Artin symbols over imaginary quadratic fields
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Let $K$ be an imaginary quadratic field with ring of integers $\mathcal{O}_K$ and $N$ be a positive integer. By $K_{(N)}$ we meanthe ray class field of $K$ modulo $N\mathcal{O}_K$. In this paper, for each prime $\mathfrak{p}$ of $K$ relatively prime to $N\mathcal{O}_K$ we explicitly describethe action of the Artin symbol $(\frac{K_{(N)}/K}{\mathfrak{p}})$on special values of modular functions of level $N$. Furthermore, weextend the Kronecker congruence relation for the elliptic modular function $j$to some modular functions of higher level.
Abstract
Let $K$ be an imaginary quadratic field with ring of integers $\mathcal{O}_K$ and $N$ be a positive integer. By $K_{(N)}$ we meanthe ray class field of $K$ modulo $N\mathcal{O}_K$. In this paper, for each prime $\mathfrak{p}$ of $K$ relatively prime to $N\mathcal{O}_K$ we explicitly describethe action of the Artin symbol $(\frac{K_{(N)}/K}{\mathfrak{p}})$on special values of modular functions of level $N$. Furthermore, weextend the Kronecker congruence relation for the elliptic modular function $j$to some modular functions of higher level.
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