Module-1: Half integral weight modular forms
Abstract: In these lectures we discuss some topics in the theory of modular forms of half-integral weight. After a brief introduction, we discuss the following topics: Hecke operators; Kohnen plus space, Shimura correspondence; and some applications.

Module-2:  Lambda-adic forms
Abstract: Modular forms are holomorphic functions on the upper-half complex plane with many symmetries. These analytic objects have been studied classically for their own sake by several mathematicians and especially by Shimura, and have many applications to number theory problems. On the other hand, Iwasawa studied the variation of the p-part of the class groups of cyclotomic fields algebraically for p a prime. Hida had the deep insight that these two disparate themes in number theory can be combined. He defined and studied p-adic families of modular forms, more formally known as Lambda-adic forms. These lectures will be an introduction to these forms, leading up to the construction of their Galois representations.

Module-4:   Hilbert Modular Forms
Abstract: We will introduce the space of Hilbert Modular forms over totally real fields. We will then go over to study the ring of Hecke operators on them.

Module-5: Under this module Professor Ken Ono of University of Virginia will deliver two lectures.
Lecture 1: Lehmer’s Conjecture and its Generalisations
Abstract: This talk will be an overview of recent works on Lehmer’s Conjecture on the nonvanishing of Ramanujan’s tau-function. There has been much work recently on Lehmer’s Conjecture and its generalization to arbitrary modular newforms. A new approach has been crafted by combining the Chaubauty-Coleman method, Diophantine approximation, Galois representations, and the “modular method”  (i.e. in the spirit of the proof of Fermat’s Last Theorem). For the first time one can identify integers which never arise as Fourier coefficients of modular newforms such as Ramanujan’s tau-function.

(This lecture will be on joint work with Jennifer Balakrishnan, Will Craig, and Wei-Lun Tsai)

Lecture 2: Gauss’ Class Number Problem
Abstract: This talk will be an overview of Gauss’ class number problem for imaginary quadratic fields. For every positive integer h, Gauss Conjectured that there are at most finitely many fundamental discriminants -D for which h(-D)=h. The class number 1 case (i.e. where h=1) is the celebrated work of A. Baker, K. Heegner, and H. Stark. For general h, this problem was not effectively solved until the work of D. Goldfeld, B. Gross, and D. Zagier in the early 1980s. However, their results fall far short of the expected asymptotic properties. In recent work with M. Griffin, the speaker has obtained stronger lower bounds by introducing a method involving “elliptic curve ideal class pairings”. The method makes use of these new pairings, effective results on the number of rational points on elliptic curves with bounded height, and some Diophantine geometry. As a bonus, these ideas lead to new effective lower bounds for the heights of non-torsion points on all rational elliptic curves in the direction of Lang’s Height Conjecture.