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.