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$p$ -adic rigidity for $\mathrm{GSp}_4$

This paper proves that certain refined noncuspidal Saito–Kurokawa automorphic representations of $\mathrm{GSp}_4(\mathbb{A}_\mathbb{Q})$ are $p$ -adically rigid, meaning they cannot be interpolated in nontrivial positive-dimensional $p$ -adic families, thereby establishing a $\mathrm{GSp}_4$ analogue of Bellaïche's rigidity theorems for $\mathrm{U}(2,1)$ .

Original authors: Charlotte Clare-Hunt

Published 2026-08-07

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Charlotte Clare-Hunt

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 a detective trying to solve a mystery about how numbers behave when you zoom in infinitely close. In the world of mathematics, there is a special branch called number theory that studies the deep, hidden patterns of whole numbers. One of its most fascinating tools is the "L-function," which acts like a cosmic barcode for complex shapes called automorphic forms. These forms are like intricate, multi-dimensional musical notes that vibrate across the universe of numbers.

For a long time, mathematicians have been trying to connect these musical notes to "Galois representations," which are essentially secret codes that describe how prime numbers interact. To do this, they invented a magical machine called an "eigenvariety." Think of an eigenvariety as a vast, shimmering landscape where every point represents a specific musical note. If you can walk smoothly from one point to another on this landscape, it means you can create a continuous "family" of notes that change slightly but stay connected. This is powerful because it allows mathematicians to use calculus to study whole numbers, leading to breakthroughs in cryptography and understanding the structure of the universe.

However, not every point on this landscape is connected to a path. Some points are "rigid." Imagine a tree growing in a forest; most trees have branches that stretch out in many directions, allowing you to climb higher or lower. But a rigid point is like a tree that has been frozen in ice, with no branches at all. You cannot move even a single step away from it without breaking the rules of the game. The question is: are there such frozen trees in the world of these special musical notes, and if so, why?

This paper, written by Charlotte Clare-Hunt, investigates a specific type of musical note called a "Saito–Kurokawa" lift. These are special because they are built by combining two simpler notes in a very specific way. The author asks a bold question: Can we find a continuous path (a "p-adic family") that connects these specific notes to their neighbors? The answer, surprisingly, is a hard "no" for a certain class of these notes.

The paper proves that for a specific set of these Saito–Kurokawa points, the landscape is completely frozen. If you try to build a family of these notes that changes smoothly, you will find that the family cannot grow; it collapses into a single, isolated point. The author shows that any attempt to create a path leads to a mathematical contradiction, much like trying to build a bridge that suddenly turns into a wall.

To reach this conclusion, the author uses a clever strategy involving "pseudocharacters," which are like shadowy outlines of the secret Galois codes. The paper argues that if a path existed, these shadows would have to split apart in a specific way. However, the author proves that for these particular notes, the shadows refuse to split. Instead, they are forced to stay stuck together in a way that violates the known laws of number theory (specifically, it would create a "Selmer group" that should be empty but isn't).

The paper explicitly rules out the idea that these specific points can be part of a larger, moving family. It does not suggest that all points are rigid; in fact, it acknowledges that other types of points (like the "cuspidal" ones) can and do move. But for the non-cuspidal Saito–Kurokawa points with certain properties, the rigidity is absolute. The author is very sure of this result, having provided a rigorous proof rather than just a guess or a simulation.

In the end, this work acts like a mapmaker who discovers a "No Entry" zone in the mathematical wilderness. It tells us that while the landscape of numbers is full of winding roads and connecting bridges, there are certain islands that stand alone, completely isolated from the rest. This helps mathematicians understand the limits of their tools and the unique, stubborn nature of these specific number patterns.

Technical Summary: p-ADIC RIGIDITY FOR GSp4

1. Problem Statement and Motivation
This paper addresses the problem of p-adic rigidity for specific noncuspidal automorphic representations of the symplectic group $GSp_4(\mathbb{A}_{\mathbb{Q}})$ . While substantial progress has been made in constructing eigenvarieties (p-adic rigid analytic spaces interpolating systems of Hecke eigenvalues) and studying p-adic families of automorphic forms, the geometry of the noncuspidal, nonholomorphic locus remains poorly understood.

The central question is whether certain "p-refined" automorphic representations can be interpolated in a nontrivial positive-dimensional p-adic family. The paper focuses on Saito–Kurokawa (SK) points, which are lifts of $GL_2$ representations to $GSp_4$ . Specifically, it investigates whether these points, when equipped with specific refinements (accessible characters of the Borel torus), admit deformations within the eigenvariety. This work serves as a $GSp_4$ analogue of Bellaïche's rigidity theorems for the unitary group $U(2, 1)$ , aiming to identify obstructions to p-adic variation on the $GSp_4$ eigenvariety.

2. Methodology and Framework
The proof strategy does not rely on the pre-existence of a global eigenvariety but rather abstracts the properties a p-adic family is expected to satisfy, adapting Bellaïche's framework for $U(2, 1)$ .

