Ray Class Invariants in Terms of Extended Form Class Groups
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Let $K$ be an imaginary quadratic field with $\mathcal{O}_K$ its ring of integers. For a positive integer $N$, let$K_{(N)}$ be the ray class field of $K$ modulo $N\mathcal{O}_K$,and let $\mathcal{F}_N$ be the field of meromorphic modular functions of level $N$whose Fourier coefficients lie in the $N$th cyclotomic field. For each $h\in\mathcal{F}_N$, we construct a ray class invariantas its special value in terms of the extended form class group,and show that the invariant satisfies the natural transformation formula via the Artin mapin the sense of Siegel and Stark. Finally, we establish an isomorphism between the extended form class group and $\mathrm{Gal}(K_{(N)}/K)$ without any restriction on $K$.
Abstract
Let $K$ be an imaginary quadratic field with $\mathcal{O}_K$ its ring of integers. For a positive integer $N$, let$K_{(N)}$ be the ray class field of $K$ modulo $N\mathcal{O}_K$,and let $\mathcal{F}_N$ be the field of meromorphic modular functions of level $N$whose Fourier coefficients lie in the $N$th cyclotomic field. For each $h\in\mathcal{F}_N$, we construct a ray class invariantas its special value in terms of the extended form class group,and show that the invariant satisfies the natural transformation formula via the Artin mapin the sense of Siegel and Stark. Finally, we establish an isomorphism between the extended form class group and $\mathrm{Gal}(K_{(N)}/K)$ without any restriction on $K$.
참고문헌 (9)
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