ویراساینس.
سند علمی

Introduction: Pre-homogeneous Spaces

MANJUL BHARGAVA

چکیده

This will be a course on arithmetic invariant theory. The goal of the course is to understand orbits of representations of algebraic groups over non-algebraically closed fields and eventually over Z. We will focus on representations that arise “naturally” from number theory, algebraic geometry, Vinberg theory, knot theory etc. 1 Let G be an algebraic group acting “integrally” on a vector space V viz. G(Z) acts on the lattice of integer points V (Z). Our goal is to classify orbits of: (1) G(C) on V (C) (2) G(Q) on V (Q) (3) G(Z) on V (Z) The first topic is Geometric Invariant Theory and the subsequent topics are Arithmetic Invariant Theory. We will start with representations over C and descend to these. Our approach in doing so is by thinking of orbits as spectra of the ring of invariants of G acting on V . Key Question: For which representations do (1), (2) and especially (3) have nice/useful answers?

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