Perfectoid overconvergent Siegel modular forms and the overconvergent Eichler--Shimura morphism

Giovanni Rosso; Hansheng Diao; Ju-Feng Wu

arxiv: 2106.00094 · v3 · submitted 2021-05-31 · 🧮 math.NT

Perfectoid overconvergent Siegel modular forms and the overconvergent Eichler--Shimura morphism

Hansheng Diao , Giovanni Rosso , Ju-Feng Wu This is my paper

Pith reviewed 2026-05-24 12:41 UTC · model grok-4.3

classification 🧮 math.NT

keywords overconvergent Siegel modular formsperfectoid methodEichler-Shimura morphismautomorphic sheavesShimura moduli spacesp-adic automorphic forms

0 comments

The pith

Generalizing the perfectoid method from Shimura curves constructs overconvergent automorphic sheaves for Siegel modular forms whose global sections are the forms and yields an explicit overconvergent Eichler-Shimura morphism.

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

The paper extends the perfectoid approach to overconvergent automorphic sheaves, previously used only for compact Shimura curves over Q, to the setting of Siegel moduli spaces. Global sections of the resulting sheaves coincide exactly with the overconvergent Siegel modular forms, and the sheaves can be compared to the earlier Andreatta-Iovita-Pilloni construction. The work also produces an explicit overconvergent Eichler-Shimura morphism that generalizes the known elliptic-modular case. A sympathetic reader cares because the construction supplies p-adic tools for studying higher-genus modular forms that were previously unavailable.

Core claim

The global sections of the overconvergent automorphic sheaves obtained by generalising the perfectoid method are precisely the overconvergent Siegel modular forms. These sheaves admit comparison with the Andreatta-Iovita-Pilloni construction. An explicit overconvergent Eichler-Shimura morphism is established for Siegel modular forms, generalising the elliptic-modular result of Andreatta-Iovita-Stevens.

What carries the argument

The perfectoid generalization of overconvergent automorphic sheaves on Siegel moduli spaces, which produces the sheaves whose global sections recover the overconvergent forms and carries the explicit Eichler-Shimura morphism.

If this is right

Overconvergent Siegel modular forms arise directly as the global sections of the perfectoid-constructed sheaves.
The new sheaves stand in direct comparison with the Andreatta-Iovita-Pilloni sheaves.
An explicit overconvergent Eichler-Shimura morphism exists for Siegel modular forms.
The morphism generalizes the one previously known only for elliptic modular forms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same perfectoid technique may extend to other families of Shimura varieties once the Siegel case is settled.
The explicit morphism could be used to attach Galois representations to overconvergent Siegel forms in a uniform way.
Comparison of the two sheaf constructions may clarify which properties of overconvergent forms are independent of the chosen method.

Load-bearing premise

The technical details of the perfectoid construction that worked for compact Shimura curves over Q carry over without major new obstructions when applied to the Siegel moduli spaces of higher genus.

What would settle it

An explicit computation in genus two showing that the global sections of the constructed sheaves fail to match the known space of overconvergent Siegel modular forms would falsify the central claim.

read the original abstract

The aim of this paper is twofold. We first present a construction of the overconvergent automorphic sheaves for Siegel modular forms by generalising the perfectoid method, originally introduced by Chojecki--Hansen--Johansson for automorphic forms on compact Shimura curves over $\mathbf{Q}$. The global sections of these automorphic sheaves are precisely the overconvergent Siegel modular forms. In particular, one can compare these automorphic sheaves with the ones constructed by Andreatta--Iovita--Pilloni. Secondly, we establish an (explicit) overconvergent Eichler--Shimura morphism for Siegel modular forms, generalising the result of Andreatta--Iovita--Stevens for the elliptic modular forms.

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.

