Abstract
It is proved that the Hilbert class field of a real quadratic field ${\Bbb Q}(\sqrt{D})$ modulo a power $m$ of the conductor $f$ is generated by the Fourier coefficients of the Hecke eigenform for a congruence subgroup of level $fD$.
Highlights
The Kronecker’s Jugendtraum is a conjecture that the maximal unramified abelian extension (The Hilbert class field) of any algebraic number field is generated by the special values of modular functions attached to an abelian variety
In the case of an arbitrary number field, a description of the abelian extensions is given by class field theory, but an explicit formula for the generators of these abelian extensions, in the sense sought by Kronecker, is unknown even for the real quadratic fields
The aim of our note is a formula for generators of the Hilbert class field of real quadratic fields based on a modularity and a symmetry of complex and real multiplication
Summary
The Kronecker’s Jugendtraum is a conjecture that the maximal unramified abelian extension (The Hilbert class field) of any algebraic number field is generated by the special values of modular functions attached to an abelian variety. Remark 1: The isomorphism (2) can be calculated using the wellknown formula for the class number of a non-maximal order + fOK of a quadratic field K = ( D) :
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