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d-gonality of modular curves and bounding torsions
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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.
Forward citations
Cited by 2 Pith papers
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Genus and Gonality of Small Curves, Dynamical Uniform Boundedness, and Bifurcation
Gonality of distinct dynatomic curves tends to infinity for non-isotrivial one-parameter rational-map families on P^1, with superlinear genus growth outside flexible Lattès families.
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Maximal curves over finite fields and a modular isogeny
Explicit genus-7 and genus-12 curves over F_{11^5} reach the Hasse-Weil-Serre upper bound, via a generalized Chen isogeny for quotients of Borel-Cartan modular curves.
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