Irreducible Hodge-Tate p-adic Representations

• Irreducible Hodge-Tate p-adic representations are one-dimensional, continuous Galois representations with well-defined Hodge–Tate weights that capture essential arithmetic properties.
• They naturally arise in the study of automorphic forms and algebraic varieties, linking local p-adic theory with global arithmetic applications.
• Their structure, characterized via Sen theory and Tate twists, offers insights into modularity, decomposition types, and geometric constructions in p-adic Hodge theory.

An irreducible Hodge–Tate $p$-adic representation is a continuous, irreducible representation of the absolute Galois group of a $p$-adic field whose Hodge–Tate weights are well defined and whose structure is governed by both $p$-adic Hodge theory and deep links with the theory of automorphic forms and arithmetic geometry. These objects play a critical role in the paper of arithmetic aspects of algebraic varieties, the Langlands program, and the structure of Galois representations arising from geometry.

1. Definitions and Fundamental Properties

Let $K$ be a $p$-adic field (i.e., a finite extension of $\mathbb{Q}_p$) with absolute Galois group $G_K = \mathrm{Gal}(\bar{K}/K)$. A $p$-adic Galois representation is a finite-dimensional $\mathbb{Q}_p$-vector space $V$ equipped with a continuous $G_K$-action. It is called Hodge–Tate if the $C_p$-linearization admits a decomposition

$$V \otimes_{\mathbb{Q}_p} C_p \simeq \bigoplus_{i \in \mathbb{Z}} C_p(-i)^{m_i}$$

where $C_p(-i)$ denotes the rank-one representation of $G_K$ of Hodge–Tate weight $i$, and the $m_i$ are the non-negative multiplicities of each weight (the Hodge–Tate weights). Such representations correspond to local systems on algebraic varieties over $K$ for which, after extension to an algebraic closure, the stalk is Hodge–Tate (Petrov, 2020).

A representation is irreducible if it admits no $G_K$-stable nontrivial subspaces. The Hodge–Tate condition is equivalent, via Sen theory, to the property that the Sen operator is semisimple with integral eigenvalues.

2. Classification and Structure of Irreducible Hodge–Tate Representations

The category of Hodge–Tate representations displays strong semisimplicity: every Hodge–Tate representation decomposes canonically as a direct sum of one-dimensional Tate twists, and the irreducible objects are precisely the one-dimensional Tate objects $C(-n) = \mathbb{Q}_p(n) \otimes_{\mathbb{Q}_p} C$ for $n \in \mathbb{Z}$ (Anschütz et al., 2022). Concretely, the corresponding vector bundles on Bhatt–Lurie’s Hodge–Tate locus are slope-stable line bundles, and irreducibility is equivalent to slope-stability in this perspective.

Criterion: A Hodge–Tate representation is irreducible if and only if it is one-dimensional. Equivalently, its Sen operator $\Theta_V$ has a single integer eigenvalue of full multiplicity (Anschütz et al., 2022).

3. Higher-Dimensional Geometric and Arithmetic Examples

While the local picture is rigid, global applications yield more intricate irreducible Hodge–Tate $p$-adic representations:

• For global Galois groups $G_F$ over a totally real or number field $F$, for many automorphic representations $\pi$ of $\mathrm{GL}_n(\mathbb{A}_F)$, the conjecturally attached (and, in many cases, known) $p$-adic Galois representations $\rho_{\pi, p}$ are irreducible, Hodge–Tate at finite places above $p$, and also often crystalline (Ramakrishnan, 2013, Duan et al., 5 Jun 2024).
• For such $\rho_{\pi, p}$ associated to regular algebraic cuspidal $\pi$ on $\mathrm{GL}_4$, irreducibility holds whenever $$p-1 > 2 \cdot \max_{v \mid p}(HT_{v, \max}-HT_{v, \min})$$, generalizing results in lower rank (Ribet, Blasius–Rogawski) to dimension $4$ (Ramakrishnan, 2013).

Context	Irreducible Hodge–Tate Rep.	Criteria/Comment
Local (over $K$)	Only $\mathbb{Q}_p(n)$	All irreducibles are one-dimensional Tate twists
Global automorphic	e.g., $\rho_{\pi, p}$ for $\pi$ cuspidal	Irreducibility under regularity and crystallinity conditions

This suggests that while the local Hodge–Tate theory is completely decomposable, global arithmetic constructions produce irreducible higher-dimensional $p$-adic representations that are Hodge–Tate and arise from geometry or automorphic forms.

