I am Pierre MATSUMI. Could you please teach me what is effective Mordell.
Assume that f(X,Y) = 0 which defines smooth affine curve with genus > 1, and that there will be the solution X=n/m, Y=n'/m' in rational number Q. Then,
Theorem(?)(Effective Mordell): max(|n|,|m|,|n'|,|m'|) < const_f for some constant const_f with some constant in positive real number.
Is this statement right?
I found some article where, if C denotes the proper smooth curve defined by f(X,Y) after compactification, effective Mordell is equivalent to the fact that there is some non-trivial function F:C ---> P^1 and the height of F(n/m,n'/m') is bounded.
I am NOT sure whether this definition is equivalent to the above Theorem(?).... Please just teach me.
Sincerely yours, Pierre MATSUMI