**MTH 519: Introduction to Modular Forms (4)**

**Pre-requisites (Desirable)**: MTH 407: Complex Analysis I

*Learning Objectives*:

The aim of this course is to introduce the theory of modular forms with minimal prerequisites. The course is intended for the students who have done the standard courses in Linear Algebra and Complex Analysis. The results and techniques from these courses will be used to understand the space of modular forms and hence the students will solidify their understandings of some basic tools learned throughout mathematics. Numerous examples of modular forms will be given which are useful in solving some classical problems in number theory. The purpose is to make the modular form theory accessible without going into the advanced algebraically oriented treatments of the subject. At the same time this course introduces the topics which are at the forefront of the current research.

*Course Contents*:

- The full modular group SL
_{2}(Z), Congruence subgroups, The upper half-plane H, Action of groups on H, Fundamental domains, The invariant metric on H - Modular forms of integral weight of level one, Eisenstein series, The Ramanujan τ-function, Dedekind η-function, Poincare series, The valence formula and dimension formula, Modular forms of integral weight of higher level
- The Petersson inner product, Hecke operators, Oldforms and newforms, Dirichlet series associated to modular forms: Convergence, Analytic continuation, Functional equation
- (if time permits) Modular forms of half-integral weight: Definition and examples, Hecke operators, Shimura-Shintani correspondences between modular forms of integral weight and half-integral weight.

*Suggested Books*:

- M. Ram Murty, M. Dewar, H. Graves, Problems in the theory of modular forms, Institute of Mathematical Sciences - Lecture Notes 1, Hindustan Book Agency, 2015.
- N. Koblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics 97, Springer, 1993.
- J. P. Serre, A course in arithmetic, Graduate Texts in Mathematics 7, Springer, 1973.
- T. M. Apostol, Modular functions and Dirichlet series in number theory, GTM 41, Springer, 1990.
- H. Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics 17, AMS, 1997.
- F. Diamond and J. Shurman, A first course in modular forms, Graduate Texts in Mathematics 228, Springer, 2005.
- T. Miyake, Modular forms, Springer Monographs in Mathematics, Springer, 2006.
- G. Shimura, Modular forms: basics and beyond, Springer Monographs in Mathematics, Springer, 2012.

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