Theory of Modular Forms

The modular group and its subgroups, Fundamental domain for SL(2, Z). Modular functions and forms, Examples of modular forms and functions such as Eisenstein series, Poincare series, Ramanujan delta function, j-function, theta functions. Space of modular forms and cusp forms, Valence formula, Dimension formula, Fourier expansions of Eisenstein series at infinity and Fourier coefficients of cusp forms, Ramanujan tau- function and Ramanujan conjectures. Hecke theory (abstract and concrete), Petersson inner product, L-functions attached to a modular form, analytic continuation and converse theorem of Hecke, Rankin-Selberg L-function. Theory of modular forms of higher levels, Atkin-Lehner-Li theory, Applications of modular forms. If time permits Modular forms of half-integral weights, Hilbert modular forms, Siegel modular forms and automorphic forms.