Desk Editor's Note private letter to a colleague

The paper generalizes the perfectoid construction of overconvergent sheaves and the Eichler-Shimura morphism from compact curves to Siegel modular forms, with explicit claims that need boundary checks.

read the letter

This paper takes the perfectoid method from Chojecki-Hansen-Johansson on compact Shimura curves and extends it to define overconvergent automorphic sheaves for Siegel modular forms. The global sections are stated to be exactly the overconvergent Siegel forms, and the sheaves are compared to the Andreatta-Iovita-Pilloni construction. It also gives an explicit overconvergent Eichler-Shimura morphism by generalizing Andreatta-Iovita-Stevens from the elliptic case. That is the core new content: two concrete extensions that were not available before for genus greater than 1. The explicit morphism and the direct comparison to prior sheaves are the parts that stand out as useful. They give people working in this area something they can actually apply rather than just an existence statement. The citation pattern is straightforward and builds on the right references without obvious gaps or self-referential loops. The main soft spot is the move to non-compact higher-dimensional moduli spaces. The base constructions relied on compactness and low dimension to control the perfectoid tower, the gluing, and the Hodge-Tate filtration. Siegel spaces have positive-dimensional boundary strata, so the cuspidal degeneration and compatibility with the perfectoid cover need separate handling. The abstract says the construction succeeds, but the result rests on whether the paper supplies independent arguments for those issues or simply extrapolates from the curve case. If the sections on the sheaves address the boundary directly, the claims hold; otherwise there is a gap in the central identification of global sections. This is for specialists already working on p-adic automorphic forms and overconvergent methods. A reader who needs explicit tools for overconvergent Siegel forms or the morphism in higher rank will find value here. It is a technical step rather than a broad shift. I would send it to peer review so referees can check the boundary arguments in detail.

Referee Report

2 major / 2 minor

Summary. The paper generalizes the perfectoid construction of overconvergent automorphic sheaves, originally due to Chojecki-Hansen-Johansson for compact Shimura curves, to the setting of Siegel modular forms on the Siegel moduli space. It asserts that the global sections of these sheaves coincide with the overconvergent Siegel modular forms (allowing comparison with the Andreatta-Iovita-Pilloni construction) and constructs an explicit overconvergent Eichler-Shimura morphism, extending the Andreatta-Iovita-Stevens result from the elliptic case.

Significance. If the generalization is carried through rigorously, the work supplies an independent perfectoid-based construction of overconvergent Siegel forms together with an explicit Eichler-Shimura map; this would furnish a useful alternative to existing approaches and open the possibility of comparing Hodge filtrations and boundary behavior across constructions.

major comments (2)

[Sections constructing the overconvergent automorphic sheaves] The central claim that the perfectoid tower and the resulting automorphic sheaves extend without new obstructions to the non-compact Siegel moduli space (genus g>1) rests on the handling of the boundary strata and the compatibility of the Hodge-Tate filtration with the perfectoid cover. The manuscript must supply a self-contained argument for these points (distinct from the one-dimensional compact case of Chojecki-Hansen-Johansson) in the sections constructing the sheaves; without it the identification of global sections with overconvergent forms is not yet load-bearing.
[Section establishing the overconvergent Eichler-Shimura morphism] The explicit overconvergent Eichler-Shimura morphism is asserted to generalize Andreatta-Iovita-Stevens. The manuscript should verify that the morphism remains well-defined on the overconvergent sheaves after the perfectoid gluing, including any necessary checks on the cuspidal degeneration; this step is load-bearing for the second main claim.

minor comments (2)

Notation for the perfectoid tower and the level structures should be introduced with explicit references to the base constructions in Chojecki-Hansen-Johansson to facilitate comparison.
The comparison with Andreatta-Iovita-Pilloni would benefit from a short table or diagram indicating which properties (e.g., overconvergence radius, Hecke action) are shown to coincide.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our generalization of the perfectoid construction to Siegel modular forms. We respond point-by-point to the major comments below.

read point-by-point responses

Referee: [Sections constructing the overconvergent automorphic sheaves] The central claim that the perfectoid tower and the resulting automorphic sheaves extend without new obstructions to the non-compact Siegel moduli space (genus g>1) rests on the handling of the boundary strata and the compatibility of the Hodge-Tate filtration with the perfectoid cover. The manuscript must supply a self-contained argument for these points (distinct from the one-dimensional compact case of Chojecki-Hansen-Johansson) in the sections constructing the sheaves; without it the identification of global sections with overconvergent forms is not yet load-bearing.

Authors: We agree that the manuscript requires a more explicit, self-contained treatment of the boundary strata and Hodge-Tate filtration compatibility for the non-compact Siegel case (g>1). The revised version will add a dedicated subsection in the construction of the sheaves that supplies this argument, distinct from the compact curve case, to make the global sections identification fully load-bearing. revision: yes
Referee: [Section establishing the overconvergent Eichler-Shimura morphism] The explicit overconvergent Eichler-Shimura morphism is asserted to generalize Andreatta-Iovita-Stevens. The manuscript should verify that the morphism remains well-defined on the overconvergent sheaves after the perfectoid gluing, including any necessary checks on the cuspidal degeneration; this step is load-bearing for the second main claim.

Authors: We acknowledge the need for explicit verification of well-definedness after perfectoid gluing, including cuspidal degeneration. The revised manuscript will include these checks in the section on the overconvergent Eichler-Shimura morphism to confirm the generalization. revision: yes

Circularity Check

0 steps flagged

No circularity: generalization rests on external prior constructions by other authors

full rationale

The paper's core claims are presented as generalizations of the perfectoid method from Chojecki-Hansen-Johansson (distinct authors) for compact Shimura curves and comparisons to Andreatta-Iovita-Pilloni and Andreatta-Iovita-Stevens for the sheaves and Eichler-Shimura morphism. No quoted equations, definitions, or steps in the provided abstract or description reduce by construction to inputs defined within this paper itself, nor do they rely on load-bearing self-citations. The derivation chain is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, ad-hoc axioms, or invented entities; the work rests on standard background results in p-adic geometry and automorphic forms whose details are not visible here.

pith-pipeline@v0.9.0 · 5663 in / 1180 out tokens · 28761 ms · 2026-05-24T12:41:07.936346+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

Overconvergent de Rham Eichler–Shimura morphisms

[AI20] Fabrizio Andreatta and Adrian Iovita. Overconvergent de Rham Eichler–Shimura morphisms . Preprint. A vailable at: https://arxiv.org/abs/2006.02185

work page arXiv 2006
[2]

p-adic families of Siegel modular cusp- forms

[AIP15] Fabrizio Andreatta, Adrian Iovita, and Vincent Pil loni. “ p-adic families of Siegel modular cusp- forms”. In: Annals of Mathematics 181.2 (2015), pp. 623–697. issn: 0003486X. url: http://www.jstor.org/stable/24522945. [AIS15] Fabrizio Andreatta, Adrian Iovita, and Glenn Steve ns. “Overconvergent Eichler–Shimura iso- morphisms”. In: Journal of the...

work page doi:10.1017/s1474748013000364 2015
[3]

Overconvergent Hilbert modular forms via perfectoid modular varieties

99 [BHW19] Christopher Birkbeck, Ben Heuer, and Chris William s. Overconvergent Hilbert modular forms via perfectoid modular varieties . Preprint. arXiv:1902.03985

work page arXiv 1902
[4]

Eigenvarieties

[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 Press, 2007, pp. 59–120. doi: 10.1017/CBO9780511721267.004. [CHJ17] Przemyslaw Chojecki, David Hansen, and Christian J ohansson. “Overconvergent Modu...

work page doi:10.1017/cbo9780511721267.004 2007
[5]

[Dia+19a] Hansheng Diao et al

url: http://www.numdam.org/item/SB_1970-1971__13__123_0/. [Dia+19a] Hansheng Diao et al. Logarithmic adic spaces: some fundamental results . Preprint. A vailable at: https://arxiv.org/pdf/1912.09836.pdf

work page arXiv 1912
[6]

Logarithmic Riemann–Hilbert correspondences for rigid varieti es

[Dia+19b] Hansheng Diao et al. Logarithmic Riemann–Hilbert correspondences for rigid varieti es. Preprint. A vailable at:https://arxiv.org/pdf/1803.05786.pdf

work page arXiv
[7]

