12F12 Inverse Galois theory
Galois representations of orthogonal rigid local systems
- We use the middle convolution introduced by Katz to construct a families of lisse sheaves on the affine line without two points. These correspond to continuous representations of the etale fundamental group, which can be specialized to compatible systems of Galois representations. This leads to the second maximally unipotent family.
Because of the geometric origin, we can show using a theorem of Barnet-Lamb, Gee, Geraghty and Taylor that they are potentially automorphic.