Most people nowadays believe that over a fixed global field, $50$% of the elliptic curves have $0$ rank, $50$% have rank $1$, and $0$% have higher rank. A significant advance in this direction has been Bhargava and Shankar's proof that for elliptic curves over $\mathbb{Q}$, the rank is bounded on average (indeed, not just the Mordell-Weil rank, but even the $2$-Selmer rank).

Has anybody put forth a similar guess for the ranks of higher dimensional abelian varieties? If so, what is the rationale behind such a guess? I am asking about the precise statistical distribution of ranks.

Chabauty's method proves that most odd degree hyperelliptic curves have only one rational point(arxiv.org/abs/1302.0061). – Chandan Singh Dalawat Feb 6 '13 at 3:08