| Autor: |
Eum, Ick Sun, Jung, Ho Yun, Koo, Ja Kyung, Shin, Dong Hwa |
| Rok vydání: |
2018 |
| Předmět: |
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| Druh dokumentu: |
Working Paper |
| Popis: |
Let $K$ be an imaginary quadratic field of discriminant $d_K$, and let $\mathfrak{n}$ be a nontrivial integral ideal of $K$ in which $N$ is the smallest positive integer. Let $\mathcal{Q}_N(d_K)$ be the set of primitive positive definite binary quadratic forms of discriminant $d_K$ whose leading coefficients are relatively prime to $N$. We adopt an equivalence relation $\sim_\mathfrak{n}$ on $\mathcal{Q}_N(d_K)$ so that the set of equivalence classes $\mathcal{Q}_N(d_K)/\sim_\mathfrak{n}$ can be regarded as a group isomorphic to the ray class group of $K$ modulo $\mathfrak{n}$. We further present an explicit isomorphism of $\mathcal{Q}_N(d_K)/\sim_\mathfrak{n}$ onto $\mathrm{Gal}(K_\mathfrak{n}/K)$ in terms of Fricke invariants, where $K_\mathfrak{n}$ is the ray class field of $K$ modulo $\mathfrak{n}$. This would be certain extension of the classical composition theory of binary quadratic forms, originated and developed by Gauss and Dirichlet. |
| Databáze: |
arXiv |
| Externí odkaz: |
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