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Singular values of multiple eta-quotients for ramified primes

Part of: Arithmetic algebraic geometry Abelian varieties and schemes Computational number theory

Published online by Cambridge University Press:  07 November 2013

Andreas Enge  and
Reinhard Schertz
Andreas Enge
Affiliation:
INRIA, LFANTCNRS, IMB, UMR 5251 Université de Bordeaux, IMB33400 Talence France email andreas.enge@inria.fr
Reinhard Schertz
Affiliation:
Universität Augsburg Germany email schertz@math.uni-augsburg.de

Abstract

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We 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.

MSC classification

Secondary: 14K22: Complex multiplication 11G15: Complex multiplication and moduli of abelian varieties 11Y40: Algebraic number theory computations

Information

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 16 , October 2013 , pp. 407 - 418
Copyright
© The Author(s) 2013 

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