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

Abstract

This is the first of a series of two papers in which we present a solution to Manin’s Real Multiplication program (Manin in: Laudal and Piene (eds) The Legacy of Niels Henrik Abel, Springer, Berlin, 2004) —an approach to Hilbert’s 12th problem for real quadratic extensions of $$\\mathbb Q$$ Q —in positive characteristic, using quantum analogs of the modular invariant and the exponential function. In this first paper, we treat the problem of Hilbert class field generation. If $$k=\\mathbb F_{q}(T)$$ k = F q ( T ) and $$k_{\\infty }$$ k ∞ is the analytic completion of k , we introduce the quantum modular invariant $$\\begin{aligned} j^\\mathrm{qt}: k_{\\infty }\\multimap k_{\\infty } \\end{aligned}$$ j qt : k ∞ ⊸ k ∞ as a multivalued, discontinuous modular invariant function. Then if $$K=k(f)\\subset k_{\\infty }$$ K = k ( f ) ⊂ k ∞ is a real quadratic extension of k and f is a fundamental unit, we show that the Hilbert class field $$H_{\\mathcal {O}_{K}}$$ H O K (associated to $$\\mathcal {O}_{K}=$$ O K = integral closure of $$\\mathbb F_{q}[T]$$ F q [ T ] in K ) is generated over K by the product of the multivalues of $$j^\\mathrm{qt}(f)$$ j qt ( f ) .