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Singular values of multiple eta-quotients for ramified primes

Andreas Enge 1 Reinhard Schertz 2
1 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
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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Preprints, Working Papers, ...
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https://hal.inria.fr/hal-00768375
Contributor : Andreas Enge <>
Submitted on : Wednesday, January 23, 2013 - 1:19:46 PM
Last modification on : Monday, May 20, 2019 - 2:30:23 PM
Long-term archiving on: : Wednesday, April 24, 2013 - 2:50:09 AM

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  • HAL Id : hal-00768375, version 1
  • ARXIV : 1301.5521

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Andreas Enge, Reinhard Schertz. Singular values of multiple eta-quotients for ramified primes. 2013. ⟨hal-00768375v1⟩

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