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The semi-stable Local Langlands Correspondence

Eknath Ghate

TL;DR

The work studies the mod $p$ reductions of 2-dimensional semistable Galois representations $V_{k,\mathcal{L}}$ with Hodge–Tate weights $(0,k-1)$ for weights $k$ in $[3,p+1]$ and primes $p\ge5$, by developing a refined lattice framework in the $p$-adic Local Langlands setting. It builds an explicit $G=\mathrm{GL}_2(\mathbb{Q}_p)$-Banach space $B_{k,\mathcal{L}}$ and its standard lattice $\tilde{\Theta}(k,\mathcal{L})$, uniformizes them, and reduces to the Iwahori mod $p$ LLC to read off the mod $p$ Galois reductions. A key parameter $\nu=v_p(\mathcal{L}-H_- -H_+)$ governs the alternating pattern of irreducible and reducible constituents in $\overline{V}_{k,\mathcal{L}}$, yielding a uniform description that recovers Breuil–Mézard and Guerberoff–Park results while extending the zig-zag phenomenon to the semi-stable setting. The approach leverages logarithmic and polylogarithmic functions, wavelet decompositions, and explicit actions of Iwahori–Hecke operators on compact inductions, ultimately translating lattice reductions into mod $p$ representations via the Iwahori LLC. The results provide a coherent, computable framework for reductions across weights $3\le k\le p+1$ and $p\ge5$, with potential extension to wider ranges via the same method.

Abstract

We start with background that goes into an Iwahori-theoretic reformulation of the mod $p$ Local Langlands Correspondence (§2). We then explain some classical $p$-adic functional analytic results (§3) that go into defining the $p$-adic Banach space (§4) attached to a two-dimensional semi-stable representation $V_{k,{\mathcal L}}$ of the Galois group of ${\mathbb Q}_p$ of weight $k$ and ${\mathcal L}$-invariant ${\mathcal L}$ under the $p$-adic Local Langlands correspondence. We then sketch how to compute the reduction of a lattice in this Banach space, which along with the Iwahori mod $p$ LLC, allows one to completely determine the mod $p$ reduction of $V_{k,{\mathcal L}}$ for all weights $3 \leq k \leq p+1$ and all ${\mathcal L}$ for $p \geq 5$ (§5). These notes are a summary of our joint work with Anand Chitrao [CG24]. Emphasis is placed on motivation and background rather than completeness.

The semi-stable Local Langlands Correspondence

TL;DR

The work studies the mod reductions of 2-dimensional semistable Galois representations with Hodge–Tate weights for weights in and primes , by developing a refined lattice framework in the -adic Local Langlands setting. It builds an explicit -Banach space and its standard lattice , uniformizes them, and reduces to the Iwahori mod LLC to read off the mod Galois reductions. A key parameter governs the alternating pattern of irreducible and reducible constituents in , yielding a uniform description that recovers Breuil–Mézard and Guerberoff–Park results while extending the zig-zag phenomenon to the semi-stable setting. The approach leverages logarithmic and polylogarithmic functions, wavelet decompositions, and explicit actions of Iwahori–Hecke operators on compact inductions, ultimately translating lattice reductions into mod representations via the Iwahori LLC. The results provide a coherent, computable framework for reductions across weights and , with potential extension to wider ranges via the same method.

Abstract

We start with background that goes into an Iwahori-theoretic reformulation of the mod Local Langlands Correspondence (§2). We then explain some classical -adic functional analytic results (§3) that go into defining the -adic Banach space (§4) attached to a two-dimensional semi-stable representation of the Galois group of of weight and -invariant under the -adic Local Langlands correspondence. We then sketch how to compute the reduction of a lattice in this Banach space, which along with the Iwahori mod LLC, allows one to completely determine the mod reduction of for all weights and all for (§5). These notes are a summary of our joint work with Anand Chitrao [CG24]. Emphasis is placed on motivation and background rather than completeness.

Paper Structure

This paper contains 42 sections, 42 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Idea of proof of Theorem \ref{['Main theorem in the second part of my thesis']}
  3. Notation
  4. Iwahori mod $p$ Local Langlands Correspondence
  5. Bruhat-Tits tree
  6. Hecke algebras
  7. The ABC theorem
  8. mod $p$ Galois representations
  9. Iwahori mod $p$ LLC
  10. Functions of a $p$-adic variable
  11. $p$-adic Banach spaces
  12. $E$-Banach spaces
  13. ONB of a $p$-adic Banach space
  14. Dual of an $E$-Banach
  15. Continuous functions on $\mathbb{Z}_p$
  16. ...and 27 more sections

Key Result

Theorem 1.1

For $k \in [3, p + 1]$ and for primes $p \geq 5$, the semi-simplification of the reduction mod $p$ of the semi-stable representation $V_{k, \mathcal{L}}$ on $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ is given by an alternating sequence of irreducible and reducible representations: where and the mod $p$ constants $\lambda_i$ are determined by the formulas:

Theorems & Definitions (83)

  • Theorem 1.1: Chitrao-Ghate CG23
  • Lemma 2.1
  • proof
  • Theorem 2.3: Anandavardhanan, Borisagar, Chitrao
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5: Iwahori mod $p$ LLC
  • proof
  • Proposition 3.1
  • ...and 73 more