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

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 $X_2(p) \rightarrow X_0(p)$, that is, a unique subcovering of $X_1(p) \rightarrow X_0(p)$ of degree $N_p := (p-1)/\gcd (p-1, 12)$. In this short paper, we show that a geometrically maximal abelian covering $X_2'(p) \rightarrow X_0(p)$ 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 Dedekind eta functions.