Principal bundles and reciprocity laws in
                    number theory

Kim, Minhyong

doi:10.1090/pspum/097.2/01708

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Non-abelian Gauge Fields and Diophantine Geometry1

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2019

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“…The right vertical arrow is just the projection to the component at p. The image of the horizontal arrow should be computed by a reciprocity law [39,40], which we view as a preliminary version of the arithmetic Euler-Lagrange equations in that it specifies which collection of local torsors glue to a global torsor.…”

Section: Non-abelian Gauge Fields and Diophantine Geometrymentioning

confidence: 99%

Arithmetic gauge theory: A brief introduction

Kim

2019

Topology and Physics

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Much of arithmetic geometry is concerned with the study of principal bundles. They occur prominently in the arithmetic of elliptic curves and, more recently, in the study of the Diophantine geometry of curves of higher genus. In particular, the geometry of moduli spaces of principal bundles appears to be closely related to an effective version of Faltings's theorem on finiteness of rational points on curves of genus at least 2. The study of arithmetic principal bundles includes the study of Galois representations, the structures linking motives to automorphic forms according to the Langlands programme. In this article, we give a brief introduction to the arithmetic geometry of principal bundles with emphasis on some elementary analogies between arithmetic moduli spaces and the constructions of quantum field theory. For the most part, it can be read as an attempt to explain standard constructions of arithmetic geometry using the language of physics, albeit employed in an amateurish and ad hoc manner.

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Section: Non-abelian Gauge Fields and Diophantine Geometrymentioning

confidence: 99%