Hodge-Tate structures and modular forms

[Fal87] Gerd Faltings. “Hodge-Tate structures and modular forms”. In: Math. Ann. 278.1-4 (1987), pp. 133–149. issn: 0025-5831. doi: 10.1007/BF01458064. url: https://doi-org.lib-ezproxy.concordia.ca/10.1007/BF 01458064. [Far11] Laurent Fargues. “La ﬁltration canonique des point s de torsion des groupes p-divisibles”. In: Annales scientiﬁques de l’École Nor...

work page doi:10.1007/bf01458064 1987
[8]

Faltings and C.-L

doi: 10.1007/978-3-662-02632-8 . [FGL08] Laurent Fargues, Alain Genestier, and Vincent Laﬀo rgue. L’isomorphisme entre les tours de Lubin-Tate et de Drinfeld . Vol

work page doi:10.1007/978-3-662-02632-8
[9]

Birkhäuser Basel, 2008

Progress in Mathematics. Birkhäuser Basel, 2008 . doi: https://doi.org/10.1007/978-3-7643-8456-2 . [FH91] William Fulton and Joe Harris. Representation Theory: a ﬁrst course . Vol

work page doi:10.1007/978-3-7643-8456-2 2008
[10]

Universal eigenvarieties, triangu line Galois representations, and p-adic Lang- lands functoriality

doi: https://doi.org/10.1007/978-1-4612-0979-9 . [Han17] David Hansen. “Universal eigenvarieties, triangu line Galois representations, and p-adic Lang- lands functoriality”. In: Journal für die reine und angewandte Mathematik (730 2017), pp. 1–64. doi: https://doi.org/10.1515/crelle-2014-0130. [Han20] David Hansen. “Vanishing and comparison theorems i n r...

work page doi:10.1007/978-1-4612-0979-9 2017
[11]

Extended eig envarieties for overconvergent cohomol- ogy

Société mathématiq ue de France, 2002, pp. 271–322. url: http://www.numdam.org/item/AST_2002__279__271_0. 100 [JN19] Christian Johansson and James Newton. “Extended eig envarieties for overconvergent cohomol- ogy”. English. In: Algebra and Number Theory 13.1 (Feb. 2019), pp. 93–158. issn: 1937-0652. doi: 10.2140/ant.2019.13.93. [Kis03] Mark Kisin. “Overco...

work page doi:10.2140/ant.2019.13.93 2002
[12]

Cohomologie cohé rente et représentations Galoisiennes

doi: https://doi.org/10.1007/978-3-540-37889-1 . [PS16] Vincent Pilloni and Benoît Stroh. “Cohomologie cohé rente et représentations Galoisiennes”. In: Annales mathématiques du Québec 40.1 (2016), pp. 167–202. issn: 2195-4763. doi: 10.1007/s40316-015-0056-0 . url: https://doi.org/10.1007/s40316-015-0056-0 . [RC21] Juan Esteban Rodríguez Camargo. Dual Eich...

work page doi:10.1007/978-3-540-37889-1 2016
[13]

Perfectoid spaces: A survey

[Sch12] Peter Scholze. “Perfectoid spaces: A survey”. In: Current Developments in Mathematics, 2012 . Ed. by David Jerison et al. International Press of Boston, In c.,

work page 2012
[14]

p-adic Hodge theory for rigid–analytic varieties

[Sch13] Peter Scholze. “ p-adic Hodge theory for rigid–analytic varieties”. In: Forum of Mathematics, Pi 1 (2013). doi: 10.1017/fmp.2013.1. [Sch15] Peter Scholze. “On torsion in the cohomology of loca lly symmetric varieties”. In: Annals of Mathematics 182.3 (2015), pp. 945–1066. [Sen88] Shankar Sen. “The Analytic Variation of p-Adic Hodge Structure”. In:...

work page doi:10.1017/fmp.2013.1 2013
[15]

