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Geometric realization of the local Langlands correspondence for representations of conductor three

Naoki Imai, Takahiro Tsushima

Abstract

We prove a realization of the local Langlands correspondence for two-dimensional representations of a Weil group of conductor three in the cohomology of Lubin-Tate curves by a purely local geometric method.

Geometric realization of the local Langlands correspondence for representations of conductor three

Abstract

We prove a realization of the local Langlands correspondence for two-dimensional representations of a Weil group of conductor three in the cohomology of Lubin-Tate curves by a purely local geometric method.

Paper Structure

This paper contains 8 sections, 13 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Lubin--Tate curve
  3. Semi-stable reduction of $\mathbf{X}_1(\mathfrak{p}^3)$
  4. Cohomology of elliptic curve
  5. Realization of correspondence
  6. Local Langlands correspondence
  7. Explicit Artin reciprocity law
  8. Explicit local Langlands correspondence

Key Result

Proposition 2.1

For a supercuspidal representation $\pi$ of $\textit{GL}_2 (K)$ over $\overline{\mathbb{Q}}_{\ell}$, we have as representations of $D^{\times}$.

Theorems & Definitions (28)

  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 18 more