Non-Critical Crystalline Representations

• Non-critical crystalline representations are p-adic Galois representations defined via filtered φ-modules with non-coinciding Hodge and Frobenius structures, ensuring reliable weight space partitioning.
• Their explicit classification in GL2 and GSp4 reveals detailed reduction patterns and deformation frameworks that mirror local analytic and automorphic structures.
• These representations underpin advancements in p-adic automorphic forms and the Langlands program, offering concrete computational and theoretical insights.

Non-critical crystalline representations are a class of $p$-adic Galois representations with crystalline period relations, distinguished by Hodge-theoretic "non-criticality" conditions that prevent certain degeneracies in the interaction between Hodge and Frobenius structures. These representations possess rich internal structure, underpin the study of $p$-adic automorphic forms, and serve as test cases for deep conjectures in the Langlands program and $p$-adic Hodge theory. The explicit classification of their reductions, both in two-dimensional settings and in higher rank (e.g., GSp$_4$), reveals subtle partitionings of weight space and deformation classes that have driven much recent research (Arsovski, 2018, Han, 4 Dec 2025).

1. Structural Definition and Non-Criticality

Consider a finite extension $E/\mathbb{Q}_p$ and an $E$–linear representation $\rho$ of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$. A crystalline representation is equipped with a filtered $\varphi$-module $(D, \varphi, \mathrm{Fil}^{\bullet}D)$ over $E$ via Fontaine's functor. For $\mathrm{GL}_2$ and $\mathrm{GSp}_4$ targets, $D$ carries the required linear or symplectic structure.

Non-criticality is a genericity condition: for $D$ with ordered Frobenius eigenbasis $(e_i)$ and filtration determined by Hodge-Tate weights, the filtration flag avoids coincidence with the $\varphi$-line flags. Explicitly, in the GSp$_4$ context, for Hodge-Tate weights $h_1>h_2>h_3>h_4$ satisfying $h_1+h_4=h_2+h_3$ and with Frobenius eigenvalues $(\alpha_1,\dots,\alpha_4)$ subject to the similitude $\alpha_1\alpha_4=\alpha_2\alpha_3$, non-criticality is equivalent to the property that no partial sum of Hodge-Tate weights coincides with a partial sum of Frobenius slopes (Han, 4 Dec 2025).

For two-dimensional crystalline representations parameterized by $(k,a)$ with $v_p(a)>0$, integer weights, and Hodge-Tate weights $(0,k-1)$, non-criticality corresponds to avoidances relating $k$ and $v_p(a)$ (dividing the weight space into explicit "non-subtle" components) (Arsovski, 2018).

2. Partitioning the Weight Space and Non-Subtle Components

In the $\mathrm{GL}_2$ case, fix an odd prime $p$ and write the weight space as $$W = \operatorname{Hom}_{\mathrm{ct}}(\mathbb{Z}_p^\times, \mathbb{C}_p^\times)$$; this decomposes into $p-1$ discs $D_s$ indexed by $s\in\{0,\dots,p-2\}$, where a classical weight $k$ corresponds to $s\equiv k-2 \bmod{p-1}$.

Given $a\in\mathbb{Z}_p$ with $v_p(a)>0$, set $v=\lfloor v_p(a)\rfloor + 1$. Each $D_s$ consists entirely of "subtle" or "non-subtle" points:

• "Subtle" if $s\in\{1,2,\dots,2v-1\}$
• "Non-subtle" otherwise, that is, $k\not\equiv3,4,\dots,2v,2v+1\,\,\bmod{p-1}$

This partitioning isolates regions where uniform modular reduction theory applies, allowing explicit classification theorems that are unencumbered by critical intersection phenomena (Arsovski, 2018).

3. Explicit Classification of Reductions

Two-Dimensional Case

Let $V_{k,a}$ be the family of two-dimensional crystalline representations with Hodge-Tate weights $(0, k-1)$ and Frobenius eigenvalues $\{\alpha, \beta\}$, $v_p(\alpha)=v_p(a)$. The reductions $\overline{V}_{k,a}$ modulo $p$ are controlled on non-subtle discs through explicit annular decompositions $R_{s,i,j}$ in each $D_s$.

Non-integer slopes ($v_p(a)\notin\mathbb{Z}$):

• $\bar V_{k,a}\cong \Irr(b_{v-1}-1)$ on the outer region $R_{s,\infty}$
• $\bar V_{k,a}\cong \Irr(b_{\max(i, v-j-1)}-1)$ on annuli $R_{s,i,j}$

where $b_i=s+i(p-1)+2$ (Arsovski, 2018).

