Singular values of multiple eta-quotients for ramified primes

TL;DR

This paper investigates conditions under which singular values of multiple eta-quotients produce class invariants, enabling faster computation of elliptic curves with complex multiplication by analyzing their algebraic properties in ring class fields.

Contribution

It generalizes the understanding of singular values of eta-quotients for ramified primes and their role as class invariants, extending to modular functions on $X_0^+(p)$.

Findings

Singular values lie in subfields of ring class fields of index $2^{k'-1}$.

Conditions identified for eta-quotients to yield class invariants.

Results facilitate faster computation of elliptic curves with complex multiplication.

Abstract

We determine the conditions under which singular values of multiple $\eta$-quotients of square-free level, not necessarily prime to~6, 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 $2^{k' - 1}$ when $k' \geq 2$ 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.