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Description
This module aims to offer an overview of the basic notions that appear in the classical theory of modular forms. These are analytic objects encoding a lot of arithmetic information which makes them a central point of study in number theory and arithmetic geometry. Modular forms also arise naturally in a variety of other research fields like transcendence proofs, differential equations and mirror symmetry.
The main objects of interest are functions on the complex upper half plane that transform in a special way under the action of SL_2(Z). Two concrete constructions of such functions will be covered: Eisenstein series and theta series. We will also show that the space of modular forms for a specific weight is finite dimensional, which makes all these functions algorithmically computable. Furthermore, we will introduce Hecke operators and show that the L-functions associated to eigenfunctions are of arithmetic nature. Throughout the module, several applications to congruences and positive definite quadratic forms will be highlighted.
Module deliveries for 2024/25 academic year
Last updated
This module description was last updated on 8th April 2024.
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