# Singular values of multiple eta-quotients for ramified primes

1 LFANT - Lithe and fast algorithmic number theory IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest 2 IMB - Institut de Mathématiques de Bordeaux 3 Institut für Mathematik Augsburg

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.

Keywords : complex multiplication class invariants eta quotients ring class fields

Author: Andreas Enge - Reinhard Schertz -

Source: https://hal.archives-ouvertes.fr/