AI 中文总结
本文研究负判别式虚二次域，定义阿代尔标架型类群，证明其与特定伽罗瓦群同构，整合经典数论概念，且该类群的抽象结构唯一确定对应虚二次域。
AI 中文摘要
设D为负判别式，令K=ℚ(√D)，记𝒬(D)为ℤ上判别式为D的本原正定二元二次型集合。我们引入阿代尔标架型集合\\(\widehat{\mathcal{Q}}(D)=\left\{(Q,\\,\gamma)\in \mathcal{Q}(D)\times\mathrm{SL}_2(\widehat{\mathbb{Z}})~|~ Q\left(\gamma\begin{bmatrix}1\\\\0\end{bmatrix}\right)\in \widehat{\mathbb{Z}}^\times\right\}\\)及其在\\(\mathrm{SL}_2(\mathbb{Z})\\)自然作用下的轨道空间\\(\widehat{C}(D)\\)。我们在\\(\widehat{C}(D)\\)上定义高斯-狄利克雷合成律的显式阿代尔类比，并赋予其由\\(\widehat{\mathcal{Q}}(D)\\)的子空间拓扑诱导的商拓扑，其中\\(\mathcal{Q}(D)\\)取离散拓扑，\\(\mathrm{SL}_2(\widehat{\mathbb{Z}})\\)取 profinite 拓扑。随后我们证明存在拓扑群同构\\(\widehat{C}(D)\simeq\mathrm{Gal}\left(K^\mathrm{ab}(\mathfrak{t}^{1/\infty})/K(\mathfrak{t})\right)\\)，其中伽罗瓦群赋予克鲁尔拓扑，\\(\mathfrak{t}\\)为正超越实数，\\(\mathfrak{t}^{1/\infty}=\{\sqrt[N]{\mathfrak{t}}~|~N\geq1\}\\)。此外，我们将\\(\widehat{C}(D)\\)的一个显式定义子群与\\(\mathrm{Gal}(K^\mathrm{ab}/K)\\)等同，并描述其对模函数特殊值的对应伽罗瓦作用，由此在单一阿代尔框架下整合经典高斯合成、有限层型类群与志村互反律。最后我们证明\\(\widehat{C}(D)\\)的抽象群结构唯一确定虚二次域K。
英文摘要
Let $D$ be a negative discriminant, and let $K=\mathbb{Q}(\sqrt{D})$. Let $\mathcal{Q}(D)$ denote the set of primitive positive definite binary quadratic forms over $\mathbb{Z}$ of discriminant $D$. We introduce the set of adelic framed forms \begin{equation*} \widehat{\mathcal{Q}}(D)= \left\{(Q,\,γ)\in \mathcal{Q}(D)\times\mathrm{SL}_2(\widehat{\mathbb{Z}})~|~ Q\left(γ\begin{bmatrix}1\\0\end{bmatrix}\right)\in \widehat{\mathbb{Z}}^\times\right\} \end{equation*} and its orbit space $\widehat{C}(D)$ under the natural action of $\mathrm{SL}_2(\mathbb{Z})$. We define an explicit adelic analogue of the Gauss-Dirichlet composition law on $\widehat{C}(D)$ and endow $\widehat{C}(D)$ with the quotient topology induced by the subspace topology on $\widehat{\mathcal{Q}}(D)$ inherited from the product topology on $\mathcal{Q}(D)\times\mathrm{SL}_2(\widehat{\mathbb{Z}})$, where $\mathcal{Q}(D)$ is discrete and $\mathrm{SL}_2(\widehat{\mathbb{Z}})$ has its profinite topology. We then prove that there is an isomorphism of topological groups \begin{equation*} \widehat{C}(D)\simeq\mathrm{Gal}\left(K^\mathrm{ab}(\mathfrak{t}^{1/\infty})/K(\mathfrak{t})\right), \end{equation*} where the Galois group is endowed with the Krull topology, $\mathfrak{t}$ is a positive transcendental real number, and $\mathfrak{t}^{1/\infty}=\{\sqrt[N]{\mathfrak{t}}~|~N\geq1\}$. Moreover, we identify an explicitly defined subgroup of $\widehat{C}(D)$ with $\mathrm{Gal}(K^\mathrm{ab}/K)$ and describe the corresponding Galois action on special values of modular functions. In this way, classical Gauss composition, finite-level form class groups, and Shimura reciprocity are brought together within a single adelic framework. Finally, we show that the abstract group structure of $\widehat{C}(D)$ uniquely determines the imaginary quadratic field $K$.
