• Corpus ID: 14292849

MODULAR CURVES AND THE CLASS NUMBER ONE PROBLEM

@inproceedings{Booher2014MODULARCA,
  title={MODULAR CURVES AND THE CLASS NUMBER ONE PROBLEM},
  author={Jeremy Booher},
  year={2014}
}

• J. Booher
• Published 2014
• Mathematics

There are several approaches. Heegner [9] gave a proof in 1952 using the theory of modular functions and complex multiplication. It was dismissed since there were gaps in Heegner’s paper and the work of Weber [18] on which it was based. In 1967 Stark gave a correct proof [16], and then noticed that Heegner’s proof was essentially correct and in fact equivalent to his own. Also in 1967, Baker gave a proof using lower bounds for linear forms in logarithms [1]. Later, Serre [14] gave a new approach based on modular curve, reducing the class number one problem to finding special points on the modular curve X ns(n). For certain values of n, it is feasible to find all of these points. He remarks that when “N = 24 An elliptic curve is obtained. This is the level considered in effect by Heegner.” Serre says nothing more, and later writers only repeat this comment. This essay will present Heegner’s argument, as modernized in Cox [7], then explain Serre’s strategy. We will work out unrecorded details of the argument for level 24, find the elliptic curve, and relate it to Heegner’s original argument. To answer the class number one problem, it is easier to generalize it to arbitrary orders in imaginary quadratic fields. In the first section, we do this and recall a few classical results about the class number of orders. Section 2 develops theory about the j invariant and the class equation. Section 3 then studies other modular functions, especially Weber’s modular functions, which are the building block’s of Heegner’s argument. Next we use Weber’s functions to calculate the j-invariants of all known imaginary quadratic fields of class number one so we can identify them based on their j invariant. Section 5 relates these modular functions to the class number using class fields and theory of complex multiplication. Heegner’s argument is presented in section 6. Changing direction, the next section deals with the question of when modular curves are defined over the rationals, and section 8 describes the connection Serre found between X ns(n) and the class number one problem. Finally, section 9 solves the class number one problem using X ns(24) and relates it to Heegner’s argument. Solving the class number one problem requires a lot of background material. Things on the level of Part III at Cambridge University will be used without proof and sometimes without explicit mention. In particular, knowledge of basic algebraic number theory, the definitions of global class field theory, the basic theory theory of modular forms and functions, and the basic theory of elliptic curves over C will be essential. Slightly less basic but very important