Subject Code: MA7L030 Subject Name:  THEORY OF MODULAR FORMS L-T-P: 3-1-0 Credit:4
Pre-requisite(s): Complex Analysis or equivalent
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.
Text/Reference Books:
  1. J-P. Serre, A course in arithmetic, Springer-Verlag, 1973.
  2. N. Kolblitz, Introduction to Elliptic curves and Modular forms, Springer-Verlag, 1984.
  3. M. Ram Murty, M. Dewar and H. Graves, Problems in the theory of modular forms, IMSc Lecture Notes in Mathematics-1, Hindustan Book Agency, 2016.
  4. F. Diamond and J. Shurman, A first course in Modular forms, GTM 228, Springer- Verlag, 2005.
  5. T. Miyake, Modular forms, Springer, 2006.
  6. G. Shimura, Introduction to arithmetic theory of automorphic functions, Princeton University Press, 1971.
  7. H. Cohen and F. Stromberg, Modular forms: A classical approach, GTM 179, American Mathematical Society, 2017.
  8. H. Iwaniec, Topics in Classical Automorphic forms, GSM 17, AMS, 1997.
  9. S. Lang, Introduction to Modular forms, Springer, 1976.
  10. A. O. L. Atkin and J. Lehner, Hecke operators on \Gamma_0(M), Math. Ann. 185 (1970), 134-160.
  11. W-C. W. Li, Newforms and functional equations, Math. Ann. 212 (1975), 285–315.