Preferably some familiarity with MA 215 (Introduction to Modular Forms) but not necessary.
Holomorphic Modular forms: motivation and introduction, Eisentein series, cusp
forms, Fourier expansion of Poincare series and Petersson trace formula, Hecke
operators and overview of newform theory, Kloosterman sums and bounds for
Fourier coefficients, Automorphic L-functions, Dirichlet-twists and Weil’s
converse theorm, Theta functions and representation by quadratic forms,
Convolution: the Rankin-Selberg method.
(Further topics if time permits: Non-holomorphic modular forms (overview),
Siegel modular forms (introduction), Elliptic curves and cusp forms, spectral
theory, analytic questions related to modular forms.)
Suggested books and references:
J.P. Serre., A Course in Arithmetic
,Springer GTM, 2007.
N.Koblitz., Introduction to Elliptic Curves and Modular Forms
,Springer GTM, 1997.
H. Iwaniec, Topics in Classical Automorphic Forms
,GTM 17, AMS,1997.
F. Diamond and J.Schurman, A First Course in Modular forms
,Springer GTM 228.