关于复上半平面的封闭测地线的代数方程及其与Hilbert 12问题的联系

The well known theory of Complex Multipliction has completely answered the question about how to give an explicite construction of abelian class fields of imaginary quadratic number fields. Now the conjectures proposed(e.g. Stark's Conjecture) and some world-famous mathematisians' work(H. Darmon, Y. Manin etc.)have point out that for other number fields, e.g. real quadratic fields, their abelian class fields can also be generated by values of some special functions which is similar to analytic j-functions appeared in the theory of Complex Multiplication. In last decade, thanks to a lot of attempt which has been made on this naturally appeared problem(the so-called "Real Multiplication"): How to generlise the theory of Complex Multiplication to general number fields, we have known some good and deep results.The difficulties in our research is obvious, for every ideal class in a real quadratic field, it's impossible to find an useful corresponding object like elliptic curves or modular curves as in the Complex Multiplication case.In our project, to develop the theory of Real Multiplication for real quadratic fields, we will try to study the closed geodesics on upper half plane, understand their algebraic properties, and use the integration method to define their invariants,finally get the explicit construction of class fields of real quadratic number fields. Any progress on the theory of Real Multiplication will be very helpful for us to understand some deep results and conjectures, for example, the unsolved Gauss Conjecture for real quadratic fields---one of the most famous conjectures in the history of mathematics.

已知的复乘理论很好地回答了虚二次代数数域的abel类域的具体构造的问题。而现有的猜想（比如Stark猜想）和一些著名的数学家的工作(比如H.Darmon和Y.Manin)则指出，对于其他的代数数域，比如实二次数域，它们的abel类域也可以由类似解析函数的一些特殊的函数的函数值生成。近年来，关于如何把复乘理论推广到一般数域的问题，即所谓的实乘理论，国外进行了大量的这方面的尝试，并且得到一些非常深刻的结果。此类研究的难点显而易见，对于实二次域中的理想类，将无法在传统的代数几何范畴中找到类似于椭圆曲线和模曲线那样满意的对象与之对应。在本项目中，我们将通过研究复上半平面上的封闭测地线的代数几何性质，并用积分方法构造其不变量，最终将其用于实二次代数数域的类域的具体构造。关于实乘理论方面的研究，将非常有利地增加人们对于数论中已知定理和一些非常著名的猜想的理解，比如类域论和关于实二次域的Gauss猜想。

类域论是代数数论的一个重要方向，其中如何具体的构造出abel数域的类域一直是这个方向重要的课题之一，本项目主要通过与已经解决的虚二次域（CM域）的情形的类比，进一步寻求实二次域的类域的具体构造问题，同时研究相关的丢番图方程的解和解析函数的数论性质。主要的阶段性的结果有：（1）使用非交换代数几何的理论得到了与经典复乘类似的几何结构---1维非交换环面，但是完全没能找到其与类域论的联系；（2）完全解决了一个相关的丢番图方程的通解的问题；（3）研究了一个与割圆域有关的zeta-函数，并且利用解析的工具得到了其一部分的性质，这部分研究尚在进行和整理中。. 本项目是一个数论与代数几何，复分析相结合的一个课题，上述问题的解决讲有利于促进这三个数学分支的交叉发展。