Modular Forms

Code: MA627 | L-T-P-C: 3-0-0-6

Prerequisites: MA521 or equivalent, and MA547 or equivalent

Preamble / Objectives (Optional): The aim of this course is to familiarize students with basic concepts, techniques, and applications of modular forms as well as with some modern results about modular forms and their applications.

Course Content/ Syllabus: Modular group, congruence subgroups, fundamental domains, extended upper half-plane; modular forms of level one, examples, Eisenstein series, some number theoretic applications, valence formula, dimension formula; Hecke operators of level one, Ramanujan's tau function; modular forms of higher level, examples, Hecke operators of higher level, Atkin-Lehner theory, Petersson inner product, newforms, Eigenforms, L-functions and some properties, relation between modular forms and elliptic curves.

Texts:

1. M. Ram Murty, M. Dewar, and H. Graves, Problems in the Theory of Modular Forms, Hindustan Book Agency, 2015.
2. F. Diamond and J. Shurman, A First Course in Modular Forms, GTM, vol. 228, Springer, 2005.

References:

1. N. Koblitz, Introduction to Elliptic Curves and Modular Forms, GTM, vol. 97, Springer, 1993.
2. J. P. Serre, A Course in Arithmetic, GTM, vol. 7, Springer, 1973.