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Let $p$ be a prime, and $J^-(p)$ be the maximal quotient of the Jacobian of the modular curve $X_0(p)$ on which the involution acts by $-1$.

Is anything known or conjectured about upper bounds for the "size" (either the rank or the dimension of the Zariski closure) of $J^-(p)(\mathbb{Q})$?

(On basis of BSD and related conjectures, we would expect these quantities to "small" in some sense. Explicit computation in SAGE gives that $J^-(p)(\mathbb{Q})$ is finite for all but 38 of the primes less than 3000.)

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Well, J_0(p) itself is known to have analytic rank at most (6.5) dim J_0(p), by a result of Kowalski and Michel:

so that's an upper bound to start with, at least if you are willing to condition on BSD. An unconditional upper bound on the rank seems hard -- I'm not sure how you would hope to rule out the existence of (e.g.) some elliptic curve of conductor p whose Mordell-Weil rank is gigantic.

The dimension of the Zariski closure is just the sum of all the A_f with positive Mordell-Weil rank, right? Here you might be able to make slightly more progress. What you are looking for is an lower bound on the number of cuspforms f in S_2(Gamma_0(p)) such that L(f,1) is nonzero. (This is unconditional, since we know that nonvanishing implies rank 0.) Of course, it's not even obvious that there is one such cuspform! One way to show the existence is to compute a suitable weighted average of the L(f,1) via Petersson formula. The average will be nonzero, which means one of the special values is nonzero. By being more clever about the many choices of average you can take, you can get lower bounds; I think Amir Akbary knows a lot about this.

But you may want something much stronger; e.g., that the sum of dim A_f over all f with L(f,1)=0 is o(dim J_0(p)). That's what one would expect. I don't know that anyone has proved anything of this kind.

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A single curve of conductor $p$ would surely not suffice: $\dim J^-(p)$ is about $p/24$, and it must be feasible to prove that the 2-Selmer bound on the ${\bf Q}$-rank of $J^-(p)$ is $o(p)$. – Noam D. Elkies May 21 '12 at 2:28
How would you prove that? – JSE May 21 '12 at 2:54
$E/2E$ injects into something that's basically a class group of the $2$-torsion field, and it must be easy to show that this is $O(p^A)$ for some finite $A$, which makes the rank $O(\log(p))$. – Noam D. Elkies May 21 '12 at 19:40
CORRECTION Thanks to Eric Larson for asking me about this and thus bringing the error to my attention: I meant that the 2-Selmer bound on the ${\bf Q}$-rank of an elliptic curve factor of $J^-(p)$ is $O(\log p)$, and hence $o(\dim J^-(p))$. The bound $O(\log p)$ for the rank of $J^-(p)$ might be correct, but I don't think we know how to prove an upper bound nearly that small. – Noam D. Elkies Jun 2 '12 at 3:47

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