Compactiﬁcation de variétés de Sieg el aux places de mauvaise réduction

[Str10] Benoît Stroh. “Compactiﬁcation de variétés de Sieg el aux places de mauvaise réduction”. In: Bulletin de la Société Mathématique de France 138.2 (2010), pp. 259–315. [SW13] Peter Scholze and Jared Weinstein. “Moduli of p-divisible groups”. In: Cambridge Journal of Mathematics 1.2 (2013). doi: https://dx.doi.org/10.4310/CJM.2013.v1.n2.a1. [SW20] Pe...

work page doi:10.4310/cjm.2013.v1.n2.a1 2010
[16]

A pairing on the cuspidal eigenvariety for GSp2g and the ramiﬁcation locus

[Wu20] Ju-Feng Wu. A pairing on the cuspidal eigenvariety for GSp2g and the ramiﬁcation locus . Preprint. https://arxiv.org/abs/2005.04776v2

work page arXiv 2005

[1] [1]

Overconvergent de Rham Eichler–Shimura morphisms

[AI20] Fabrizio Andreatta and Adrian Iovita. Overconvergent de Rham Eichler–Shimura morphisms . Preprint. A vailable at: https://arxiv.org/abs/2006.02185

work page arXiv 2006

[2] [2]

p-adic families of Siegel modular cusp- forms

[AIP15] Fabrizio Andreatta, Adrian Iovita, and Vincent Pil loni. “ p-adic families of Siegel modular cusp- forms”. In: Annals of Mathematics 181.2 (2015), pp. 623–697. issn: 0003486X. url: http://www.jstor.org/stable/24522945. [AIS15] Fabrizio Andreatta, Adrian Iovita, and Glenn Steve ns. “Overconvergent Eichler–Shimura iso- morphisms”. In: Journal of the...

work page doi:10.1017/s1474748013000364 2015

[3] [3]

Overconvergent Hilbert modular forms via perfectoid modular varieties

99 [BHW19] Christopher Birkbeck, Ben Heuer, and Chris William s. Overconvergent Hilbert modular forms via perfectoid modular varieties . Preprint. arXiv:1902.03985

work page arXiv 1902

[4] [4]

Eigenvarieties

[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 Press, 2007, pp. 59–120. doi: 10.1017/CBO9780511721267.004. [CHJ17] Przemyslaw Chojecki, David Hansen, and Christian J ohansson. “Overconvergent Modu...

work page doi:10.1017/cbo9780511721267.004 2007

[5] [5]

[Dia+19a] Hansheng Diao et al

url: http://www.numdam.org/item/SB_1970-1971__13__123_0/. [Dia+19a] Hansheng Diao et al. Logarithmic adic spaces: some fundamental results . Preprint. A vailable at: https://arxiv.org/pdf/1912.09836.pdf

work page arXiv 1912

[6] [6]

Logarithmic Riemann–Hilbert correspondences for rigid varieti es

[Dia+19b] Hansheng Diao et al. Logarithmic Riemann–Hilbert correspondences for rigid varieti es. Preprint. A vailable at:https://arxiv.org/pdf/1803.05786.pdf

work page arXiv

[7] [7]

Hodge-Tate structures and modular forms

[Fal87] Gerd Faltings. “Hodge-Tate structures and modular forms”. In: Math. Ann. 278.1-4 (1987), pp. 133–149. issn: 0025-5831. doi: 10.1007/BF01458064. url: https://doi-org.lib-ezproxy.concordia.ca/10.1007/BF 01458064. [Far11] Laurent Fargues. “La ﬁltration canonique des point s de torsion des groupes p-divisibles”. In: Annales scientiﬁques de l’École Nor...

work page doi:10.1007/bf01458064 1987

[8] [8]

Faltings and C.-L

doi: 10.1007/978-3-662-02632-8 . [FGL08] Laurent Fargues, Alain Genestier, and Vincent Laﬀo rgue. L’isomorphisme entre les tours de Lubin-Tate et de Drinfeld . Vol

work page doi:10.1007/978-3-662-02632-8

[9] [9]

