Weakly reductive groups and Galois representations

Ever since Wiles’ work on Fermat’s Last Theorem, a central problem in algebraic number theory has been to study lifting mod-p representations of global Galois groups to p-adic representations of such groups.  A key step is the corresponding lifting problem for local Galois groups.  Most prior work in this direction has focused on representations valued in specific “classical matrix groups” such as general linear or orthogonal or symplectic groups, with each type of matrix group treated in its own way.

In joint work with Jeremy Booher and Shiang Tang, we introduce new unified methods to build such local lifts (on Galois groups of $\ell$-adic fields with $\ell \ne p$) valued in arbitrary reductive groups without any “bigness” assumption on the image of the mod-p representation. This requires extending some of the basic theory of reductive groups to a wider class of group schemes that we call weakly reductive, and as an application we build global lifts valued in the exceptional group G_2