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.)
Basics of number theory, complex analysis, preferably some familiarity with MA
215 (Introduction to Modular Forms) but not necessary.
A Course in Arithmetic by J.P. Serre. Springer GTM 7.
Introduction to Elliptic Curves and Modular Forms by N.Koblitz. Springer GTM
Topics in Classical Automorphic Forms by H. Iwaniec. GTM 17, AMS,1997.
A First Course in Modular forms, by F. Diamond and J.Schurman. Springer GTM