Ja Kyung Koo

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

In this paper we show that the Peterson root multiplicity formula and the Freudenthal weight multiplicity formula can be extended to the case of generalized Kac-Moody superalgebras. Applying these to some modular functions, we derive interesting relations among the Fourier coefficients. In particular, it will be shown that the Fourier coefficients of the elliptic modular function j − 744 can be determined only by the first three ones.

We find the full normalizer of γ1(N) in psl2(R).

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.

Let $K$ be an imaginary biquadratic field and $K_1$, $K_2$ be its imaginary quadratic subfields. For integers $N>0$, $μ\geq 0$ and an odd prime $p$ with $\gcd(N,p)=1$, let $K_{(Np^μ)}$ and $(K_i)_{(Np^μ)}$ for $i=1,2$ be the ray class fields of $K$ and $K_i$, respectively, modulo $Np^μ$. We first present certain class fields $\widetilde{K_{N,p,μ}^{1,2}}$ of $K$, in the sense of Hilbert, which are generated by Siegel-Ramachandra invariants of $(K_i)_{(Np^{μ+1})}$ for $i=1,2$ over $K_{(Np^μ)}$

We first find a sufficient condition for a product of theta constants to be a Siegel modular function of a given even level. And, when $K_{(2p)}$ denotes the ray class field of $K=\mathbb{Q}(e^{2πi/5})$ modulo $2p$ for an odd prime $p$, we describe a subfield of $K_{(2p)}$ generated by the special value of certain theta constant by using Shimura's reciprocity law.