The completed Kirillov model and local-global compatibility for functions on Igusa varieties

Sean Howe

REVIEW 2 major objections 6 minor 1 cited by

The completed Kirillov model and local-global compatibility for functions on Igusa varieties

T0 review · 2 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read For the ordinary GL2 Igusa variety, the local p-adic automorphic piece attached to a modular form is pinned, up to finitely many logarithmic twists, between two Galois-determined bounds, and is exact when the Galois representation is…

desk verdict Genuinely new structural results on p-adic automorphic forms; the proof is coherent and the main soft spots are explicit gaps, not hidden flaws. read the letter →

arxiv 2506.24089 v1 pith:K4TTBRCV submitted 2025-06-30 math.NT

Sean Howe This is my paper · ORCID

classification math.NT MSC 11F7011F3311F8011F8511G1822E50

keywords KirillovmodelIgusavarietieslocal-globalcompatibilityp-adicautomorphicformsordinarymodularGaloisrepresentationscoinvariantsFouriertheory

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 establishes a local-global compatibility statement for the space of cuspidal $p$-adic automorphic functions on the ordinary Caraiani-Scholze Igusa variety attached to $\mathrm{GL}_2$. The main result, Theorem B, says that the local piece $W_\pi$ attached to a classical cuspidal automorphic representation $\pi$ of weight at least $2$ always contains the functions vanishing at $0$ and infinity together with one indicator character per character of the associated Galois representation $\rho_p$, and is contained in that same space extended by finitely many logarithmic twists; when $\rho_p$ is irreducible, the two bounds coincide. The proof works by describing the whole space of cuspidal functions as a completion of the smooth Kirillov model of classical cusp forms (Theorem A) and by identifying the ordinary $p$-adic modular forms of the classical theory with the topological coinvariants of the $\tilde{\mu}_{p^\infty}$ action (Theorem C). These results give a concrete way to read local Galois information off a natural space of $p$-adic automorphic forms, and they suggest a conjectural analogue for more general Igusa varieties.

What carries the argument

Two identifications carry the argument. First, Theorem A: after fixing a compatible system of roots of unity, the $q$-expansion map $\mathrm{Kir}$ composed with the evaluation map $\mathrm{eval}_k$ identifies the cuspidal functions $V_b^{\mathrm{cusp}}$ on the $\mathrm{GL}_2$ ordinary Caraiani-Scholze Igusa formal scheme with the completion of the smooth Kirillov model of the classical cusp forms $S_k$ inside the bounded continuous functions on $A_f^\times$, the Kirillov model being the classical realization of a smooth $\mathrm{GL}_2(Q_p)$-representation on functions on $Q_p^\times$ with the action of the mirabolic subgroup given by translations and dilations. Second, Theorem C: the ordinary projector $e$ on the Mantovan space $V_{\mathrm{Mant}}^{\mathrm{cusp}}$ is a $T(Q_p) \times \mathrm{GL}_2(A_f^{(p)})$-equivariant isomorphism onto the topological coinvariants $(V_b^{\mathrm{cusp}})_{\tilde{\mu}_{p^\infty}}$, where the action of the universal cover $\tilde{\mu}_{p^\infty}$ is understood by $p$-adic Fourier duality as an action of the bounded continuous functions on $Q_p$. Theorem C turns the finiteness theorem for the ordinary Hecke algebra into the admissibility of the coinvariant space (Corollary D), and the completed Kirillov model converts the classical descriptions of Kirillov models and Jacquet modules for $\mathrm{GL}_2(Q_p)$ into the explicit lower bound; the upper bound then follows by feeding the admissible Banach structure through the characters of $Q_p^\times$ and using the description of Galois representations attached to ordinary $p$-adic eigenforms.

What would settle it

Compute $W_\pi$ for a classical cuspidal eigenform of weight at least $2$ whose local component $\pi_p$ is supercuspidal (equivalently, whose Galois representation $\rho_p$ is irreducible): the theorem predicts $W_\pi$ is exactly $S(Q_p^\times, C_p)$, the functions on $Q_p^\times$ that vanish at $0$ and $\infty$. If this computation produces any further vector, such as a non-vanishing indicator function $1_{Z_p}\cdot\chi$ for some character $\chi$, the central compatibility claim fails.

