Colloquium abstracts

Najmuddin Fakhruddin
TIFR
October 7, 2021

Lifting Galois representations:  Representations of the absolute Galois group of $mathbb{Q}$ (or more generally a global field) over $p$-adic fields have played an important role in number theory ever since Deligne associated such representations to classical modular eigenforms (extending works of Eichler and Shimura) around fifty years ago. The work of Wiles and Taylor in the mid-nineties showed that the reductions of such representations over finite fields could be used as a bridge to connect seemingly unrelated objects, leading to the resolution of many outstanding conjectures, e.g., the Shimura--Taniyama conjecture and the Sato--Tate conjecture. In this talk, I will discuss the question of when a Galois representation over a finite field can be lifted to a representation over a $p$-adic field, in the context of representations with target an arbitrary split reductive group $G$, but our results are new even in the case $G = mathrm{GL}_2$. (Based on joint work with Chandrashekhar Khare and Stefan Patrikis.)