The following question came to me earlier as a "side question"; something I'd like to know, but which is not totally necessary for what I'm thinking about or doing:

Consider the genus 32 curve $X_0(389)$, and denote its Jacobian variety as $J_0(389)$.

I am interested in finding an upper bound for the Mordell-Weil rank of $J_0(389)(\mathbb{Q}(i))$.

After thinking about this for some time, I turned to Google, which threw up [1]. Apparently, assuming the Birch-Swinnerton-Dyer conjecture, there is an absolute constant $C > 0$ such that for all primes $q$ sufficiently large, we have

$ \mbox{rank } J_0(q)(\mathbb{Q}) \leq C \mbox{ dim} J_0(q) $.

[Ideally this equation would be in the center]

The point of that paper is to show that $C = 6.5$ will do (existence of $C$ having been proved in an earlier paper by the same authors), but, "assuming [also] the Riemann Hypothesis for automorphic $L$-functions, Iwaniec, Luo and Sarnak have recently proved that one could take $C = \frac{99}{100}$".

Now I reckon 389 is "sufficiently large", which means (if you believe those conjectures) an upper bound for the rank over $\mathbb{Q}$ is 31. But this feels like it is way too big, maybe because there are generally so few rational points on modular curves.

Does anyone know how to get a better upper bound? Or am I wrong in hoping for a smaller upper bound?

Furthermore, since I'm interested in $\mathbb{Q}(i)$-rank, there is the following:

What is the biggest the rank can jump by when going from $J_0(389)(\mathbb{Q})$ to $J_0(389)(\mathbb{Q}(i))$?

I guess this last question can be asked in greater generality, replacing the Js with any abelian variety $A/\mathbb{Q}$. Can the rank jump be arbitrarily large when passing from $\mathbb{Q}$ to $\mathbb{Q}(i)$? Or is there a bound in terms of the dimension of $A$, say?

It's my bedtime now, so I'll pick this thread up in 9 or so hours.

[1]: "Explicit Upper Bound for the (Analytic) rank of $J_0(q)$". E. Kowalski, P. Michel. Israel J. Math, 2000. Preprint Available here