On Bhargava’s representations and Vinberg’s invariant theory

Manjul Bhargava has recently made a great advance in the arithmetic theory of elliptic curves. Together with his student, Arul Shankar, he determines the average order of the Selmer group Sel(E,m) for an elliptic curve E over Q, when m = 2, 3, 4, 5. We recall that the Selmer group is a finite subgroup of H(Q, E[m]), which is defined by local conditions. Their result (cf. [1, 2]) is that the average order of Sel(E,m) is σ(m) = (the sum of the divisors d of m) in these four cases (where σ(m) = 3, 4, 7, 6 respectively). Since the Selmer group contains the subgroup E(Q)/mE(Q), they are able to conclude that the average rank of elliptic curves over Q is bounded above by a constant which is less than 1. We expect that the average rank is equal to 1/2, although this is the first result which proves that the average rank is bounded!