Watch

Extended reading notes

Core claim

The central discovery is that the eigenspace $W_\pi = \mathrm{Hom}_{\mathrm{GL}_2(A_f^{(p)})}(\pi^{(p)}, V_b^{\mathrm{cusp}})$ attached to an irreducible subrepresentation $\pi$ of the classical cuspidal modular forms of weight $k \ge 2$ satisfies, for some non-negative integer $M$, $$S(Q_p^\times, C_p) + \sum_{\chi \subset \rho_p} C_p \cdot 1_{Z_p} \cdot \chi \subseteq W_\pi \subseteq S(Q_p^\times, C_p) + \sum_{\chi \subset \rho_p}\sum_{a,b \le M} C_p \cdot 1_{Z_p} \cdot \chi_{a,b},$$ where the $\chi$ are the characters of $Q_p^\times$ obtained from the subrepresentations of $\rho_p$ via local class field theory, and $\chi_{a,b}(p^k,\zeta,t) = \chi(p)^k k^a \chi(\zeta)\chi(t)\log(t)^b$. In the irreducible (non-ordinary) case the two containments collapse to an equality, so $W_\pi$ is exactly $S(Q_p^\times, C_p)$. Thus the local representation of the Igusa variety at $p$ carries essentially the same information as the Galois representation, with the only indeterminacy being finitely many logarithmic twists in the reducible case.

Load-bearing premise

The upper bound leans on a published finiteness theorem for a certain algebra of Hecke operators, and the lower bound in one case leans on an external theorem producing an extra modular form; if either external result failed, the two-sided bound on $W_\pi$ would not follow.

Editorial extensions

If this is right

If Theorem B is correct, then for any classical cuspidal $\pi$ of weight $\ge 2$ with irreducible $\rho_p$, the local piece $W_\pi$ is exactly the vanishing-at-infinity space $S(Q_p^\times, C_p)$, so supercuspidal local components are read off completely from the Galois side.
In the ordinary potentially crystalline case, $W_\pi$ determines $\rho_p$; in the potentially semistable non-crystalline case it determines $\rho_p^{\mathrm{ss}}$; and under Conjecture 1.2.1 it determines $\rho_p$ in all reducible cases.
Theorem C identifies the classical ordinary $p$-adic modular forms with the $\tilde{\mu}_{p^\infty}$-coinvariants, and the paper notes this may yield a new derivation of the finiteness theorem for the ordinary Hecke algebra from the smooth admissibility of Jacquet modules.
Conjecture 8.2.1 predicts that the same coinvariant construction on any Caraiani-Scholze Igusa variety yields an admissible Banach representation of the associated Levi subgroup, which would give a systematic supply of $p$-adic automorphic representations beyond $\mathrm{GL}_2$.

Reading between the lines

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

The logarithmic twists $\chi_{a,b}$ in the upper bound have the shape of powers of $p$-adic logarithms, hinting that $W_\pi$ is a $p$-adic interpolation of the Jacquet module of $\pi_p$; computing $W_\pi$ for a specific Steinberg or principal-series representation could reveal whether the bound $M$ is sharp.
The equivalent mod $p$ form of Conjecture 8.2.1 suggests that the Galois characters detected by Theorem B should already be visible in the mod $p$ coinvariants, offering a characteristic-$p$ testing ground for local-global compatibility.
The local construction of §8.3, realizing the appearing characters as bounded sections on a quotient of a local Shimura variety, suggests a concrete recipe for building local representations for higher-rank Shimura varieties; a natural next test is to see which bounded sections survive the coinvariant quotient in a rank-two example.

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.

Desk Editor's Note

read the letter

Main take: the paper earns a serious referee. The genuinely new pieces are Theorem A, identifying cuspidal functions on the GL2 ordinary Caraiani-Scholze Igusa variety with a completion of the smooth Kirillov model, and Theorem C, identifying a Hida-ordinary variant with topological coinvariants. Theorem B then uses these to give a two-sided containment for W_pi, with the ordinary Galois characters appearing up to finitely many logarithmic twists, and it proves the clean conjecture in the irreducible case. These are not repackaged versions of [17]; the completed-Kirillov description and the coinvariants story are new statements, and the proof has real machinery in it.

