Singular values of multiple eta-quotients for ramified primes

Abstract

AbstractWe determine the conditions under which singular values of multiple $\eta $-quotients of square-free level, not necessarily prime to six, yield class invariants; that is, algebraic numbers in ring class fields of imaginary-quadratic number fields. We show that the singular values lie in subfields of the ring class fields of index ${2}^{{k}^{\prime } - 1} $ when ${k}^{\prime } \geq 2$ primes dividing the level are ramified in the imaginary-quadratic field, which leads to faster computations of elliptic curves with prescribed complex multiplication. The result is generalised to singular values of modular functions on ${ X}_{0}^{+ } (p)$ for $p$ prime and ramified.