Editing Complex multiplication (section)

==Singular moduli==
The points of the upper half-plane ''τ'' which correspond to the period ratios of elliptic curves over the complex numbers with complex multiplication are precisely the imaginary quadratic numbers.{{sfn|Silverman|1986|p=339}} The corresponding [[Elliptic modular function|modular invariant]]s ''j''(''τ'') are the '''singular moduli''', coming from an older terminology in which "singular" referred to the property of having non-trivial endomorphisms rather than referring to a [[singular curve]].{{sfn|Silverman|1994|p=104}}

The [[modular function]] ''j''(''τ'') is algebraic on imaginary quadratic numbers ''τ'':{{sfn|Serre|1967|p=293}} these are the only algebraic numbers in the upper half-plane for which ''j'' is algebraic.<ref>{{cite book | first=Alan | last=Baker | author-link=Alan Baker (mathematician) | title=Transcendental Number Theory | publisher=[[Cambridge University Press]] | year=1975 | isbn=0-521-20461-5 | zbl=0297.10013 | page=56 }}</ref>

If Λ is a lattice with period ratio ''τ'' then we write ''j''(Λ) for ''j''(''τ'').  If further Λ is an ideal '''a''' in the ring of integers ''O<sub>K</sub>'' of a quadratic imaginary field ''K'' then we write ''j''('''a''') for the corresponding singular modulus.  The values ''j''('''a''') are then real algebraic integers, and generate the [[Hilbert class field]] ''H'' of ''K'': the [[field extension]] degree [''H'':''K''] = ''h'' is the class number of ''K'' and the ''H''/''K'' is a [[Galois extension]] with [[Galois group]] isomorphic to the [[ideal class group]] of ''K''.  The class group acts on the values ''j''('''a''') by ['''b'''] : ''j''('''a''') → ''j''('''ab''').

In particular, if ''K'' has class number one, then ''j''('''a''') = ''j''(''O'') is a rational integer: for example, ''j''('''Z'''[i]) = ''j''(i) = 1728.