Ludvig Olsson: Moduli spaces of G-bundles and Galois representations

Abstract

This thesis consists of two papers.

In paper I we construct the moduli space of \(G\)-bundles over a smooth projective curve \(C\) when \(G\) is disconnected. This construction is used to prove a result about the nondensity of essentially finite \(G\)-bundles. For a connected reductive group \(G\), we show that there is a finite family of subgroups \(H(i)\), with connected component a torus, such that each essentially finite \(G\)-bundle is obtained as the extension by scalars of a bundle on one of the \(H(i)\). This result, obtained from a classic result by Jordan on finite subgroups of \(\mathrm{GL}(n)\), implies that the set of essentially finite \(G\)-bundles in the connected component of the moduli space of \(G\)-bundles is contained in a union of subspaces of positive codimension.

In paper II, we consider continuous, \(\ell\)-adic, finitely ramified Galois \(G\)-representation for \(G\) a split reductive group, that are not necessarily de Rham at \(\ell\). Using a modularity lifting construction we can freely control the Hodge-Tate quasi-cocharacter of our representation, implying it can be almost any quasi-cocharacter.