What the paper does well: the q-expansion embedding Kir is treated carefully, with flat cokernel and a bootstrap from Hida-Shimura density; Theorem C is an honest kernel/fiber-at-zero comparison; and the upper bound in Section 7 is a real argument through admissible Banach representations and Hida's finiteness theorem. I traced the external dependencies and agree with the stress test: Hida's finiteness and Breuil-Emerton [1] are used under their stated hypotheses, and I found no circularity. The paper is self-aware about what it does not prove.

Soft spots, in proportion: Theorem B's upper bound contains an undetermined M and generalized-eigenspace terms chi_{a,b} with a,b <= M. The author says directly that the argument cannot exclude them; this is a weakness relative to Conjecture 1.2.1, not a hidden error. Proposition 3.2.1 depends on [12], which is in preparation; this is a real verification gap in a foundational Fourier statement, though the content is standard. Section 8 is openly provisional and should stay labeled as such; it is a conjecture section with motivation, not a load-bearing part of the main results. The citation pattern is fine: [17] is the natural predecessor, and the in-preparation citations should be flagged to the editor but are not a reason to reject.

Who it is for: p-adic automorphic forms researchers, especially those working on Hida theory, Igusa varieties, and local-global compatibility. A serious referee should engage; I recommend sending it to review, asking the author to provide a published substitute or proof for Proposition 3.2.1 and to discuss the dependence of M in Theorem B more explicitly. Conditional accept seems right.

Referee Report

2 major / 6 minor

Summary. This paper studies the space of cuspidal functions on the ordinary Caraiani–Scholze Igusa formal scheme for GL2 and relates it to classical and p-adic automorphic forms. Theorem A (5.3.1) identifies V_{b,C_p}^{cusp} with the completion, in the sup-norm Banach space of bounded functions on A_f^×, of the smooth Kirillov model of classical cuspidal forms of weight k ≥ 2. Theorem C (6.3.1) identifies Hida-ordinary p-adic modular forms with the topological coinvariants of the μ_{p^∞}-action, and Corollary D derives admissibility of these coinvariants from Hida's finiteness theorem. Theorem B then gives a weak local-global compatibility statement: for an irreducible π ⊆ S_{k,C_p}, the space W_π = Hom_{GL_2(A_f^{(p)})}(π^{(p)}, V_{b,C_p}^{cusp}) contains S(Q_p^×, C_p) plus one indicator character for each Galois character appearing in ρ_p, and is contained in that lower bound up to finitely many logarithmic twists. The paper concludes with a conjectural generalization to arbitrary Caraiani–Scholze Igusa varieties.

Significance. The paper's main theorems are substantive and largely proved by detailed, explicit q-expansion computations. Theorem A gives a clean structural description of a natural p-adic automorphic space, Theorem C exposes a new relation between Hida theory and topological coinvariants, and Theorem B is the first local-global compatibility statement of this kind for completed Igusa-variety functions, with explicit and uniform Banach-space bounds. The paper is careful about normalizations, especially the Kirillov and Galois normalizations in §2.6. The external dependencies (Hida finiteness, Kisin's theorem, Breuil–Emerton) are invoked in ways that appear consistent with their stated hypotheses; the stress-test concern about those dependencies does not land as a substantive objection. The conjectural section is clearly labeled and provides a concrete testable framework for future work.

major comments (2)

[Section 3, Proposition 3.2.1] The integral p-adic Fourier isomorphisms are cited to [12, Theorem 7.0.1], which is listed as 'In preparation'. This proposition is load-bearing: it is the mechanism by which the geometric actions of Z_p(1), μ_{p^∞}, and tilde-μ_{p^∞} are converted into the C(Q_p/Z_p, Z_p), C(Z_p, Z_p), and C(Q_p, Z_p) actions used in Definitions 3.4.2 and 3.5.3, in Lemma 3.5.4, and in the proof of Theorem C. The manuscript should either include a proof of Proposition 3.2.1 or replace the reference by a published source before the paper can be considered complete.
[Section 7.3, the identity W_π/S(Q_p^×, C_p) = W_{π, tilde-μ_{p^∞}}] This equality requires that the relation submodule of W_π for the C(Q_p, C_p)-action is exactly S(Q_p^×, C_p). For a general closed submodule of C^{bdd}_∞ containing S, this is not automatic. Since the admissibility argument for W_π/S depends on Corollary D applied to the coinvariants of the ambient space, the proof should justify the equality, for instance by showing that compactly supported elements of S are relations in W_π via multiplication by indicator functions of clopen sets not containing 0.

