929 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
Arithmetic properties of orders in imaginary quadratic fields
Let be an imaginary quadratic field. For an order in
and a positive integer , let be the ray class field of
modulo . We deal with various subjects related to
, mainly about Galois representations attached to elliptic
curves with complex multiplication, form class groups and -functions for
orders
Class fields arising from the form class groups of order O and level N
Let be an imaginary quadratic field and be an order in .
We construct class fields associated with form class groups which are
isomorphic to certain -ideal class groups in terms of the theory
of canonical models due to Shimura. By utilizing these form class groups, we
first derive a congruence relation on special values of a modular function of
higher level as an analogue of Kronecker's congruence relation. Furthermore, as
an application of such class fields, for a positive integer we examine
primes of the form with some additional conditions.Comment: 30 page
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