Adelic framed form class groups and explicit class field theory

Abstract

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,\,\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\} \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 $K$.