minor comments (6)

[Title/header] The running header contains a typo: 'COMP A TIBILITY' should read 'COMPATIBILITY'.
[Lemma 2.6.7] In the proof, the phrase 'we may replace π with tilde-π' is terse; the twist tilde-π should be defined explicitly so that the normalization of χ_ord is unambiguous.
[Section 4.5] The phrase 'p-torsion free (i.e. flat over Z_p)' is not an equivalence for general modules; either add a finiteness hypothesis or rephrase to avoid a misleading implication.
[Sections 7.2–7.3] The notation C^{bdd}_∞(Q_p^×, C_p) is introduced informally, and the transition to functions on Q_p used in §7.3 is not fully spelled out; clarifying the extension-by-zero convention would remove ambiguity.
[Section 6.1–6.3] The ordinary projector e = lim_n U_p^{n!} is stated to exist in Lemma 6.2.1, but the text could state explicitly that the limit is taken in the strong operator topology; this is implied by the argument but would be clearer if said directly.
[Remark 3.2.4 and reference [13]] Reference [13] is cited only in a remark and is also listed as 'In preparation'; since it is not used in the proofs, it could be removed or explicitly marked as speculative.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: Theorem B is derived from external published inputs (Hida, Katz, Kisin, Breuil-Emerton) plus direct q-expansion computations; self-citations supply foundational constructions rather than restating the conclusion.

full rationale

I traced the argument from Theorem A through Theorem C and Corollary D to Theorem B. Theorem A is proved by bootstrap from Hida/Shimura density and Katz's q-expansion principle; the 'completion of the Kirillov model' is a proved identification, not a definitional equivalence. Theorem C is a direct q-expansion computation showing that the kernel of the ordinary projector e equals the kernel of the fiber-at-0 coinvariant map; the statement is new but the proof is self-contained once the Up-action and Kir map are fixed. Corollary D is a genuine deduction from Hida's finiteness theorem for the ordinary Hecke algebra, an external published result. The lower bound in Theorem B uses the classical Jacquet/Kirillov computation of (pi_Kir)^\wedge plus Breuil-Emerton [1, Theorem 1.1.3] to supply the split-reducible companion vector; the upper bound uses Corollary D, Kisin's Proposition 7.1.1, and the structure theory of admissible Banach representations of 1+2pZ_p. No step fits a parameter to the predicted object or defines the target W_pi in terms of the characters chi_subset_rho_p it is meant to predict. The self-citations [17] and [19] provide the construction of the tilde-mu_p^infty action and an alternative density argument; they are published foundational inputs, not hidden restatements of Theorem B. One flagged gap: Proposition 3.2.1 is cited to [12, Theorem 7.0.1], an in-preparation work by the same authors; this is a verification gap for a standard p-adic Fourier isomorphism, not a circular reduction. I therefore find no circular step and give score 2 to reflect the minor self-citation/gap without any load-bearing circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central results rest on standard published results in p-adic automorphic forms and on two external theorems, Hida finiteness and Breuil-Emerton companions, that are not proved in this paper. There are no fitted numerical parameters and no new postulated particles, forces, or conserved quantities. The only notable input-sourcing concern is the in-preparation reference [12] for the Fourier isomorphisms in Section 3.

assumptions (7)

