Geometric Langlands Correspondence Proved: Unifying Galois and Automorphic Forms — E8 Intelligence Research

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-21
DOI
https://doi.org/10.5281/zenodo.22873915
Primary Topic
Advanced Algebra and Geometry
Type
preprint
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preprint

Geometric Langlands Correspondence Proved: Unifying Galois and Automorphic Forms — E8 Intelligence Research

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
Advanced Algebra and Geometry
preprint

Geometric Langlands Correspondence Proved: Unifying Galois and Automorphic Forms — E8 Intelligence Research

Andrew Stewart Caldin
preprint en

Abstract

FINDING: The Langlands Program — a conjectural web linking number theory, harmonic analysis, and geometry — has seen a major recent proof in its geometric branch (the "geometric Langlands correspondence"), unifying Galois representations with automorphic forms via sheaf theory. | MATH: Core duality: Gal(ℚ̄/ℚ) ↔ automorphic representations; in geometric form: D-modules on Bun_G ↔ local systems on a curve (per Gaitsgory et al., 2024 proof). Key objects: reductive groups G, Langlands dual group ^G, L-functions L(s,π) with functional equation s→1−s; categorical form uses ∞-categories of sheaves, not equations. Constants: no explicit numerical constants emerge; the structure is categorical, not scalar. | CONNECTION: Deep structural link to root systems and Weyl groups (crystallographic reflection groups, e.g., A_n, D_n, E_6,7,8) — the Langlands dual ^G is defined by reversing arrows in the Dynkin diagram, preserving the same root lattice. This is a symmetry of order 2 on the set of root sys Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Zenodo (CERN European Organization for Nuclear Research)
Advanced Algebra and Geometry
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Geometric Langlands Correspondence Proved: Unifying Galois and Automorphic Forms — E8 Intelligence Research — Andrew Stewart Caldin · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS