By Mazur's theorems (Reference:- Modular curves and the Eisenstein ideal, 1977), we know that the only rational points of X_0(N) for N any prime > 163 are the two cusps (o) and (oo) (|X_0(N)(Q)| = 2 ).

Is there any bound for |X_0(N)(Q)| for all N?X_0(N)(Q)|, where N is an arbitrary +ve integer?.

More Generally,

Is there a bound on the number of rational points on the modular curve i.e. for |X_0(N)(K)|, where K is some number field. We know that |X_0(N)(K)| is finite.

By Mazur's theorems (Reference:- Modular curves and the Eisenstein ideal, 1977), we know that the only rational points of X_0(N) for N any integer prime > 163 are the two cusps (o) and (oo) (|X_0(N)(Q)| = 2 ).

Is there any bound for |X_0(N)(Q)| for all N?.

More Generally,

Is there a bound on the number of rational points on the modular curve i.e. for |X_0(N)(K)|, where K is some number field. We know that |X_0(N)(K)| is finite.

By Mazur's theorems (Reference:- Modular curves and the Eisenstein ideal, 1977), we know that the only rational points of X_0(N) for N any integer > 163 are the two cusps (o) and (oo). oo) (|X_0(N)(Q)| = 2 ).

Is there a bound exists for on the number of rational points on the modular curve i.e. for |X_0(N)(K)|, where K is some number field. We know that |X_0(N)(K)| is finite.

By Mazur's theorems (Reference:- Modular curves and the Eisenstein ideal, 1977), we know that the only rational points of X_0(N) for N any integer > 163 are the two cusps (o) and (oo).

Is there a bound exists for the number of rational points on the modular curve i.e. for |X_0(N)(K)|, where K is some number field. We know that |X_0(N)(K)| is finite.