It's conjectured that, asymptotically, half of elliptic curves have rank 0, half have rank 1, and elliptic curves of rank $\geq 2$ have density 0. But what if we disregard elliptic curves of rank 0 or 1: Are there any conjectures about the average rank of elliptic curves of rank $\geq 2$?


More generally, for any integer $n \geq 2$, what value should one expect for the average rank of elliptic curves of rank $\geq n$?



Average rank of elliptic curves, excluding those of low rank
A very simple random matrix heuristic says, based on the function field model, that the rank of a random elliptic curve is the dimension of the invariant subspace of a random element of $O(n)$ for large $n$.


We can easily compute the dimension of the space of matrices that preserve exactly $k$-dimensional subspace. We can write each such matrix as a nondegenerate invariant subspace of dimension $k$ plus an orthogonal matrix with no fixed vectors of dimension $n-k$. The dimension of the space of $k$-dimensional subspaces is $k(n-k)$, and nondegenerate ones are generic. The dimension of the space of matrices in $O(n-k)$ that with no fixed points is $(n-k)(n-k-1)/2$, because that's a dense set of one of the connected components. So the total number is $k(n-k)+(n-k)(n-k-1)/2 = (n-k)(n+k-1)/2$.


This dimension takes the same value at $k=0$ and $k=1$ but takes rapidly decreasing values after that. So this heuristic suggests that most elliptic curves of rank at least $n$ have rank exactly $n$.


