Applications of the theory of elliptic functions in number theory

Abstract

The aim of this thesis is to present some striking applications of Number Theory, essentially based on the powerful machinery of Elliptic Modular Functions and Class Field Theory. One of these applications is the explicit determination of all imaginary quadratic fields with class-number one, famous as the 10th discriminant problem. In my discussion of this problem, I have followed the work of K. Heegner and others, based on the results of H. Weber found in his "Lehrbuch der Algebra". I have presented the results of Weber adopting up to date methods, since Weber's proofs were rather computational using complicated, lengthily presented properties of theta functions. For this purpose I have been rescued by Group Theory, which has been used throughout to prove the critical results of Weber. Thus, I have shortened the ground work which I needed for further exploration. A second application very closely related to the above is the identification of elliptic curves with infinitely many rational points, or, what is essentially the same thing, cubic equations with infinitely many rational solutions. In the first part of this work I have provided a systematic development of a pertinent background for the objectives outlined above.