Definitions of Families and Points:
- A p-adic family $X(\phi)$ is defined as a reduced, separated, irreducible rigid analytic space over $\mathbb{Q}_p$ equipped with a weight map, a Hecke algebra homomorphism, and a Zariski-dense set of classical points $Z$ .
- A Saito–Kurokawa (SK) point $z_0 = (\pi, \psi_R)$ corresponds to a noncuspidal SK representation $\pi$ with an accessible refinement $\psi_R$ .
- The paper distinguishes between four possible refinements ( $\psi_1, \dots, \psi_4$ ) of the local SK factor. The main results apply specifically to $\psi_2$ - and $\psi_3$ -refined points, which are analogous to "anti-ordinary" points in the $U(2, 1)$ setting.
Key Technical Inputs:
1. Arthur's Classification: The structural role of Arthur's classification of discrete automorphic representations for $GSp_4$ is utilized to analyze the reducibility of the associated Galois representations.
2. Pseudocharacters and Reducibility: The paper employs the theory of pseudocharacters (following Taylor and Bellaïche–Chenevier) to analyze the generic pseudocharacter $T_\eta$ attached to a family. It investigates whether $T_\eta$ is $(2, 1, 1)$ -reducible (decomposing into a 2-dimensional and two 1-dimensional representations).
3. Kisin Property: An interpolation property for crystalline Frobenius eigenvalues is assumed to hold at the point $z_0$ . This ensures that extensions arising from the family are crystalline at $p$ .
4. Monodromy Control (SK–P2): A condition is imposed to control the Galois monodromy away from $p$ . Specifically, for primes $\ell \neq p$ where the local factor is a twist of Steinberg, the monodromy operator in the family must lie in the Zariski closure of the conjugacy class of the monodromy at $z_0$ .
5. Bloch–Kato Selmer Groups: The proof culminates in a contradiction derived from the vanishing of specific Selmer groups, $H^1_f(\mathbb{Q}, \rho_\mu(2))$ and $H^1_f(\mathbb{Q}, \mathbb{Q}_p(1))$ .

3. Key Results

SK Rigidity (Proposition 6.3):
If a p-adic family $X(1)$ passes through a $\psi_i$ -refined SK point $z_0$ with $i \in \{2, 3\}$ (and satisfying a slope condition $v(\alpha(z_0)) \neq k(z_0)-2$ if $i=3$ ), and if the set of SK points accumulates at $z_0$ , then the family $X(1)$ must be a single point. This establishes that $\psi_2$ and $\psi_3$ -refined SK points do not deform in families containing other SK points.
Non-(2, 1, 1)-Reducibility (Proposition 7.10):
For a positive-dimensional family through an SK point, the generic pseudocharacter $T_\eta$ cannot be $(2, 1, 1)$ -reducible. The proof uses the fact that type (e)/(f) points (which are $(1,1,1,1)$ -reducible) and SK points (which are $(2,1,1)$ -reducible) cannot accumulate on each other in a way that preserves the generic reducibility type without contradiction.
Complete Rigidity (Theorem 8.4):
Let $z_0$ be a noncuspidal $\psi_i$ -refined SK point ( $i \in \{2, 3\}$ ) satisfying the Steinberg condition (St) and the slope condition. If $X(1)$ is a p-adic family through $z_0$ satisfying the Kisin property and the monodromy control condition (SK–P2), then $X(1)$ is a point.

Proof Sketch: The argument assumes $\dim X > 0$ . This implies $T_\eta$ is not $(2, 1, 1)$ -reducible. However, the existence of the family forces the existence of a non-split extension in either $\text{Ext}_T(\epsilon^{-2}, \epsilon^{-1})$ or $\text{Ext}_T(\epsilon^{-2}, \rho_\mu)$ .
- Extensions in $\text{Ext}_T(\epsilon^{-2}, \rho_\mu)$ are shown to be unramified outside $p$ and crystalline at $p$ . Twisting by $\epsilon^2$ yields a class in $H^1_f(\mathbb{Q}, \rho_\mu(2))$ , which vanishes because $L(1/2, \mu) \neq 0$ .
- Extensions in $\text{Ext}_T(\epsilon^{-2}, \epsilon^{-1})$ similarly yield a class in $H^1_f(\mathbb{Q}, \mathbb{Q}_p(1))$ , which vanishes by Kummer theory.
- The vanishing of these Selmer groups contradicts the existence of a nontrivial extension, implying the family must be zero-dimensional.

4. Significance and Claims
The paper claims to establish the p-adic rigidity of a specific class of noncuspidal automorphic representations on $GSp_4$ .

Analogy to $U(2, 1)$ : It provides a $GSp_4$ analogue of Bellaïche's rigidity theorems, identifying a similar obstruction to p-adic variation for "anti-ordinary" (specifically $\psi_2$ and $\psi_3$ ) refined points.
Distinction from Cuspidal Points: The paper explicitly notes that cuspidal SK points are not rigid (they do deform in positive-dimensional families, e.g., semi-ordinary families). The rigidity result is specific to the noncuspidal case.
Role of Refinement: The results highlight that the choice of refinement is decisive. The paper expects that $\psi_1$ and $\psi_4$ -refined points (analogous to ordinary points) do deform, and the rigidity theorems specifically exclude these cases.
Obstruction Identification: The work identifies the vanishing of specific Bloch–Kato Selmer groups (driven by the non-vanishing of the central L-value $L(1/2, \mu)$ and Kummer theory) as the fundamental obstruction to p-adic variation for these noncuspidal points.

The paper remains modest regarding the necessity of its hypotheses (such as the noncuspidality and the specific monodromy condition), noting that while the method suggests rigidity for certain cuspidal cases, the monodromy conditions required for the proof may not hold for generic cuspidal points.

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