# Maximal abelian extension of $X_0(p)$ unramified outside cusps

## TL;DR

This paper constructs a new maximal abelian extension of the modular curve $X_0(p)$ unramified outside cusps, extending Mazur's work by using generalized Dedekind eta functions to produce a cyclic cover of degree $2N_p$.

## Contribution

It introduces a new cyclic abelian covering of $X_0(p)$ unramified outside cusps, expanding the understanding of modular curve extensions beyond Mazur's established covers.

## Key findings

- The maximal abelian cover $X_2'(p)$ is cyclic of degree $2N_p$.
- Construction utilizes generalized Dedekind eta functions.
- Extends Mazur's results on unramified abelian coverings.

## Abstract

Let $p$ be a prime number. Mazur proved that a geometrically maximal unramified abelian covering of $X_0(p)$ over $\mathbb Q$ is given by the Shimura covering $X_2(p) \to X_0(p)$, that is, a unique subcovering of $X_1(p) \to X_0(p)$ of degree $N_p := (p-1)/\gcd(p-1, 12)$. In this short paper, we show that a geometrically maximal abelian covering $X_2'(p) \to X_0(p)$ of $X_0(p)$ over $\mathbb Q$ unramified outside cusps is cyclic of degree $2N_p$. The main ingredient for the construction of $X_2'(p)$ is the generalized Dedelind eta functions.