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arXiv · alg-geom/9603024

d-gonality of modular curves and bounding torsions

Khac Viet Nguyen·Masa-Hiko Saito
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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.

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Khac Viet Nguyen, Masa-Hiko Saito. 1996-03-29. d-gonality of modular curves and bounding torsions. https://arxiv.org/abs/alg-geom/9603024

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