Integer slopes ($v_p(a)=v-1\in\mathbb{Z}$):

• On $R_{s,\infty}$: $\bar V_{k,a}\cong \Reds_v(0, A_{k,v})$ with $A_{k,v}=(s-k+2)^{v-1}(s-v+2)a^{p-1}$
• On $R_{s,i,j}$ with $i+j<v-1$: $\bar V_{k,a}\cong \Reds_v(j, A_{k,v,i,j})$ with $$A_{k,v,i,j}=(-1)^{v+i+j+1}(v-j-1)(v-j-2)(s-v+j+2)^{v-j-1}a^p$$
• On $i+j>v-1$: $\bar V_{k,a} \cong \Irr(b_i-1)$

$\Reds_v(j,\alpha)$ denotes the unique (up to scalar) non-split extension in $\mathrm{Ext}^1_G(\omega^{s+2j-2v+2},1)$ with extension class parameter $\alpha$ (Arsovski, 2018).

Higher-Dimensional Generalization (GSp$_4$)

For GSp$_4$, non-critical crystalline representations with regular Hodge-Tate weights $(h_1,h_2,h_3,h_4)$ and generic Frobenius parameters $(\alpha_1, \alpha_2, \alpha_3, \alpha_4)$ (with certain genericity and non-critical intersection conditions as above) admit explicit classification by triple $(\vec\alpha, \vec h; a, b)$, where $(a, b)\in E^2$ parametrizes the Hodge filtration in a canonical form (Han, 4 Dec 2025).

These parameters are functorially encoded in a locally analytic, length-5 ("three layered") representation $\pi_\mathrm{min}(\rho)$ of GSp$_4(\mathbb{Q}_p)$, which both determines and is determined by $\rho$. There exists a universal deformation theory in which $\pi_\mathrm{min}(\rho)$ recovers all trianguline deformations (Han, 4 Dec 2025).

4. Deformation Theory and Local-Global Compatibility

The deformation-theoretic structure of non-critical crystalline representations is highly regular. In GSp$_4$, the genericity assumption ensures all trianguline and parabolic deformation functors are formally smooth of expected dimension; their intersection pattern mirrors the local Weyl group chamber structure. The explicit construction of $\pi_\mathrm{min}(\rho)$ provides a categorical equivalence between the Galois side (filtered $(\varphi,\Gamma)$-modules with GSp$_4$-structure and Hodge-parameter invariants) and analytic representations on the automorphic side (Han, 4 Dec 2025).

The local-global compatibility is realized by embedding $\pi_\mathrm{min}(\rho)^{\oplus r}$ into the completed cohomology of a suitable definite unitary group, with precise control of multiplicities and eigenvariety local geometry at non-critical points (Han, 4 Dec 2025).

5. Mechanisms Underlying Irreducibility and Reducibility

Key to the understanding of non-critical crystalline representations is the role of the $p$-adic slope $v_p(a)$:

• In non-integer slope cases, combinatorial analysis of Hecke operators on mod-$p$ principal series modules reveals that no nontrivial subquotients remain, enforcing irreducibility (Arsovski, 2018).
• In integer-slope cases, there is a unique subrepresentation on which the operator acts non-invertibly, leading to nontrivial extensions parameterized by explicit scalars determined by the highest-weight structure in the socle of the induced module. This mechanism precisely locates and quantifies reducibility within the "inner annuli" of the non-subtle weight disks (Arsovski, 2018).

Analogous mechanisms extend to higher rank, where the non-criticality and genericity constraints on the parameters ensure irreducibility and well-behaved local deformation theory.

6. Applications, Examples, and Computational Aspects

In $v=1$ cases ($0 < v_p(a) < 1$), every non-critical (i.e., non-subtle) weight yields an irreducible reduction, recovering the explicit Buzzard–Gee results. For $v=2$ ($1 \leq v_p(a) < 2$), all non-subtle disks yield irreducible reductions, generalizing to Bhattacharya–Ghate's theorems, while for integer slopes, explicit classification of reducible extensions in the innermost annulus matches classifications by Bhattacharya–Ghate–Rozensztajn. Rozensztajn's algorithm, implemented in Sage, allows computational verification of these reductions for any specific $(p, k, a)$ (Arsovski, 2018).

In the GSp$_4$ setting, explicit test cases with chosen Hodge-Tate weights and Frobenius slopes are worked out in detail, confirming that only two among the eight middle constituents in the Jordan–Hölder series reflect the anisotropic Hodge filtration, matching the theoretical predictions for the minimal representation $\pi_\mathrm{min}(\rho)$ (Han, 4 Dec 2025).

7. Extensions, Conjectures, and Broader Structural Insights

The framework for non-critical crystalline representations in GSp$_4$ extends, via local deformation theory, to general reductive groups $G$ equipped with a $G$-structure on $(\varphi,\Gamma)$-modules. This approach is conjectured to realize a broader, group-theoretic correspondence between crystalline deformation data and analytic representation theory, in agreement with local and global versions of the Langlands conjectures. Under mild locally and globally vanishing Selmer group assumptions, eigenvarieties are proven smooth at non-critical classical points, and the universal family of representations provides a natural geometric model for the family of local Galois representations (Han, 4 Dec 2025).