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Arithmetic level raising for certain quaternionic unitary Shimura variety

This paper establishes an arithmetic level raising theorem for the symplectic group of degree four in the ramified case by analyzing the supersingular locus of a quaternionic unitary Shimura variety, a result that serves as a crucial step toward the Beilinson-Bloch-Kato conjecture for specific Rankin-Selberg motives.

Original authors: Haining Wang

Published 2026-05-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Haining Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a massive, multi-dimensional puzzle made of numbers and shapes. This puzzle represents the deep connections between two different worlds of mathematics: geometry (shapes and spaces) and number theory (properties of numbers like primes).

This paper, written by Haining Wang, is a significant step in solving a specific, very difficult part of this puzzle. Here is a breakdown of what the author did, using simple analogies.

The Big Picture: The "Level Raising" Puzzle

In the world of modular forms (a special type of function used to study numbers), mathematicians have a rule called Level Raising. Think of a "level" as the complexity or the "resolution" of a pattern.

The Old Rule: Usually, if you have a pattern at a low resolution (Level $N$ ), you can't just magically create a new, more complex pattern at a higher resolution (Level $N \times p$ ) that looks exactly the same in the most important ways, unless a very specific mathematical "lock" clicks open.
The Lock: This lock is a specific condition involving prime numbers. If the numbers line up just right (a specific congruence), the lock opens, and you can "raise the level" to create a new, more complex pattern that shares the same DNA as the old one.

For simple shapes (like a line or a circle), mathematicians figured out how to do this a long time ago. But for complex, 4-dimensional shapes (which this paper deals with), it has been a mystery, especially when the shapes have "rough spots" or singularities.

The Setting: A Bumpy Landscape

The author studies a specific type of geometric object called a Shimura Variety.

The Analogy: Imagine a landscape. In most places, the ground is smooth and flat. But in this specific landscape, there are a few spots where the ground is crumpled or bumpy (singularities).
The Problem: When you try to study the "vibrations" (cohomology) of this landscape, the bumps make the math very messy. Usually, mathematicians avoid these bumps. Wang decided to study the bumps directly because that's where the secret to "Level Raising" is hidden.

The Method: The "Supersingular" Map

To solve the puzzle, the author had to map out the Supersingular Locus.

The Analogy: Imagine the landscape has a hidden underground network of caves. These caves only exist under very specific, "superspecial" conditions.
The Discovery: Wang proved that these caves are not random; they are organized in a very precise, grid-like pattern. He showed that the "vibrations" of the entire landscape are actually generated by the vibrations coming from these specific caves.
The Matrix: He built a giant calculator (a "Level Raising Matrix") to see how the vibrations in the caves connect to the vibrations on the surface. He proved that if the "lock" (the level raising condition) clicks, the connection is strong enough to prove the existence of the new, higher-level pattern.

The Main Result

The paper proves a Level Raising Theorem for a specific group of numbers called GSp4 (which relates to 4-dimensional symplectic geometry).

What it means: If you have a specific number pattern (an automorphic representation) and you find a prime number that satisfies the "lock" condition, you can mathematically prove that there must exist a new, more complex pattern that is related to the first one.
Why it matters: This is a key step toward proving the Beilinson–Bloch–Kato conjecture. Think of this conjecture as the "Grand Unified Theory" for a specific family of number problems. It predicts how the number of solutions to certain equations relates to the values of special functions (L-functions). This paper provides the necessary bridge to cross a difficult river on the way to that theory.

The "Ihara's Lemma" Connection

The author compares his result to a famous rule called Ihara's Lemma.

The Analogy: Ihara's Lemma is like a rule that says, "If you want to understand the whole forest, you only need to look at the trees that are growing on the rocky, hard-to-reach cliffs."
The New Rule: Wang proved a version of this rule for his 4-dimensional landscape. He showed that even though the landscape is huge and complex, all the important information about the "new" patterns is actually stored in the "bumpy" supersingular parts of the landscape.

Summary

In short, Haining Wang built a bridge between two different mathematical worlds by:

Studying the "bumpy" parts of a complex 4-dimensional shape.
Mapping out the hidden "caves" (supersingular locus) inside that shape.
Proving that if a specific number condition is met, you can mathematically guarantee the existence of a new, more complex pattern (Level Raising).
Using this to take a giant step toward solving a major unsolved problem in number theory (the Beilinson–Bloch–Kato conjecture).

The paper is a technical tour de force that turns a chaotic, bumpy mathematical landscape into a structured, predictable map, allowing mathematicians to finally "raise the level" of their understanding in this specific, high-dimensional context.

Title: Arithmetic Level Raising on Certain Quaternionic Unitary Shimura Varieties
Author: Haining Wang

1. Problem Statement and Context

The paper addresses the problem of establishing an arithmetic level raising theorem for the symplectic group of degree four, $GSp_4$ , in the ramified case. This work is situated within the broader framework of the Beilinson–Bloch–Kato (BBK) conjecture for Rankin–Selberg motives associated with orthogonal groups, specifically in the context of the Gan–Gross–Prasad (GGP) conjecture.

While level raising theorems are well-understood for $GL_2$ (via Ribet's work) and have been extended to certain unitary Shimura varieties (e.g., by Liu, Taylor, and others), extending these results to higher-rank groups like $GSp_4$ presents significant geometric and representation-theoretic challenges. Specifically, the author aims to prove that if a cuspidal automorphic representation $\pi$ of $GSp_4(\mathbb{A}_\mathbb{Q})$ satisfies specific congruence conditions at a prime $p$ (the "level raising condition"), one can construct a new automorphic representation $\Pi$ of higher level (ramified at $p$ ) that shares the same residual Galois representation.

