Let `$F$`

be a global field and `$\Gamma_F$`

its absolute Galois group. Given
a continuous representation `$\bar{\rho}: \Gamma_F \to G(k)$`

, where `$G$`

is a split
reductive group and `$k$`

is a finite field, it is of interest to know when `$\bar{\rho}$`

lifts
to a representation `$\rho: \Gamma_F \to G(O)$`

, where `$O$`

is a complete discrete
valuation ring of characteristic zero with residue field `$k$`

. One would also like to control
the local behaviour of `$\rho$`

at places of `$F$`

, especially at primes dividing `$p = \mathrm{char}(k)$`

(if `$F$`

is a number field). In this talk I will give an overview of a method developed in joint work with
Chandrashekhar Khare and Stefan Patrikis which allows one to construct such lifts in many cases.

- All seminars.
- Seminars for 2022

Last updated: 22 Sep 2023