Singular values of multiple eta-quotients for ramified primes

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.
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Submitted on : Friday, July 19, 2013 - 6:20:06 PM
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Andreas Enge, Reinhard Schertz. Singular values of multiple eta-quotients for ramified primes. LMS Journal of Computation and Mathematics, London Mathematical Society, 2013, 16, pp.407-418. ⟨http://dx.doi.org/10.1112/S146115701300020X⟩. ⟨10.1112/S146115701300020X⟩. ⟨hal-00768375v2⟩

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