An explicit geometric Langlands correspondence for the projective line minus four points

2019/06/07 by Niels uit de Bos, de Bos, Niels uit · 3 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1906.03240

openalex publication_date 2019/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This article deals with the tamely ramified geometric Langlands correspondence for GL₂ on P_Fq¹, where q is a prime power, with tame ramification at four distinct points D = \∞, 0,1, t\ ⊂ P¹(F_q). We describe in an explicit way (1) the action of the Hecke operators on a basis of the cusp forms, which consists of q elements; and (2) the correspondence that assigns to a pure irreducible rank 2 local system E on P¹ ∖ D with unipotent monodromy its Hecke eigensheaf on the moduli space of rank 2 parabolic vector bundles. We define a canonical embedding P¹ into this module space and show with a new proof that the restriction of the eigensheaf to the degree 1 part of this moduli space is the intermediate extension of E.

Cited by

• Graphs of Hecke operators in mixed ramification
• Eigenforms and graphs of Hecke operators with wild ramification
• Analytic Langlands correspondence for PGL(2) on P¹ with parabolic\n structures over local fields

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• Analytic Langlands correspondence for PGL(2) on P¹ with parabolic\n structures over local fields
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