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On Ribet's Lemma for $\mathrm{GL}_2$ modulo prime powers

Amit Ophir, Ariel Weiss

Abstract

Let $ρ\colon G\to \mathrm{GL}_2(K)$ be a continuous representation of a compact group $G$ over a complete discretely valued field $K$, with ring of integers $\mathcal O$ and uniformiser $π$. We prove that $\operatorname{tr}ρ$ is reducible modulo $π^n$ if and only if $ρ$ is reducible modulo $π^n$. More precisely, there exist characters $χ_1,χ_2 \colon G\to(\mathcal O/π^n\mathcal O)^{\times}$ such that $\det(t - ρ(g))\equiv (t-χ_1(g))(t-χ_2(g))\pmod{π^n}$ for all $g\in G$, if and only if there exists a $G$-stable lattice $Λ\subset K^2$ such that $Λ/π^nΛ$ contains a $G$-invariant, free, rank one $\mathcal O/π^n\mathcal O$-submodule. Our result applies in the case that $ρ$ is not residually multiplicity free, in which case it answers a question of Bellaïche--Chenevier. As an application, we prove an optimal version of Ribet's Lemma, which gives a condition for the existence of a $G$-stable lattice $Λ$ that realises a non-split extension of $χ_2$ by $χ_1$

On Ribet's Lemma for $\mathrm{GL}_2$ modulo prime powers

Abstract

Let be a continuous representation of a compact group over a complete discretely valued field , with ring of integers and uniformiser . We prove that is reducible modulo if and only if is reducible modulo . More precisely, there exist characters such that for all , if and only if there exists a -stable lattice such that contains a -invariant, free, rank one -submodule. Our result applies in the case that is not residually multiplicity free, in which case it answers a question of Bellaïche--Chenevier. As an application, we prove an optimal version of Ribet's Lemma, which gives a condition for the existence of a -stable lattice that realises a non-split extension of by

Paper Structure

This paper contains 17 sections, 19 theorems, 77 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Bruhat--Tits tree of PGL(2, K)
  3. Ribet's Lemma
  4. Isogenies of elliptic curves
  5. Preliminaries
  6. The G-invariant subtree
  7. The shape of the invariant building
  8. Conjugate characters modulo modulo powers of the uniformiser
  9. The invariants n and m
  10. Boundary points
  11. Proof of Theorem $\ref{['thm:main-intro']}$
  12. A generalisation of Ribet's Lemma
  13. Proof of Theorem $\ref{['thm:ribet-intro']}$
  14. Linearly extendable characters and a counterexample
  15. The shape of $\mathcal{X}(\rho)$
  16. ...and 2 more sections

Key Result

Theorem 1.2

The following are equivalent:

Figures (3)

  • Figure 1: A ball of radius $3$ in the Bruhat--Tits tree for $\mathbf{Q}_3$
  • Figure 2: A generalised ball of radius $3$ in the Bruhat--Tits tree for $\mathbf{Q}_3$
  • Figure 3: A band in the Bruhat--Tits tree for $\mathbf{Q}_3$, with diameter $7$ and radius $2$. The solid points form the nerve $S$.

Theorems & Definitions (56)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Lemma 2.3: Bellaiche-arbres*Prop. 11
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • ...and 46 more