Bielliptic intermediate modular curves

TL;DR

This paper classifies bielliptic properties of intermediate modular curves $X_ riangle(N)$, identifies an exceptional automorphism, and determines which have infinitely many quadratic points over $Q$, advancing understanding of their automorphisms and rational points.

Contribution

It provides a complete classification of bielliptic intermediate modular curves and identifies cases with exceptional automorphisms and infinite quadratic points.

Findings

Identified which $X_ riangle(N)$ are bielliptic.

Discovered an intermediate modular curve with exceptional automorphisms.

Listed all $X_ riangle(N)$ with infinitely many quadratic points over $Q$.

Abstract

We determine which of the modular curves $X_\Delta(N)$, that is, curves lying between $X_0(N)$ and $X_1(N)$, are bielliptic. Somewhat surprisingly, we find that one of these curves has exceptional automorphisms. Finally we find all $X_\Delta(N)$ that have infinitely many quadratic points over $\mathbb{Q}$.