E 8 and the average size of the 3‐Selmer group of the Jacobian of a pointed genus‐2 curve

Abstract: We prove that the average size of the 3‐Selmer group of a genus‐2 curve with a marked Weierstrass point is 4. We accomplish this by studying rational and integral orbits in the representation associated to a stably Z / 3 Z ‐graded simple Lie algebra of type E 8 . We give new techniques to construct integral orbits, inspired by the proof of the fundamental lemma and by the twisted vertex operator realisation of affine Kac–Moody algebras