Reduction mod $p$ of semi-stable representations of some super-Breuil weights

Abstract: We determine the mod $p$ reductions of the semi-stable representations $V_{k, \mathcal{L}}$ of weight $k \in [p + 5, 2p]\cup[2p + 6, 3p + 1]$ and $v_p(\mathcal{L}) < 1-k/2$ for primes $p \geq 5$. In particular, this shows that the techniques introduced in [CG24] involving the $p$-adic and mod $p$ local Langlands correspondences can be used to compute the reduction of $V_{k, \mathcal{L}}$ outside the range $k \in [3, p + 1]$. Moreover, this shows that the bound on $v_p(\mathcal{L})$ given by Bergdall-Levin-Liu [BLL23] can be improved, at least for weights $k \in [2p + 6, 3p + 1]$.

• The paper extends mod p reduction results by demonstrating that for specific super-Breuil weights and vₚ(L) < -r/2, the mod p reduction of semi-stable representations is irreducible.
• It employs a novel combinatorial filtration method using congruence relations and Stirling numbers to systematically eliminate non-contributing subquotients.
• The improved bounds on L-invariants provide critical theoretical insights and a practical framework for advancing p-adic and mod p local Langlands correspondence.

Mod $p$ Reduction of Semi-stable Representations for Super-Breuil Weights

Overview and Objectives

This paper investigates the reduction modulo $p$ of two-dimensional irreducible semi-stable representations $V_{k, L}$ of $$\operatorname{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$$, focusing on weights $k$ in the ranges $[p+5, 2p]\cup[2p+6, 3p+1]$ and $p$-adic $L$-invariants satisfying $v_p(L) < 1 - k/2$. The main contribution includes both the extension of reduction results beyond the classical Fontaine-Laffaille range and the improvement of existing bounds on $v_p(L)$, particularly those established by Bergdall-Levin-Liu.

The approach is rooted in the $p$0-adic and mod $p$1 local Langlands correspondence, exploiting and generalizing the filtration method and congruence techniques from prior work by the authors. The results confirm that, for specific parameter ranges and improved bounds on $p$2, the mod $p$3 reduction of $p$4 is irreducible of a prescribed form, answering optimality questions left open in previous literature.

Technical Framework

Let $p$5 and $p$6 a finite extension of $p$7 containing certain necessary elements. For $p$8, $p$9, and $V_{k, L}$0, the representation $V_{k, L}$1 is the semi-stable, non-crystalline, irreducible two-dimensional representation of Hodge-Tate weights $V_{k, L}$2 and $V_{k, L}$3-invariant $V_{k, L}$4. The previous reductions for $V_{k, L}$5 were known in the Fontaine-Laffaille range ($V_{k, L}$6) or under strong bounds on $V_{k, L}$7 tied to factorials, primarily via the machinery of Breuil-Kisin modules.

This work employs the $V_{k, L}$8-adic and mod $V_{k, L}$9 local Langlands correspondence. A key instrument is the analysis of the canonical $$\operatorname{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$$0-equivariant filtration on symmetric powers and its translation to the $$\operatorname{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$$1-equivariant filtration on the associated spaces via the surjection $$\operatorname{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$$2. The nontrivial work lies in identifying which filtration sub-quotient on $$\operatorname{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$$3 survives in mod $$\operatorname{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$$4 reduction, leveraging detailed congruence relations and recurrence properties involving Stirling numbers and $$\operatorname{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$$5-adic combinatorics.

Main Results and Methods

Structure of the Proof

The main technical result is that for $$\operatorname{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$$6 in $$\operatorname{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$$7 and $$\operatorname{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$$8, the reduction $$\operatorname{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$$9 is irreducible, coinciding with the socle induced representation $k$0. This is demonstrated by:

• Construction of Filtration: Identifying sub-quotients $k$1, and showing that all but one (the socle) die in reduction.
• Elimination of Sub-quotients: Development of congruence relations to systematically eliminate each non-contributing sub-quotient, including both "shallow" ($k$2 small) and "deep" ($k$3 large) contributors, using power series expansions, Taylor arguments, and identities for binomials modulo $k$4 (e.g., via Lucas' theorem and its extensions).
• Improvement of Bounds: In the higher weight range $k$5, the bound $k$6 from previous literature is improved to $k$7, showing that previously implicit “computational” optimizations are in fact systematic and theoretical.

Three distinct technical “methods” (“good”, “bad”, “ugly”) are introduced for the stratified elimination of sub-quotients, each corresponding to different divisibility conditions on $k$8 and designed to handle edge-cases arising from combinatorial structure.

Numerical and Structural Outcomes

• Main Theorem: For $k$9, $[p+5, 2p]\cup[2p+6, 3p+1]$0 or $[p+5, 2p]\cup[2p+6, 3p+1]$1, and $[p+5, 2p]\cup[2p+6, 3p+1]$2, then $[p+5, 2p]\cup[2p+6, 3p+1]$3 is isomorphic, as a $[p+5, 2p]\cup[2p+6, 3p+1]$4-module, to $[p+5, 2p]\cup[2p+6, 3p+1]$5.
• The results strictly generalize the reductions computed by Bergdall-Levin-Liu, sharpening the bound on $[p+5, 2p]\cup[2p+6, 3p+1]$6 for higher weights.
• The analysis confirms the irreducibility of the mod $[p+5, 2p]\cup[2p+6, 3p+1]$7 reduction in this weight and $[p+5, 2p]\cup[2p+6, 3p+1]$8-invariant regime, in contrast to certain reducible cases at low weights.

Implications and Future Perspectives

The technical innovation, beyond the improvement of numeric bounds, lies in the robustness of the combinatorial filtration and congruence method, which circumvents the limitations of explicit Breuil-Kisin module computations and provides a template for further generalizations. Practical consequences for the calculation of local Galois representations in modular and automorphic contexts are immediate: the improved ranges and bounds enable more uniform and systematic control over mod $[p+5, 2p]\cup[2p+6, 3p+1]$9 correspondences in the non-Fontaine-Laffaille regime.

Theoretically, these results address and settle the optimality of prior bounds for $p$0 and bridge computational observations (notably those of Pollack and Bergdall-Levin-Liu) with general theory. The methods also offer a potential framework for tackling reductions in even higher weight ranges, $p$1, or situations where the representation is not strictly semi-stable.

Further directions include extension to families of modular forms, explicit description of reductions for other classes of Breuil weights (e.g., deeper in the socle filtration), and the potential impact on the p-adic local Langlands program for higher-dimensional groups or representations with more complex Hodge-Tate weights.

Conclusion

The paper delivers a precise and systematic computation of mod $p$2 reductions of semi-stable representations for previously inaccessible weights and establishes improved, theoretically optimal bounds on the $p$3-adic size of $p$4-invariants required for irreducibility. The methodology synthesizes $p$5-adic representation theory, combinatorics, and congruence calculus, confirming the effectiveness and adaptability of the local Langlands filtration approach. The results both answer open questions regarding the sharpness of previous bounds and establish a platform for further advances in the arithmetic of local representations.