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.