Gonality of the modular curve 𝑋₀(𝑁)

Abstract

In this paper we determine the Q \mathbb {Q} -gonalities of the modular curves X 0 ( N ) X_0(N) for all N > 145 N>145 . We determine the C \mathbb {C} -gonality of many of these curves and the Q \mathbb {Q} -gonalities and C \mathbb {C} -gonalities for many larger values of N N . Using these results and some further work, we determine all the modular curves X 0 ( N ) X_0(N) of gonality 4 4 , 5 5 and 6 6 over Q \mathbb {Q} . We also find the first known instances of pentagonal curves X 0 ( N ) X_0(N) over C \mathbb {C} .