The first half of this course (by Dr. Xiaoyu Zhang) will be a continuation of the course Modular Form 1 of the last Winter semester. We will continue to introduce some basic notions and results in the theory of modular forms on the upper half plane, in particular Hecke operators, Petersson products, oldforms and newforms (the theory of Atkin-Lehner), the Eichler-Shimura isomorphism. After that, we will study the algebraicity of certain Dirichlet L-functions via Eisenstein series, certain standard L-functions of modular forms and then pass to the construction of p-adic interpolations of these L-functions using modular symbols. If times permits, we will study the p-adic interpolation of certain families of modular forms (the Hida theory) and briefly discuss some of its applications.

        In the second half (by Dr. Jie Lin), we will talk about automorphic forms for GL_2 over the real field and over the ring of adeles. We will start from Mass forms, their Fourier-Whittaker expansions, and the spectrum. We will then talk about the ring of adeles and the automorphic forms over it. In particular, we will discuss the adelic lifting of classical forms. In the end, we will introduce the theory of automorphic representations.

References:

        [1] F. Diamond and J. Shurman, A first course in modular forms, GTM 228, Springer, 2005.

        [2] H. Hida, Elementary theory of L-functions and Eisenstein series, London Mathematical Society Student Texts 26, 1993.

        [3] D. Bump, Automorphic forms and representations, CSAM 55, Cambridge University Press, 1997.

        [4] D. Goldfeld and J. Hundley, Automorphic Representations and L-functions for the General Linear Group, CSAM 128, Cambridge University Press, 2011.

 Prerequisites: Complex analysis, Modular form 1 (basic definitions, Hecke operators, basic of L-functions), Complex elliptic curves