Quantum Drinfeld Modules I: Quantum Modular Invariant and Hilbert Class Fields

2016/07/08 by Demangos, L., Gendron, T. M.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1607.03027

Abstract

This is the first of a series of two papers in which we present a solution to Manin's Real Multiplication program -- an approach to Hilbert's 12th problem for real quadratic extensions of ℚ -- in positive characteristic, using quantum analogs of the exponential function and the modular invariant. In this first paper, we treat the problem of Hilbert class field generation. If k=\mathbbF_q(T) and k_∞ is the analytic completion of k, we introduce the quantum modular invariant j^{\rm qt}: k_∞\multimap k_∞ as a multivalued, modular invariant function. Then if K=k(f)⊂ k_∞ is a real quadratic extension of k where f is a quadratic unit, we show that the Hilbert class field H_O_K (associated to O_K= integral closure of \mathbbF_q[T] in K) is generated over K by the product of the multivalues of j^{\rm qt}(f).

Related

• Quantum Drinfeld Modules and Ray Class Fields of Real Quadratic Global Function Fields
• Quantum arithmetic of Drinfeld modules
• Simple modules over finite quantum groups and their Drinfel'd doubles
• Quantum modular forms and singular combinatorial series with repeated roots of unity
• Quantum Modular Forms from Real Quadratic Double Sums