DescriptionThe theory of modular forms is one of the most fascinating branches of mathematics with connections to Number Theory, Complex Analysis, Algebraic Geometry, Topology, and even Physics. Despite the importance of modular forms, and their appearence in so many fields of mathematics, it is usually quite challenging to provide explicit examples of them. One of the few available tools for this, is through the theory of Eisenstein series, introduced by the German mathematician Gotthold Eisenstein in the 19th century.The main goal of this project is the study of Eisenstein series with a view towards applications in Number Theory. The project should be of interest to students wishing to learn more about Modular Forms (beyond the material covered in the Elliptic Functions IV course), and explore some connections with other parts of Number Theory. Depending on interest the project may take various directions. The following list is by all means not complete. 1) The Rankin-Selberg method and Eisenstein series, 2) Special values of Hecke L-series and nearly holomorphic Eisenstein series, 3) Half-integral weight Eisenstein series, 4) The first and second Kronecker limit formula, 5) Higher rank Eisenstein series,
ResourcesThere are many reference for Modular Forms that go into a detailed study of Eisenstein series. A good starting point is the book of Miyake [2] and especially Chapter 7, or the first two sections of chapter 5 of the book of Hida [1]. More references will be given at the beginning of the project.
[2] T. Miyake, Modular Forms, Springer Monograps in Mathematics.
Pre-requisites
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