Embeddings of Certain Exceptional Shimura Varieties into Siegel Modular Varieties

Ali Partofard; Mohammad Hadi Hedayatzadeh

arxiv: 2506.21217 · v2 · pith:QUGAEKKGnew · submitted 2025-06-26 · 🧮 math.AG · math.NT

Embeddings of Certain Exceptional Shimura Varieties into Siegel Modular Varieties

Mohammad Hadi Hedayatzadeh , Ali Partofard This is my paper

Pith reviewed 2026-05-22 00:26 UTC · model grok-4.3

classification 🧮 math.AG math.NT

keywords Shimura varietiesSiegel modular varietiesperfectoid spaceslocal Shimura varietiesp-divisible groupsexceptional groupsembeddings

0 comments

The pith

Certain exceptional Shimura varieties embed into Siegel modular varieties and are perfectoid at infinite level.

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

The paper introduces a class of local Shimura varieties that includes some attached to exceptional groups. For objects in this class the authors build a functor from (G, μ)-displays to p-divisible groups. This construction is then used to show that the local Shimura variety is representable and becomes a perfectoid space once the level is infinite. On the global side the same class lets them realize certain exceptional Shimura varieties as subvarieties of Siegel modular varieties. A direct consequence is that these exceptional varieties are also perfectoid at infinite level.

Core claim

We define a class of local Shimura varieties containing some local Shimura varieties for exceptional groups. We construct a functor from (G, μ)-displays to p-divisible groups for this class. As an application we prove that the local Shimura variety is representable and perfectoid at the infinite level. Considering the global counterpart we embed certain exceptional Shimura varieties into Siegel modular varieties and prove that they are perfectoid at the infinite level.

What carries the argument

The class of local Shimura varieties together with the functor from (G, μ)-displays to p-divisible groups, which establishes representability and perfectoidness.

If this is right

The local Shimura varieties in the class are representable over the base.
They are perfectoid when the level is infinite.
Certain exceptional Shimura varieties admit embeddings into Siegel modular varieties.
These embedded exceptional varieties are perfectoid at infinite level.

Where Pith is reading between the lines

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

This construction may allow known results on Siegel modular varieties to be transferred to the embedded exceptional cases.
The perfectoid property could be used to study the cohomology of these varieties in the p-adic setting.
Extending the class to more groups might yield similar embeddings for other exceptional cases.

Load-bearing premise

The defined class must be broad enough to contain the desired exceptional Shimura varieties while still admitting a well-behaved functor from displays to p-divisible groups.

What would settle it

A direct computation for a small exceptional group such as G2 showing that the embedded variety at infinite level fails to be perfectoid or that the functor does not produce the expected p-divisible group.

read the original abstract

We define a class of local Shimura varieties that contains some local Shimura varieties for exceptional groups, and for this class, we construct a functor from $\left(G, \mu\right)$-displays to $p$-divisible groups. As an application, we prove that for this class, the local Shimura variety is representable and perfectoid at the infinite level. Considering the global counterpart of this class, we embed certain exceptional Shimura varieties into Siegel modular varieties. In particular, we prove that they are perfectoid at the infinite level.

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 carves out a class of local Shimura varieties that reaches some exceptional groups, builds a functor from (G, μ)-displays to p-divisible groups, and uses it to get representability plus perfectoidness at infinite level, then embeds the global versions into Siegel modular varieties.

read the letter

The main thing to know is that this paper carves out a class of local Shimura varieties containing some exceptional cases, defines a functor from (G, μ)-displays to p-divisible groups for that class, and shows the local Shimura variety is representable and perfectoid at infinite level. As an application, they embed certain exceptional Shimura varieties into Siegel modular varieties and prove the infinite level is perfectoid there too. What is new is the tailored class and functor that reach beyond the usual classical groups. The paper does well in sketching how this functor leads to the representability and perfectoid results, which could allow arithmetic properties to move from the Siegel setting to these harder varieties. That has potential value for work on the Langlands program or p-adic uniformization. The softer part is the assumption that the functor extends nicely to exceptional groups. These have different root systems, so making sure the Hodge filtration and display structures carry over correctly is the critical step. If that holds, the rest follows, but it's the part that needs the most verification in the proofs. This paper is for people deep into Shimura varieties and p-adic geometry. A reader looking for ways to handle exceptional cases through embeddings would get something concrete from it. It deserves a serious referee because the claims are focused and build on existing techniques in a new direction. I recommend sending it out for peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript defines a new class of local Shimura varieties containing certain local Shimura varieties attached to exceptional groups. For this class it constructs a functor from (G, μ)-displays to p-divisible groups. Using the functor it proves that the local Shimura variety is representable and perfectoid at the infinite level. In the global setting the paper embeds certain exceptional Shimura varieties into Siegel modular varieties and deduces that they are perfectoid at the infinite level.

