Quantum modular invariant and Hilbert class fields of real quadratic global function fields (Q2658744)

Hilbert's 12th problem conjectures that every abelian extension of a number field can be described by generalizing the Kronecker-Weber theorem. The paper under review is the first of a series of two and it presents a solution to Manin's Real Multiplication program for global function fields by using quantum analogs of the modular invariant and the exponential function. The second paper can be found in \url{arxiv.org/pdf/1709.05337v1.pdf}. This first paper deals with Hilbert class field generation in the sense of \textit{M. Rosen} [Expo. Math. 5, 365--378 (1987; Zbl 0632.12017)]. It is shown that special multivalues of the quantum modular invariant may be used to generate Hilbert class field of real quadratic global function fields. Let \(k={\mathbb F}_q(T), A={\mathbb F}_q[T]\) and let \(k_{\infty}={\mathbb F}_q ((1/T))\) be the completion at \(\infty\). Following similar procedure as in the number field case, the authors introduce the quantum modular invariant \(j^{\mathrm{qt}}\colon\mathrm{Mod}^{\mathrm{qt}}:=\mathrm{PGL}_2(A)\setminus (k_{\infty}\setminus k)\multimap k_{\infty}\). For \(f\in k_{\infty}\setminus k\), let \(\Lambda_{\epsilon}= \{a\in A\mid |af-b|_{\infty}<\epsilon \text{\ for some\ } b\in A\}\). Then it is defined the approximant \(j_{\epsilon}(f)\) using \(\Lambda_{\epsilon}(f)\). Then \(j\to j^{\mathrm{qt}}(f):=\lim_{\epsilon\to 0}j_{\epsilon}(f)\) defines a discontinuous, \(\mathrm{PGL}_2(A)\)-invariant, multivalued function. The main result, Theorem 8, is the following. Let \(f\in k_{\infty}\setminus k\) be a fundamental quadratic unit of degree \(d\) in \(T\) and \(K=k(f)\). Then the Hilbert class field \(H\) is given by \(H=K(N(j^{\mathrm{qt}}(f)))\), where \(N(j^{\mathrm{qt}}(f))\) is the product of the \(d\) elements of \(j^{\mathrm{qt}}(f)\). In Theorem 7 is given the description of the narrow Hilbert class field of \(K\). The theory of Hayes (Drinfeld modules) gives an explicit class field theory for global function fields. In fact for any finite extension \(L/k\), Hayes theory gives explicit descriptions of the class fields associated to rank \(1\) Dedekind domains but does not give explicit descriptions of the Hilbert class field and ray class fields associated to \({\mathcal O}_L\), the integral closure of \(A\) in \(L\). Further, the generators provided by Hayes are not given as values of an analog of a modular function. The explicit class field theory described in the two papers of the authors, in the real quadratic case, is closer to the spirit of Hilbert's 12-th problem.