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

L. Demangos, T. M. Gendron

• 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 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=Fq(T) and k∞ is the analytic completion of k, we introduce the quantum modular invariant jqt:k∞⊸k∞ as a multivalued, discontinuous modular invariant function. Then if K=k(f)⊂k∞ is a real quadratic extension of k and f is a fundamental unit, we show that the Hilbert class field HOK (associated to OK= integral closure of Fq[T] in K) is generated over K by the product of the multivalues of jqt(f).