domain assumption Hida's finiteness theorem: the ordinary Hecke algebra with p inverted is a finite rank module over Z_p[[Z_p^*]][1/p].
Used in Proposition 6.4.1 to prove Corollary D and in the upper bound of Theorem B.
domain assumption Katz's q-expansion principle and Hida's density of classical cusp forms in p-adic modular forms.
Used in Proposition 5.1.5 and Lemma 5.2.4 to prove injectivity of Kir and density of the evaluation maps for Theorem A.
domain assumption Breuil-Emerton's theory of overconvergent companions, specifically [1, Theorem 1.1.3].
Used in Section 7.4 to construct the missing indicator function 1_{Z_p}*chi in the split reducible case of Theorem B.
domain assumption Kisin's theorem on Galois representations attached to ordinary p-adic modular forms ([25, section 6.13]).
Used in Proposition 7.1.1 to show that unit-eigenvalue characters in W_pi appear as subrepresentations of rho_p.
domain assumption Integral p-adic Fourier theory for mu_{p^infinity}, Z_p(1), and tilde-mu_{p^infinity} (Proposition 3.2.1).
Provides the C(Q_p,Z_p)-action and topological coinvariants; the paper cites [12, Theorem 7.0.1], which is in preparation, though a proof sketch is given.
domain assumption Jacquet-Langlands results on Kirillov models and Jacquet modules of smooth irreducible GL2(Q_ell) representations (Theorem 2.2.1).
Used to compute the completed Kirillov model in Lemma 7.2.3.
domain assumption Construction and affineness of Caraiani-Scholze Igusa formal schemes ([4], [5]).
Defines the spaces V_b and their group actions; assumed as background for both the main theorems and the conjectural generalization.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The completed Kirillov model and local-global compatibility for functions on Igusa varieties." pith.science (2026). https://pith.science/paper/K4TTBRCV

@misc{pith2026250624089,
  author       = {Pith},
  title        = {Pith review of: The completed Kirillov model and local-global compatibility for functions on Igusa varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K4TTBRCV}},
  note         = {Machine review of arXiv:2506.24089}
}

abstract

We describe the cuspidal functions $\mathbb{V}_b^{\mathrm{cusp}}$ on the ordinary Caraiani-Scholze Igusa variety for $\mathrm{GL}_2$ as a completion of the smooth Kirillov model for classical cuspidal modular forms, and identify a variant of Hida's ordinary $p$-adic modular forms with the coinvariants of an action of $\tilde{\mu}_{p^\infty}$ on $\mathbb{V}_b^{\mathrm{cusp}}$. As a consequence of these results, we establish a weak local-global compatibility theorem for eigenspaces in $\mathbb{V}_b^{\mathrm{cusp}}$ associated to classical cuspidal modular forms. Based on these results, we conjecture an analog of Hida theory and an associated local-global compatibility for functions on more general Caraiani-Scholze Igusa varieties, which are natural spaces of $p$-adic automorphic forms.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

$p$-adic Maass--Shimura operators on $\mu$-ordinary Igusa varieties
math.NT 2026-07 accept novelty 7.0 of 10

The authors construct a canonical formal group action on μ-ordinary Igusa varieties for Hodge type Shimura data that integrates Maass-Shimura operators and, via p-adic Fourier theory, yields a unified algebra action e...

Reference graph

Works this paper leans on

34 extracted references · 27 canonical work pages · cited by 1 Pith paper

[17]

A unipotent circle action on p-adic modular forms

Sean Howe. A unipotent circle action on p-adic modular forms. Trans. Amer. Math. Soc. Ser. B, 7:186–226, 2020

work page 2020
[1]

Repr´ esentationsp-adiques ordinaires de GL 2(Qp) et compatibilit´ e local-global.Ast´ erisque, (331):255–315, 2010

Christophe Breuil and Matthew Emerton. Repr´ esentationsp-adiques ordinaires de GL 2(Qp) et compatibilit´ e local-global.Ast´ erisque, (331):255–315, 2010

work page 2010
[12]

p-adic Fourier theory in families

Andrew Graham, Pol van Hoften, and Sean Howe. p-adic Fourier theory in families. In preparation

work page
[2]

Automorphic forms and representations , volume 55 of Cambridge Studies in Advanced Mathematics

Daniel Bump. Automorphic forms and representations , volume 55 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1997

work page 1997
[3]

The conjectural connections between automorphic representa- tions and Galois representations

Kevin Buzzard and Toby Gee. The conjectural connections between automorphic representa- tions and Galois representations. In Automorphic forms and Galois representations. Vol. 1 , volume 414 of London Math. Soc. Lecture Note Ser. , pages 135–187. Cambridge Univ. Press, Cambridge, 2014. 5Emphasis on “‘hope”: what follows is very speculative! 36 SEAN HOWE

work page 2014
[4]

