What is known and not known about rational points of modular curves? What are some good references?

4$\begingroup$ Could you make the question more specific? Do you mean $\mathbb{Q}$ or an arbitrary number field or what? Which modular curves $X_0(N),X_1(N),X(N)$ or some other? Good reference is Mazur, B., Modular curves and the Eisenstein ideal. Inst. Hautes Études Sci. Publ. Math. No. 47 (1977), 33186. $\endgroup$ – Felipe Voloch Apr 25 '11 at 18:42

3$\begingroup$ Loic Merel's work from the mid90s generalises some of Mazur's results to arbitrary number fields. See also Edixhoven's Seminaire Bourbaki talk from around the same time. But this question is ridiculously vague. $\endgroup$ – Kevin Buzzard Apr 25 '11 at 18:54

3$\begingroup$ "What is known?" A lot; "and not known?" Also a lot. $\endgroup$ – Olivier Apr 25 '11 at 20:49

1$\begingroup$ A nice reference for Merel's result is the Clay Proceedings: math.univbpclermont.fr/~rebolledo/pagefichiers/… It also touches on related work of Mazur, Kamienny, Kenku, Momose, Parent and others. If we allow ourselves to move slightly outside the classical modular curves, we get lots of other interesting results, including the BiluParent result on $X_{split}(p)$. $\endgroup$ – stankewicz Apr 25 '11 at 21:57
Dear Dick,
I am going to interpret rational points to mean points over $\mathbb Q$.
Given this, Mazur's Eisenstein ideal paper, his rational isogenies paper, and various surveys he wrote around that time (midtolate 1970s) give a good description (and also prove the most of the key results).
Ogg also wrote surveys in the early 70s, making conjectures which Mazur then proved, which are very helpful. You'll easily find them on mathscinet (as you will with Mazur's papers and surveys).
Best wishes,
Matt