testing

Just want to see how this works. Hmmm... well, I should put something mathematical in here anyways. So here is a question I'm wondering about right now. Let X_E(N) be the moduli space of elliptic curves $F$ along with an isomorphims $E[N]->F[N]$, ane let J_E(N)=Jac(X_E(N)). Note that elliptic curves isogeneous to E correspond to rational points on X_E(N), so we call these the trivial points on X_E(N). It is conjectured that for any E there is a constant c(E) such that for N>c(E) we have X_E(N) has only trivial rational points. We call this Frey-Mazur conjecture (although, Frey-Mazur conjecture is usually stated with c(E)=23 or so, and certainly independant of E). Unfortunately the standard methods used in proving X_1(N)(Q) has only trivial solutions don't apply here, since we can't find a quotient of J_E(N) with rank 0. In fact generically we expect J_E(N) to be irreducible over rationals, and there is no obvious reason why J_E(N) should have small rank. In fact, in the cases done by Poonen-Schaefer-Stoll for J_E(7) they already found an example with rank 3. I think Brown's work in extending their work to J_E(11) and J_E(13) seems to find other seemingly innnocent examples with fairly high ranks.

So here's my question: for a given E, how large can the rank of J_E(N)(Q) get as N gets larger? Is this value bounded? Are there examples where it grows as N gets larger? In theory, if the rank J_E(N)(Q) is less than genus of X_E(N) we can find all rational points of X_E(N)(Q), so if they don't grow too fast we should be fine.