Prerequisite

Complex analysis in one variable

Abstract

This is an undergraduate and first year graduate course on arithmetic of elliptic curves and modular forms with an eye toward generalizations for Calabi-Yau varieties, and in particular Calabi-Yau threefolds. The course will consist of two sub courses. The first one will be dedicated to many well-known topics related to elliptic curves and it is supposed to be in the undergraduate level. A basic knowledge of complex analysis in one variable would be sufficient to follow it. The topics include: Modular and congruence groups, modular forms of a given weight, cusp forms, Eisenstein series, theta series, Weierstrass pi function, elliptic curves in Weierstrass format, elliptic curves as group, rank of elliptic curves, Mordell-Weil theorem, Hecke operators, Fourier expansions, Growth of the coefficients, L-functions of modular forms and elliptic curves, Birch Swinnerton-Dyer conjecture, functional equation of L-functions, Old forms and new forms, modular elliptic curves, Galois representations and modular forms, application to congruent numbers, Arithmetic modularity of elliptic curves and its relation with Fermat's last theorem. The second part of the course will be dedicated to Calabi-Yau threefolds. For this a basic knowledge of algebraic geometry, algebraic topology and complex analysis in several variables is necessary. An attempt to generalize modular forms for Calabi-Yau varieties, has resulted in the author's books ``Modular and automorphic forms & beyond" published in 2022 and ``Gauss-Manin connection in disguise: Calabi-Yau modular forms" published in 2017. These two books will contain the ingredients of the second part of the course. I will try to fill many details which are missing in these books. The origin of this generalization partially comes from many q-expansion computations in theoretical physics and in particular string theory. In one hand we want to collect many classical topics related to elliptic curves, seen as one dimensional compact Calabi-Yau varieties, and (elliptic) modular forms. This includes the arithmetic modularity theorem which relates the $L$-functions of elliptic curves to those of modular forms. On the other hand we have an eye on the generalization of all these topics into the framework of arbitrary dimensional Calabi-Yau varieties and the corresponding Calabi-Yau modular forms. Obs. Each week I will define the content of the course for the next week, in particular, I will determine whether in the next two lectures I am going to talk about the first or the second part of the course.

Lecturer Intro.

Hossein Movasati is an Iranian-Brazilian mathematician who since 2006 has worked at the Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro. He began his mathematical career working on holomorphic foliations and differential equations on complex manifolds, and gradually moved to study Hodge theory and modular forms and the role of these in mathematical physics, and in particular mirror symmetry.