Birkhäuser Basel, 2008

Progress in Mathematics. Birkhäuser Basel, 2008 . doi: https://doi.org/10.1007/978-3-7643-8456-2 . [FH91] William Fulton and Joe Harris. Representation Theory: a ﬁrst course . Vol

work page doi:10.1007/978-3-7643-8456-2 2008

[10] [10]

Universal eigenvarieties, triangu line Galois representations, and p-adic Lang- lands functoriality

doi: https://doi.org/10.1007/978-1-4612-0979-9 . [Han17] David Hansen. “Universal eigenvarieties, triangu line Galois representations, and p-adic Lang- lands functoriality”. In: Journal für die reine und angewandte Mathematik (730 2017), pp. 1–64. doi: https://doi.org/10.1515/crelle-2014-0130. [Han20] David Hansen. “Vanishing and comparison theorems i n r...

work page doi:10.1007/978-1-4612-0979-9 2017

[11] [11]

Extended eig envarieties for overconvergent cohomol- ogy

Société mathématiq ue de France, 2002, pp. 271–322. url: http://www.numdam.org/item/AST_2002__279__271_0. 100 [JN19] Christian Johansson and James Newton. “Extended eig envarieties for overconvergent cohomol- ogy”. English. In: Algebra and Number Theory 13.1 (Feb. 2019), pp. 93–158. issn: 1937-0652. doi: 10.2140/ant.2019.13.93. [Kis03] Mark Kisin. “Overco...

work page doi:10.2140/ant.2019.13.93 2002

[12] [12]

Cohomologie cohé rente et représentations Galoisiennes

doi: https://doi.org/10.1007/978-3-540-37889-1 . [PS16] Vincent Pilloni and Benoît Stroh. “Cohomologie cohé rente et représentations Galoisiennes”. In: Annales mathématiques du Québec 40.1 (2016), pp. 167–202. issn: 2195-4763. doi: 10.1007/s40316-015-0056-0 . url: https://doi.org/10.1007/s40316-015-0056-0 . [RC21] Juan Esteban Rodríguez Camargo. Dual Eich...

work page doi:10.1007/978-3-540-37889-1 2016

[13] [13]

Perfectoid spaces: A survey

[Sch12] Peter Scholze. “Perfectoid spaces: A survey”. In: Current Developments in Mathematics, 2012 . Ed. by David Jerison et al. International Press of Boston, In c.,

work page 2012

[14] [14]

p-adic Hodge theory for rigid–analytic varieties

[Sch13] Peter Scholze. “ p-adic Hodge theory for rigid–analytic varieties”. In: Forum of Mathematics, Pi 1 (2013). doi: 10.1017/fmp.2013.1. [Sch15] Peter Scholze. “On torsion in the cohomology of loca lly symmetric varieties”. In: Annals of Mathematics 182.3 (2015), pp. 945–1066. [Sen88] Shankar Sen. “The Analytic Variation of p-Adic Hodge Structure”. In:...

work page doi:10.1017/fmp.2013.1 2013

[15] [15]

Compactiﬁcation de variétés de Sieg el aux places de mauvaise réduction

[Str10] Benoît Stroh. “Compactiﬁcation de variétés de Sieg el aux places de mauvaise réduction”. In: Bulletin de la Société Mathématique de France 138.2 (2010), pp. 259–315. [SW13] Peter Scholze and Jared Weinstein. “Moduli of p-divisible groups”. In: Cambridge Journal of Mathematics 1.2 (2013). doi: https://dx.doi.org/10.4310/CJM.2013.v1.n2.a1. [SW20] Pe...

work page doi:10.4310/cjm.2013.v1.n2.a1 2010

[16] [16]

A pairing on the cuspidal eigenvariety for GSp2g and the ramiﬁcation locus

[Wu20] Ju-Feng Wu. A pairing on the cuspidal eigenvariety for GSp2g and the ramiﬁcation locus . Preprint. https://arxiv.org/abs/2005.04776v2

work page arXiv 2005