Bielliptic quotient modular curves of 𝑋₀(𝑁)

Abstract

Let N ≥ 1 N\geq 1 be a non-square free integer and let W N W_N be a nontrivial subgroup of the group of the Atkin-Lehner involutions of X 0 ( N ) X_0(N) such that the modular curve X 0 ( N ) / W N X_0(N)/W_N has genus at least two. We determine all pairs ( N , W N ) (N,W_N) such that X 0 ( N ) / W N X_0(N)/W_N is a bielliptic curve and the pairs ( N , W N ) (N,W_N) such that X 0 ( N ) / W N X_0(N)/W_N has an infinite number of quadratic points over Q \mathbb {Q} .