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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.

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I don't know about a general conjecture, but for instance for hyperelliptic curves with a rational Weierstrass point, Bhargava and Gross have a bound for the average rank of the Jacobian. See arxiv.org/pdf/1208.1007.pdf – Abhinav Kumar Feb 6 '13 at 1:13
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By the way, the Bhargava-Gross bound on the number of points on such curves has been greatly improved by Bjorn Poonen and Michael Stoll in 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
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Thank you very much for those references. The bounds for the hyperelliptic Jacobians in the Bhargava-Gross paper being the same as for elliptic curves, it could perhaps not be too wild to believe that (contrary to what I expected) the same distribution persists for abelian varieties: for a given global field $F$ and a given dimension $g$, that $50$% of the $g$-dimensional abelian varieties over $F$ have $0$ rank, $50$% have rank $1$, and $0$% have higher rank? – Vesselin Dimitrov Feb 6 '13 at 8:07
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Here is another example of possible bias when considering special families of abelian varieties : consider the abelian varieties $A_f$ associated to weight 2 newforms $f$. The Mordell-Weil group $A_f(\mathbf{Q})$ is a module over the Hecke algebra which is an order of a number field of degree $d=\dim A_f$. Thus in this setting I think one expects the average rank to be $d/2$. It would be nice to have a clean conjecture for a generic abelian variety over a number field (a generic abelian variety has trivial ring of endomorphisms, unlike $A_f$). – François Brunault Feb 8 '13 at 8:08
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Thanks for the reference. I guess we may ask about the rank distribution among abelian varieties with given Mumford-Tate group. – François Brunault Feb 8 '13 at 12:42

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