Local systems in algebraic-arithmetic geometry

Time: Thursday, December 21, 2023 (14:00, GMT 07)

Venue: 301 Lecture hall, A5 Building

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https://us06web.zoom.us/j/82927090825?pwd=sCz1LoTwwU9lBgM74B7Q1G1jytdD3m.1

Meeting ID: 829 2709 0825
Passcode: 123456

Abstract: Riemann proved that a topological cover of a Riemann surface is endowed with an analytic structure of a Riemann surface so that the covering is analytic. This is Riemann’s existence theorem. A Riemann surface underlies an algebraic structure which realizes it as the complex points of an algebraic curve. The topological cover is not only analytic, but algebraic as well. This goes back to Grothendieck in the proper case, to Deligne in general, who introduces the notion a regular singularities at infinity. Poincaré defined the topological fundamental group of topological manifold, in particular of Riemann surfaces. Grothendieck defined the étale fundamental group, which bridges Riemann’s existence theorem with Galois theory of fields. Those groups, defined with the help of a base point, are difficult to understand, our knowledge is limited. To study them we consider their linear representations, modulo isomorphisms as we want an invariant of the variety only, not of the chosen base point. A linear representation modulo isomorphism is a local system.

Our two hours lecture shall make a small journey through some properties of local systems which intertwine topological, analytic, arithmetic properties, some of them being, in the current state of understanding, dreams.


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