I will talk about recent work with Jack Thorne in which we find the average size of the 3-Selmer group for a family of genus-2 curves by analyzing a graded Lie algebra of type $E_8$. I will focus on the role representation theory plays in our proofs.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/9073