Real Multiplication

Hilbert 12th Problem asks:For any given number field, how to construct explicitly the generators of its abelian extensions? The first unsolved case is about real quadratic fields. Our ideas comes from the progress that already made in this direction, we will mainly focus on the moduli aspect of the so called "Real Multiplication" theory. The structure of this thesis is organized as follows:In chapter one, we recall firstly class field theory, and the two solved cases for explicit class field theory:rational number field and imaginary quadratic fields. We briefly summarize them in unified and generalizable forms. When we try to give a similar geometric context for real quadratic fields, a critical obstacle occurs is that the topological space engaged is not Hausdorff. Yuri Manin introduced Noncommutative Geometry to treat this problem, we recall the main results in this direction, which is part of our ideas in this thesis.In chapter two, using formal power series, we define formal moduli space of real quadratic tori and its affine coordinates ring and strict formal modular forms. Mean-while, we give some examples of formal modular forms, including the so called "formal Eisenstein series", and strict formal modular forms, and give an explicit method for calculating an important example of strict formal modular forms—relative Eisenstein series.In chapter three, we prove that the formal Eisenstein series has a standard q-expansion. The main result of the thesis is that we assign a series of PSL2(Z) to each PSL2(Z)-equivalent class of real quadratic numbers.In chapter four, we recall Darmon's work on the construction of class fields of real quadratic fields via p-adic methods, his work can be viewed as an important evidence for the following conjecture we make:The formal moduli space can be naturally viewed as the spectrum or formal spectrum of some ring of formal power series.In the last chapter, we indicate an independent result, An expression of prime numbers (1(mod 8)) is proved, and it can be used in the computations of K2Q. The proof of this expression can be viewed as a complementary to Minkowski's Theorem in Geometry of Numbers.