Singular values of multiple eta-quotients for ramified primes

2013/01/23 by Enge, Andreas, Schertz, Reinhard
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1301.5521

Abstract

We determine the conditions under which singular values of multiple η-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' ≥ 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 X₀⁺ (p) for p prime and ramified.

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