This is a geometrically flavoured introduction to the theory of modular forms. We will start with a standard introduction to some basic analytic aspects concerning modular forms and to their interpretation as sections of line bundles on modular curves.
Then, our main goal will be to explain how one can attach certain 2-dimensional cohomology groups to Hecke eigenforms. In this course, we will only deal with algebraic de Rham and Betti cohomology, but this can also serve to build geometric intuition on the l-adic setting, which gives rise to the famous l-adic representations attached to modular forms.
We will finish with a discussion on the Eichler-Shimura isomorphism, periods of modular forms, and, depending on time, Manin’s theorem on the critical values of L-functions of modular forms.
This video is part of the London Mathematical Society‘s Online Graduate Lecture Series. These are supported by the LMS, and organized jointly by the ROW Diophantine Geometry Seminar, the ERLASS Arithmetic Statistics Seminar, and the Northern Number Theory Seminar.
