I will explain how various results in arithmetic statistics by Bhargava, Gross, Shankar, and others on 2-Selmer groups of Jacobians of (hyper)elliptic curves can be organized and reproved using the theory of graded Lie algebras, following the earlier work of Thorne. This gives uniform proof of these results and yields new theorems for certain families of non-hyperelliptic curves. I will also mention some applications to rational points on certain families of curves.

The talk will involve a mixture of representation theory, number theory
and algebraic geometry and I will assume no familiarity with arithmetic
statistics.