Let $A$ be an abelian variety defined over $\mathbb{Q}$ of dimension $g \geq 1$. We shall denote by $A(\mathbb{Q})$ the Mordell-Weil group of rational points in $A$, and denote by $r = r_A$ the rank of $A(\mathbb{Q})$ (known to be finite due to Mordell (in the case $g = 1$) and Weil (for arbitrary $g$)).
In the case $g = 1$, $A = E$ is an elliptic curve, and it is a well-known question whether there exist elliptic curves with arbitrarily large Mordell-Weil rank. The record so far is a curve known to have rank at least 28 due to Elkies.
In general, one can find $A$ with arbitrarily large rank. For instance, simply take an elliptic curve $E$ with positive rank and the consider the product $E \times \cdots \times E$. However, it is a non-trivial question as to whether one can find abelian varieties $A$ with rank arbitrarily large with respect to the dimension.
My question is this: let $g \geq 1$ be a positive integer. For any positive number $C$, can we find an abelian variety $A/\mathbb{Q}$ with dimension $g$ such that the rank $r$ of $A$ satisfies $r > Cg$?
Since this question seems to cover the case of arbitrarily large rank of elliptic curves, a more modest version which is consistent with both a positive and negative answer to the boundedness of rank of elliptic curves is the following.
Does there exist a function $u(x)$ with the property that $\lim_{x \rightarrow \infty} u(x) = \infty$ such that for any $g \geq 1$ one can find an abelian variety $A/\mathbb{Q}$ such that $r_A > g u(g)$? Say, if we take $u(x) = \log x$?