4. Explicit Constructions: Automorphic, Modular, and Geometric Origins

A principal source of irreducible Hodge–Tate $p$-adic representations is the étale cohomology of smooth projective varieties over number fields or the Galois representations associated with modular (and, more generally, automorphic) forms.

For $\pi$ a regular algebraic, cuspidal automorphic representation of $\mathrm{GL}_4/\mathbb{Q}$ associated to a holomorphic Siegel modular cusp form of genus $2$ and parallel weight $k > 3$, the $4$-dimensional Galois representation $$\rho_{\pi, p} : \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}_4(\mathbb{Q}_p)$$ is irreducible for every $p$ with $p-1 > 2(k-1)$ (Ramakrishnan, 2013). The proof uses explicit local-global compatibility, parity considerations, and automorphy lifting.

In higher dimensions, similar irreducibility criteria for strictly compatible systems of $\ell$-adic Galois representations apply, provided the Hodge–Tate weights are not all equal (i.e., avoiding the so-called parallel-weight case) (Duan et al., 5 Jun 2024).

5. Decomposition and Parity Results

The structure of possible decompositions of Hodge–Tate $p$-adic representations attached to automorphic forms was systematically analyzed in (Ramakrishnan, 2013). Key findings include:

• No Even 2-dimensional Subquotients: For $n \leq 4$, with $\pi$ quasi-regular and $\rho$ local–globally compatible, the semisimplification $\rho^{ss}$ contains no even, irreducible, 2-dimensional subrepresentation.
• Matching of Decomposition Types: If $\pi$ is regular and $\rho$ is crystalline at all $v | p$ (with $p-1$ suitably large compared to Hodge–Tate weights), the decomposition type of $\rho^{ss}$ matches the isobaric type of $\pi$; cuspidal $\pi$ yield irreducible $\rho$ (Ramakrishnan, 2013).

This framework uses parity at infinity, functorial transfer (involving exterior-square and symmetric-square lifts), and automorphy-lifting theorems to exclude improper decompositions.

6. Geometric and Twisting Behavior

Global irreducibility of $p$-adic Galois representations can, in various contexts, persist after twisting. Any geometrically irreducible $\overline{\mathbb{Q}}_p$-local system on a smooth variety over a $p$-adic field becomes de Rham (and thus Hodge–Tate) after twisting by a suitable character (Petrov, 2020). The essential geometric mechanism is rigidity of the Sen operator’s eigenvalues: geometric irreducibility forces Hodge–Tate weights to lie in a single coset of $\mathbb{Z}$, so after an appropriate twist the representation is Hodge–Tate, and indeed de Rham.

For representations arising from algebraic geometry (e.g., the transcendental part of étale cohomology of algebraic surfaces), algorithmic criteria based on the behavior of local Frobenius polynomials and patterns of Hodge–Tate weights can be used to verify (often for all but finitely many places) irreducibility in strictly compatible systems (Duan et al., 5 Jun 2024).

7. Modularity and Reduction Aspects

Galois representations arising from geometry may be semi-stable but non-crystalline, as in the case of irreducible 2-dimensional semi-stable non-crystalline representations with Hodge–Tate weights $(0,k-1)$. These are classified by a Fontaine–Mazur $\mathcal{L}$-invariant. The reduction modulo $p$ can yield irreducible representations exactly when the local $\mathcal{L}$-invariant is sufficiently large, with full explicitness for $k \geq p$ (Bergdall et al., 2020).

A plausible implication is that the moduli of Hodge–Tate structures interact subtly with $p$-adic variation and reductions, linking the intricacies of the boundary between crystalline, semi-stable, and general de Rham settings to patterns of irreducibility.

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Irreducible Hodge–Tate $p$-adic representations, while locally rigid and entirely decomposed into Tate twists, admit a rich and subtle arithmetic theory in the global context when constructed from geometry or automorphic representations, reflecting deep structures in $p$-adic Hodge theory, the Langlands program, and the arithmetic of algebraic varieties (Ramakrishnan, 2013, Petrov, 2020, Anschütz et al., 2022, Duan et al., 5 Jun 2024, Bergdall et al., 2020).