Schertz conjectured that every finite abelian extension of imaginary quadratic fields can be generated by the norm of the Siegel–Ramachandra invariants. We present a conditional proof of his conjecture by means of the characters on class groups and the second Kronecker limit formula.

Let K be an imaginary quadratic field different from $\open{Q}(\sqrt {-1})$ and $\open{Q}(\sqrt {-3})$ . For a positive integer N, let KN be the ray class field of K modulo $N {\cal O}_K$ . By using the congruence subgroup ± Γ1(N) of SL2(ℤ), we construct an extended form class group whose operation is basically the Dirichlet composition, and explicitly show that this group is isomorphic to the Galois group Gal(KN/K). We also present an algorithm to find all distinct form classes and show how to multiply two form classes. As an application, we describe Gal(KNab/K) in terms of these extended form class groups for which KNab is the maximal abelian extension of K unramified outside prime ideals dividing $N{\cal O}_K$ .

We investigate certain families of meromorphic Siegel modular functions on which Galois groups act in a natural way. By using Shimura's reciprocity law we construct some algebraic numbers in the ray class fields of CM-fields in terms of special values of functions in these Siegel families.

We generate ray-class fields over imaginary quadratic fields in terms of Siegel–Ramachandra invariants, which are an extension of a result of Schertz. By making use of quotients of Siegel–Ramachandra invariants we also construct ray-class invariants over imaginary quadratic fields whose minimal polynomials have relatively small coefficients, from which we are able to solve certain quadratic Diophantine equations.

We provide a concrete example of a normal basis for a finite Galois extension which is not abelian. More precisely, let $\mathbb{C}(X(N))$ be the field of meromorphic functions on the modular curve $X(N)$ of level $N$ . We construct a completely free element in the extension $\mathbb{C}(X(N))/\mathbb{C}(X(1))$ by means of Siegel functions.

We investigate two kinds of Fricke families, those consisting of Fricke functions and those consisting of Siegel functions. In terms of their special values we then generate ray class fields of imaginary quadratic fields over the Hilbert class fields, which are related to the Lang–Schertz conjecture.

We show that every modular form on Γ0(2n) (n ⩾ 2) can be expressed as a sum of eta-quotients, which is a partial answer to Ono's problem. Furthermore, we construct a primitive generator of the ring class field of the order of conductor 4N (N ⩾ 1) in an imaginary quadratic field in terms of the special value of a certain eta-quotient.

We show that a weakly holomorphic modular function can be written as a sum of modular units of higher level. Furthermore, we find a necessary and sufficient condition for a meromorphic Siegel modular function of degree g to have neither a zero nor a pole on a certain subset of the Siegel upper half-space .

Let , where η(τ) is the Dedekind eta function. We show that if τ0 is an imaginary quadratic argument and m is an odd integer, then is an algebraic integer dividing This is a generalization of a result of Berndt, Chan and Zhang. On the other hand, when K is an imaginary quadratic field and θK is an element of K with Im(θK) > 0 which generates the ring of integers of K over ℤ, we find a sufficient condition on m which ensures that is a unit.

We estimate the genus of the modular curves X1(N).

We study the genus theory for function fields which is the analogue of the classical genus theory developed by Hasse and Fröhlich.

We give holomorphic differentials of some algebraic function field K of complex dimension one which is a generalisation of a hyperelliptic field.