scholarly journals On orders in number fields: Picard groups, ring class fields and applications

2015 ◽  
Vol 58 (8) ◽  
pp. 1627-1638 ◽  
Author(s):  
Chang Lv ◽  
YingPu Deng
Keyword(s):  
Number Fields ◽  
Picard Groups ◽  
Class Fields

scholarly journals Singular values of multiple eta-quotients for ramified primes

2013 ◽  
Vol 16 ◽  
pp. 407-418 ◽  
Author(s):  
Andreas Enge ◽  
Reinhard Schertz
Keyword(s):  
Complex Multiplication ◽  
Singular Values ◽  
Number Fields ◽  
Algebraic Numbers ◽  
Modular Functions ◽  
Class Invariants ◽  
Free Level ◽  
Yield Class ◽  
Imaginary Quadratic Number ◽  
Class Fields

AbstractWe 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.

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