d-gonality of modular curves and bounding torsions

Abstract

We study the problem of $d$-gonality of the modular curve $X_0(N)$. As a result, we can give an upperbound of the level $N$ by means of $d$. This generalizes Ogg's result on hyperelliptic modular curves ($d = 2$). As a corollary of this result, we prove an analogue of the strong Uniform Boundedness Conjecture for elliptic curves defined over the function fields of curves. If a base curve is $d$-gonal, we can bound orders of torsions of Mordell-Weil groups in terms of $d$ uniformly.