On the generic part of the cohomology of compact unitary shimura varieties

Ana Caraiani and Peter Scholze. On the generic part of the cohomology of compact unitary shimura varieties. Annals of Mathematics , 186(3):649–766, 2017

2017
[5]

On the generic part of the cohomology of non-compact unitary Shimura varieties

Ana Caraiani and Peter Scholze. On the generic part of the cohomology of non-compact unitary Shimura varieties. Ann. of Math. (2) , 199(2):483–590, 2024

2024
[6]

Sur les repr´ esentationsl-adiques associ´ ees aux formes modulaires de Hilbert

Henri Carayol. Sur les repr´ esentationsl-adiques associ´ ees aux formes modulaires de Hilbert. Ann. Sci. ´Ecole Norm. Sup. (4) , 19(3):409–468, 1986

work page 1986

Show all 34 references

[7]

Casselman

W. Casselman. On representations of GL 2 and the arithmetic of modular curves. pages 107–
[8]

P. Deligne. Formes modulaires et repr´ esentations de GL(2). pages 55–105. LNM, Vol. 349, 1973

1973
[9]

Formes modulaires et repr´ esentationsl-adiques

Pierre Deligne. Formes modulaires et repr´ esentationsl-adiques. In S´ eminaire Bourbaki. Vol. 1968/69: Expos´ es 347–363, volume 175 of LNM., pages Exp. No. 355, 139–172. Springer, Berlin, 1971

1968
[10]

Repr´ esentations l-adiques potentiellement semi-stables

Jean-Marc Fontaine. Repr´ esentations l-adiques potentiellement semi-stables. Number 223, pages 321–347. 1994. P´ eriodesp-adiques (Bures-sur-Yvette, 1988)

1994
[11]

Gouvˆ ea.Arithmetic of p-adic modular forms , volume 1304 of LNM

Fernando Q. Gouvˆ ea.Arithmetic of p-adic modular forms , volume 1304 of LNM. Springer- Verlag, Berlin, 1988

1988
[13]

Towards a Fourier theory for Banach- Colmez spaces

Andrew Graham, Pol van Hoften, and Sean Howe. Towards a Fourier theory for Banach- Colmez spaces. In preparation
[14]

Galois representations into GL 2(Zp[[X]]) attached to ordinary cusp forms

Haruzo Hida. Galois representations into GL 2(Zp[[X]]) attached to ordinary cusp forms. Invent. Math. , 85(3):545–613, 1986

1986
[15]

On p-adic Hecke algebras for GL2

Haruzo Hida. On p-adic Hecke algebras for GL2. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) , pages 434–443. Amer. Math. Soc., Providence, RI, 1987

1986
[16]

p-adic automorphic forms on Shimura varieties

Haruzo Hida. p-adic automorphic forms on Shimura varieties. Springer Monographs in Math- ematics. Springer-Verlag, New York, 2004

2004
[18]

Slope classicality in higher Coleman theory via highest weight vectors in com- pleted cohomology

Sean Howe. Slope classicality in higher Coleman theory via highest weight vectors in com- pleted cohomology. Proc. Natl. Acad. Sci. USA , 119(45):Paper No. e2208249119, 3, 2022

2022
[19]

The spectral p-adic Jacquet-Langlands correspondence and a question of Serre

Sean Howe. The spectral p-adic Jacquet-Langlands correspondence and a question of Serre. Compos. Math., 158(2):245–286, 2022

2022
[20]

Admissible pairs and p-adic Hodge structures II: The bi-analytic Ax-Lindemann theorem

Sean Howe and Christian Klevdal. Admissible pairs and p-adic Hodge structures II: The bi-analytic Ax-Lindemann theorem. arXiv:2308.11064

arXiv
[21]

Jacquet and R

H. Jacquet and R. P. Langlands. Automorphic forms on GL(2). LNM, Vol. 114. 1970

1970
[22]

Nicholas M. Katz. Higher congruences between modular forms. Ann. of Math. (2) , 101:332– 367, 1975

1975
[23]

Nicholas M. Katz. p-adic L-functions via moduli of elliptic curves. In Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) , volume Vol. 29 of Proc. Sympos. Pure Math. , pages 479–506. Amer. Math. Soc., Providence, RI, 1975

