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 $X_2(p)\to X_0(p)$, 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 $X_2^{\prime }(p)\to X_0(p)$ of $X_0(p)$ over $\mathbb{Q}$ unramified outside cusps is cyclic of degree $2N_p$. The main ingredient for the construction of $X_2^{\prime }(p)$ is the generalized Dedelind eta functions.

arXiv comment12 pages