This project concerns mainly algebraic number theory, that is the study of fundamental problems about the integers using algebraic tools. To be more precise, we are interested in aspects related to the Langlands program, which conjecturally relates arithmetic objects, like Galois representations, with their analytic avatar, automorphic forms. It can be considered as a generalization of Wiles​ proof of Fermat​s last theorem.
One direction of research that we will develop in this project treats p-adic aspects of the Langlands program, like the study of p-adic L-functions, p- adic variation and p-adic comparison theorems in families. For example using methods introduced by Peter Scholze to prove new cases of the Eichler-Shimura isomorphisms. These themes are related to modularity results, a central and extremely active area of research in modern number theory. For example, we recently have seen spectacular results like the proof of potential modularity of elliptic curve over totally real or CM fields.
Another direction of research is precisely related to the study of Elliptic curves connected to modularity problems and arithmetic consequences. In this second direction the computational aspect will have a great importance.
We plan to reach the goals of the project through international collaborations between the participants of the project and other researchers in south America and France. Moreover, workshops for Students will be organized in Chile and Uruguay. Finally, this project will help in the organization of a summer school followed by a research conference, that will be organized in Chile.