A series of lectures will be given by the ESI Junior Research Fellow Abhiram M Kidambi (MPI MiSc, Leipzig) at the following dates:

Monday, April 8, 2024, 15:00 - 16:30
Wednesday, April 10, 2024, 15:00 - 16:30
Monday, April 15, 2024, 15:00 - 16:30
Wednesday, April 17, 2024, 15:00 - 16:30
Monday, April 22, 2024, 15:00 - 16:30
Wednesday April 24, 2024, 15:00 - 16:30
Friday, April 26, 2024, 15:00 - 16:30
Monday, April 29, 2024, 15:00 - 16:30

The following topics will be considered:

1. Elliptic functions
2. Elliptic curves
3. Modular forms (Automorphic forms for SL(2,Z))
4. Generalizations of modular forms to abelian varieties
5. L-functions associated to modular forms, Dirichlet characters and elliptic curves
6. Applications in mathematical problems: Modularity Theorem, Riemann Hypothesis, the Birch & Swinnerton-Dyer conjectures (and  very brief overview of the Langlands program)
7. Applications in physics
8. Applications in computing and cryptography

Abstract:

This is an introductory course on the theory of automorphic forms and L-functions. These objects play a central role in modern mathematics and are crucial to making progress towards some of the biggest problems in mathematics. For most of the course, we will deal with the simplest automorphic forms viz., modular forms. The course will begin with introducing the theory of elliptic functions and elliptic curves. We will then see how modular forms arise from the theory of elliptic functions. We will then study the key properties of modular forms (the structure of spaces, algebra etc). We will then consider generalizations of modular forms to other abelian varieties. We shall then turn focus to L-functions which we motivate through the theory of Hecke eigenforms. L-functions serve as a bridge between modular forms and arithmetic. Following study of some fundamental properties of L-functions, We shall then consider two kinds of L-functions (the Dirichlet L-functions and L-functions of elliptic curves).

For further details:

www.esi.ac.at/events/e552/