsp-adiques associées aux représentations cuspidales de GSp4/Q’. In:Formes automorphes (II) - Le cas du groupeGSp(4). Ed. by Jacques Tilouine, Henri Carayol, Michael Harris and Marie-France Vignéras. ...

    work page 1993

  30. [30]

    Société math- ématique de France, 2005, pp. 151–176. [Urb11] Eric Urban. ‘Eigenvarieties for reductive groups’. In:Annals of Mathematics174.3 (2011), pp. 1685–1784. 92 [Wei05] Rainer Weissauer. ‘Four dimensional Galois representations’. In:Formes automorphes (II) - Le cas du groupeGSp(4). Ed. by Jacques Tilouine, Henri Carayol, Michael Harris and Marie- F...

    work page 2005

  31. [31]

    Société mathématique de France, 2005, pp. 67–150. [Zag81] Don Zagier. ‘Sur la conjecture de Saito-Kurokawa (d’après H. Maass)’. In:Seminar on Number Theory, Paris 1979–80. Vol

    work page 2005

  32. [32]

    Progr. Math. Birkhäuser, Boston, MA, 1981, pp. 371–394. [Zav24] Bogdan Zavyalov. ‘Quotients of admissible formal schemes and adic spaces by finite groups’. In:Algebra Number Theory18.3 (2024), pp. 409–475. M.M. Concordia University Department of Mathematics and Statistics Montréal, Québec, Canada E-mail address:muhammad.manji@concordia.ca F.E.T. School of...

    work page 1981