The question in the title naturally breaks up in two parts, namely the torsion part and the rank part. I already read about some results on both the torsion and the rank part. And I want to know whether this is currently still the state of the art, or whether there has been some recent progress in these questions.

First the torsion part, as far as I know the only results on the torsion parts are for the case $N$ is a prime, and for prime $N$ the results are quite good.

The structure of $J_0(N)(\mathbb Q)_{tors}$ is determined by Bary Mazur in his 'Modular Curves and the Eisenstein Ideal' paper. For $J_1(N)(\mathbb Q)_{tors}$ there exists a conjecture by Conrad, Edixhoven and Stein which states that it can be generated by the cusps. This conjecture was recently proved (up to two torsion) by Ohta in his 'Eisenstein ideals and the rational torsion subgroups of modular Jacobian varieties' paper.

**Are there any results on the structure of $J_0(N)(\mathbb Q)_{tors}$ or $J_1(N)(\mathbb Q)_{tors}$ for composite N?**

Now for the rank part I am only interested in for which $N$ it is $0$. Barry Mazur already proved in the same paper as mentioned earlier that for $N = 37, 43, 54, 61, 67$ and all primes $N \geq 73$ one has that $J_0(N)$ has positive rank. Now this makes me suspect that there will be only finitely many $N$ such that $J_0(N)$ has rank zero.

**Are there only finitely many $N$ such that the rank of $J_0(N)(\mathbb Q)$ is $0$, and if so is this list known?**

I have the feeling that I can answer the rank question myself by doing some computations, but I would much rather give a reference to the literature if it exists.