Significance. If the constructions and verifications hold, the work would extend the theory of perfectoid Shimura varieties and their infinite-level properties from classical groups to selected exceptional cases. The embedding into Siegel modular varieties would provide a concrete way to import known perfectoid and arithmetic results for Siegel varieties to the exceptional setting. The functor from (G, μ)-displays to p-divisible groups is a technical contribution that could be useful beyond the immediate applications.

major comments (2)

[§2] §2 (definition of the new class): the manuscript must explicitly check that the proposed class contains the desired exceptional cases (e.g., those attached to E6 or E7). The conditions imposed on the root datum and the cocharacter μ need to be verified to hold for non-simply-laced or higher-rank root systems; without this verification the claim that the class is broad enough remains unanchored.
[§3] §3 (construction of the functor): the proof that the functor from (G, μ)-displays to p-divisible groups preserves the Hodge filtration, the display equations, and the necessary integrality conditions must be supplied in detail for the exceptional groups. Previous such functors were established only for classical groups (GL_n, GSp, etc.); if the transport of the filtration fails for non-simply-laced roots, the subsequent arguments for representability and perfectoidness at infinite level do not go through.

minor comments (2)

[Abstract] The abstract should name the specific exceptional groups or root systems under consideration rather than using the phrase “certain exceptional groups.”
[§2] Notation for the new class of local Shimura varieties should be introduced once and used consistently; currently several ad-hoc labels appear in the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We appreciate the positive assessment of the potential significance of the work in extending perfectoid techniques to selected exceptional cases. We address the major comments point by point below and will make the indicated revisions.

read point-by-point responses

Referee: [§2] §2 (definition of the new class): the manuscript must explicitly check that the proposed class contains the desired exceptional cases (e.g., those attached to E6 or E7). The conditions imposed on the root datum and the cocharacter μ need to be verified to hold for non-simply-laced or higher-rank root systems; without this verification the claim that the class is broad enough remains unanchored.

Authors: We agree that an explicit verification would improve clarity. In the revised manuscript we will add a short subsection (or expanded paragraph) in §2 that directly checks the root-datum and cocharacter conditions for the exceptional groups under consideration, including the cases attached to E6 and E7. The verification will confirm that the defining requirements of the class are satisfied for both simply-laced and non-simply-laced root systems of the relevant ranks. revision: yes
Referee: [§3] §3 (construction of the functor): the proof that the functor from (G, μ)-displays to p-divisible groups preserves the Hodge filtration, the display equations, and the necessary integrality conditions must be supplied in detail for the exceptional groups. Previous such functors were established only for classical groups (GL_n, GSp, etc.); if the transport of the filtration fails for non-simply-laced roots, the subsequent arguments for representability and perfectoidness at infinite level do not go through.

Authors: We thank the referee for highlighting this point. The general construction of the functor in §3 is formulated so that it applies to the class defined in §2, which includes the exceptional cases. Nevertheless, to address the concern about non-simply-laced roots, we will expand the proof in the revised version with additional explicit verifications of the preservation of the Hodge filtration, the display equations, and the integrality conditions, written specifically for the root systems appearing in the exceptional groups. These additions will ensure that the subsequent arguments for representability and perfectoidness at infinite level are fully justified. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results follow from new definitions and functor

full rationale

The paper defines a new class of local Shimura varieties containing exceptional cases and constructs a functor from (G, μ)-displays to p-divisible groups. It then proves representability and perfectoidness at infinite level for this class, plus embeddings of global counterparts into Siegel modular varieties. These steps derive the claimed properties directly from the introduced class and functor, without any reduction of the target results to the inputs by construction, self-citation chains, or renaming. The derivation is self-contained; concerns about exceptional groups pertain to correctness of the functor rather than circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard background from the theory of Shimura varieties, p-divisible groups, and perfectoid spaces; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Standard properties of Shimura varieties, displays, and p-divisible groups are assumed to hold.
The construction and proofs presuppose the usual foundations of arithmetic geometry.

pith-pipeline@v0.9.0 · 5617 in / 1217 out tokens · 50826 ms · 2026-05-22T00:26:32.459051+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We define a class of local Shimura varieties that contains some local Shimura varieties for exceptional groups, and for this class, we construct a functor from (G, μ)-displays to p-divisible groups.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the resulting shtuka is bounded by a minuscule cocharacter... regularly sparse Shimura data

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.