Modularity of Some Geometric Objects - Definitions, Reasons and State of the Art

Explore the intricacies of modularity in geometric objects through this comprehensive mathematics seminar. Delve into the definitions of zeta functions and L-functions, with a particular focus on plane projective curves. Examine the expected properties and conjectural relationships of these functions with analytic counterparts. Gain insights into the crucial role of Galois representations in establishing correspondences between geometric and analytic objects. Progress from introductory concepts to advanced topics, including elliptic curves, modular forms, and higher-dimensional cases such as Calabi-Yau varieties. Conclude with an examination of the Consani-Scholten example, providing a thorough overview of the current state of research in this fascinating area of mathematics.

Intro
A global baby example
General local picture
Global picture
Back to elliptic curves
Modular forms
Elliptic Curves II
Generalizations
Higher dimensional cases: Calabi-Yau varieties
The Consani-Scholten example