2. Methodology

The proof relies on a combination of arithmetic geometry, the theory of Shimura varieties, and the Jacquet–Langlands correspondence. The core methodology involves:

Geometric Setup: The author utilizes a quaternionic unitary Shimura variety $Sh(B, K_{Pa})$ of dimension 3, associated with an indefinite quaternion algebra $B$ ramified at $p$ and another prime $q$ . The integral model of this variety, $X_{Pa}(B)$ , has isolated ordinary quadratic singularities in its special fiber.
Supersingular Locus Analysis: A critical step is the detailed description of the supersingular locus of the special fiber. The author employs the Rapoport–Zink uniformization theorem to relate this locus to a Rapoport–Zink space. The structure of this locus is analyzed using Bruhat–Tits stratification, showing that its irreducible components are parametrized by Shimura sets (definite quaternionic unitary groups) and are isomorphic to Deligne–Lusztig varieties.
Monodromy and Cohomology: The proof heavily utilizes the Picard–Lefschetz formula to compute the monodromy action on the cohomology of the special fiber. The author refines this formula to identify the graded pieces of the monodromy filtration with specific intersection cohomology groups. This allows for the computation of the action of the monodromy operator $N$ on the nearby cycle cohomology.
Level Raising Matrices: The author constructs a level raising matrix ( $T_{lr}$ ) and a supersingular matrix ( $T_{ss}$ ). These matrices relate the cohomology of the Shimura variety to the cohomology of the supersingular locus (which corresponds to the Shimura set). The entries of these matrices are computed using Hecke operators and intersection numbers on the singular fibers.
Galois Deformation Theory: To handle torsion issues and establish freeness results, the paper employs techniques from Galois deformation rings (following the work of Taylor–Wiles and recent advances by LTXZZ). This is used to prove that the relevant cohomology groups are free over universal deformation rings, which is essential for deducing the existence of the level-raised form.

3. Key Contributions and Results

A. Arithmetic Level Raising Theorem for $GSp_4$ (Theorem 1.4 & 9.4)
The main result establishes an isomorphism between a singular quotient of the Galois cohomology of the nearby cycles of the Shimura variety and the cohomology of a Shimura set (the "old" forms):
$H^1_{sing}(\mathbb{Q}_{p^2}, H^3_c(Sh(B, K_{Pa}), \mathcal{O}_\lambda/\lambda^m(2))_{\mathfrak{m}}) \xrightarrow{\sim} \mathcal{O}_\lambda/\lambda^m[Sh(B, K_H)]_{\mathfrak{m}}$
Here, $\mathfrak{m}$ is a maximal ideal corresponding to a residual Galois representation $\rho_{\pi, \lambda}$ . This isomorphism serves as an analogue of Ihara's lemma for Shimura varieties with bad reduction.

B. Level Raising for Automorphic Representations (Theorem 1.6 & 9.11)
As a direct application, the paper proves that if $\pi$ is a cuspidal automorphic representation of $GSp_4(\mathbb{A})$ of "general type" (non-endoscopic, non-CAP) satisfying the level raising condition at $p$ , there exists a new automorphic representation $\Pi$ such that:

$\Pi$ is also of general type.
The local component $\Pi_p$ is of type IIa (a ramified representation).
The residual Galois representations are isomorphic: $\rho_{\pi, \lambda} \cong \rho_{\Pi, \lambda}$ .

C. Geometric Results on Supersingular Loci

The author proves that the intersection cohomology of the special fiber, localized at a generic maximal ideal, is generated by Tate cycles coming from the supersingular locus (Theorem 1.5).
The paper establishes the weight-monodromy conjecture for certain interior cohomology groups of the Shimura variety (Corollary 5.19).
It is shown that the cohomology of the special fiber is torsion-free after localization at a generic maximal ideal (Corollaries 9.7, 9.8, 9.10), a result achieved by adapting methods from Galois deformation theory.

D. Matrix Computations
The paper provides explicit computations of the level raising matrix and the supersingular matrix in terms of Hecke operators ( $T_{p,0}, T_{p,1}, T_{p,2}$ ) and their determinants. These computations are crucial for verifying that the relevant maps are isomorphisms under the generic level raising conditions.

4. Significance and Claims

The paper claims that this result is a key step towards the Beilinson–Bloch–Kato conjecture for certain Rankin–Selberg motives associated with orthogonal groups ( $SO_4 \times SO_5$ ) within the GGP framework.

Analogue of Ihara's Lemma: The main theorem is presented as a higher-rank analogue of Ihara's lemma for Shimura curves ramified at $p$ , extending the scope of level raising techniques beyond $GL_2$ and unitary groups.
Bridge to BBK: The author notes that in a sequel, this result will be used to show that the non-vanishing of certain period integrals implies the vanishing of specific Bloch–Kato Selmer groups, thereby establishing one direction of the BBK conjecture in rank 0 and rank 1 cases.
Level Lowering: The results are also intended to facilitate the proof of level lowering theorems for $GSp_4$ (as referenced in the companion article [Wang22b]), which is a necessary component for the full arithmetic GGP conjecture.

The author explicitly states that the restriction to $GSp_4$ over $\mathbb{Q}$ (rather than a totally real field) is due to the current limitations of the Rapoport–Zink uniformization theorem for the specific Shimura varieties studied, though the author believes the geometric description of the supersingular locus would generalize. The paper does not claim to solve the full BBK conjecture but rather provides the necessary arithmetic level raising machinery to attack specific cases.