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 \(\mathbb Q\)—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 \(k=\mathbb F_{q}(T)\) and \(k_{\infty }\) is the analytic completion of k, we introduce the quantum modular invariant

as a multivalued, discontinuous modular invariant function. Then if \(K=k(f)\subset k_{\infty }\) is a real quadratic extension of k and f is a fundamental unit, we show that the Hilbert class field \(H_{\mathcal {O}_{K}}\) (associated to \(\mathcal {O}_{K}=\) integral closure of \(\mathbb F_{q}[T]\) in K) is generated over K by the product of the multivalues of \(j^\mathrm{qt}(f)\).

这是两篇论文系列中的第一篇，其中我们提出了Manin的实数乘法程序的解决方案（Manin in：Laudal和Piene（eds）Niels的遗产Legacy of Niels Henrik Abel，施普林格，柏林，2004年）-希尔伯特第12版的方法\（\ mathbb Q \）的实二次扩展的正问题，使用模不变式和指数函数的量子模拟。在第一篇论文中，我们讨论了希尔伯特类字段生成的问题。如果\（k = \ mathbb F_ {q}（T）\）和\（k _ {\ infty} \）是k的解析完成，我们引入量子模不变量

作为一个多值，不连续的模块化不变函数。然后，如果\（K = k（f）\子集k _ {\ infty} \）是k的实二次扩展，而f是基本单位，则表明希尔伯特类字段\（H _ {\ mathcal {O} _ {K}} \） （相关联于\（\ mathcal {ö} _ {K} = \）的积分闭合\（\ mathbb F_ {q} [T] \）在ķ上方产生）ķ通过的产物的多值\（j ^ \ mathrm {QT}（F）\） 。