601 research outputs found
Ray class invariants over imaginary quadratic fields
Let be an imaginary quadratic field of discriminant less than or equal to
-7 and be its ray class field modulo for an integer greater
than 1. We prove that singular values of certain Siegel functions generate
over by extending the idea of our previous work. These generators
are not only the simplest ones conjectured by Schertz, but also quite useful in
the matter of computation of class polynomials. We indeed give an algorithm to
find all conjugates of such generators by virtue of Gee and Stevenhagen
Ray class fields generated by torsion points of certain elliptic curves
We first normalize the derivative Weierstrass -function appearing in
Weierstrass equations which give rise to analytic parametrizations of elliptic
curves by the Dedekind -function. And, by making use of this
normalization of we associate certain elliptic curve to a given
imaginary quadratic field and then generate an infinite family of ray class
fields over by adjoining to torsion points of such elliptic curve. We
further construct some ray class invariants of imaginary quadratic fields by
utilizing singular values of the normalization of , as the -coordinate
in the Weierstrass equation of this elliptic curve, which would be a partial
result for the Lang-Schertz conjecture of constructing ray class fields over
by means of the Siegel-Ramachandra invariant
A modularity criterion for Klein forms, with an application to modular forms of level
We find some modularity criterion for a product of Klein forms of the
congruence subgroup and, as its application, construct a basis of
the space of modular forms for of weight . In the process we
face with an interesting property about the coefficients of certain theta
function from a quadratic form and prove it conditionally by applying Hecke
operators
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