1974
[24]

Katz and Barry Mazur

Nicholas M. Katz and Barry Mazur. Arithmetic moduli of elliptic curves , volume 108 of Annals of Mathematics Studies . Princeton University Press, Princeton, NJ, 1985

1985
[25]

Overconvergent modular forms and the Fontaine-Mazur conjecture

Mark Kisin. Overconvergent modular forms and the Fontaine-Mazur conjecture. Invent. Math., 153(2):373–454, 2003

2003
[26]

B(G) for all local and global fields

Robert Kottwitz. B(G) for all local and global fields. arXiv:1401.5728, 2014

2014 arXiv
[27]

Isocrystals with additional structure

Robert E Kottwitz. Isocrystals with additional structure. Compositio Mathematica , 56(2):201–220, 1985

1985
[28]

R. P. Langlands. Modular forms and ℓ-adic representations. In Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) , volume Vol. 349 of Lecture Notes in Math. , pages 361–500. Springer, Berlin-New York, 1973

1972
[29]

Modular forms and p-adic Hodge theory

Takeshi Saito. Modular forms and p-adic Hodge theory. Invent. Math., 129(3):607–620, 1997

1997
[30]

p-adic geometry

Peter Scholze. p-adic geometry. Proceedings of the ICM 2018 , 2018

2018
[31]

Moduli of p-divisible groups

Peter Scholze and Jared Weinstein. Moduli of p-divisible groups. Camb. J. Math. , 1(2):145– 237, 2013

2013
[32]

Stacks Project

The Stacks Project Authors. Stacks Project. https://stacks.math.columbia.edu, 2018. THE COMPLETED KIRILLOV MODEL 37

2018
[33]

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

1988
[34]

A PEL-type Igusa Stack and the p-adic geometry of Shimura Varieties

Mingjia Zhang. A PEL-type Igusa Stack and the p-adic geometry of Shimura Varieties. arXiv:2309.05152, 2023

2023 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [17]

A unipotent circle action on p-adic modular forms

Sean Howe. A unipotent circle action on p-adic modular forms. Trans. Amer. Math. Soc. Ser. B, 7:186–226, 2020

work page 2020

[2] [1]

Repr´ esentationsp-adiques ordinaires de GL 2(Qp) et compatibilit´ e local-global.Ast´ erisque, (331):255–315, 2010

Christophe Breuil and Matthew Emerton. Repr´ esentationsp-adiques ordinaires de GL 2(Qp) et compatibilit´ e local-global.Ast´ erisque, (331):255–315, 2010

work page 2010

[3] [12]

p-adic Fourier theory in families

Andrew Graham, Pol van Hoften, and Sean Howe. p-adic Fourier theory in families. In preparation

work page

[4] [2]

Automorphic forms and representations , volume 55 of Cambridge Studies in Advanced Mathematics

Daniel Bump. Automorphic forms and representations , volume 55 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1997

work page 1997

[5] [3]

The conjectural connections between automorphic representa- tions and Galois representations

Kevin Buzzard and Toby Gee. The conjectural connections between automorphic representa- tions and Galois representations. In Automorphic forms and Galois representations. Vol. 1 , volume 414 of London Math. Soc. Lecture Note Ser. , pages 135–187. Cambridge Univ. Press, Cambridge, 2014. 5Emphasis on “‘hope”: what follows is very speculative! 36 SEAN HOWE

work page 2014

[6] [4]

On the generic part of the cohomology of compact unitary shimura varieties

Ana Caraiani and Peter Scholze. On the generic part of the cohomology of compact unitary shimura varieties. Annals of Mathematics , 186(3):649–766, 2017

2017

[7] [5]

On the generic part of the cohomology of non-compact unitary Shimura varieties

Ana Caraiani and Peter Scholze. On the generic part of the cohomology of non-compact unitary Shimura varieties. Ann. of Math. (2) , 199(2):483–590, 2024

2024

[8] [6]

Sur les repr´ esentationsl-adiques associ´ ees aux formes modulaires de Hilbert

Henri Carayol. Sur les repr´ esentationsl-adiques associ´ ees aux formes modulaires de Hilbert. Ann. Sci. ´Ecole Norm. Sup. (4) , 19(3):409–468, 1986

work page 1986