247 research outputs found

    Ray class invariants over imaginary quadratic fields

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    Let KK be an imaginary quadratic field of discriminant less than or equal to -7 and K(N)K_{(N)} be its ray class field modulo NN for an integer NN greater than 1. We prove that singular values of certain Siegel functions generate K(N)K_{(N)} over KK 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

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    Let KK be an imaginary quadratic field. For an order O\mathcal{O} in KK and a positive integer NN, let KO, NK_{\mathcal{O},\,N} be the ray class field of O\mathcal{O} modulo NON\mathcal{O}. We deal with various subjects related to KO, NK_{\mathcal{O},\,N}, mainly about Galois representations attached to elliptic curves with complex multiplication, form class groups and LL-functions for orders

    Class fields arising from the form class groups of order O and level N

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    Let KK be an imaginary quadratic field and O\mathcal{O} be an order in KK. We construct class fields associated with form class groups which are isomorphic to certain O\mathcal{O}-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 nn we examine primes of the form x2+ny2x^2+ny^2 with some additional conditions.Comment: 30 page
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