Conference abstracts

Session A3 - Computational Number Theory

July 12, 14:50 ~ 15:30 - Room B6

Congruences, graphs and modular forms

Samuele Anni

University of Heidelberg, Germany   -   samuele.anni@gmail.com

The theory of congruences between modular forms is a central topic in contemporary number theory, lying at the basis of the proof of Mazur's theorem on torsion in elliptic curves, Fermat's Last Theorem, and Sato-Tate, amongst others. Congruences are a display of the interplay between geometry and arithmetic. In order to study them, in a joint work with Vandita Patel (University of Warwick), we are constructing graphs encoding congruence relations between newforms. These graphs have extremely interesting features: they help our understanding of the structure of Hecke algebras, and they are also a new tool in the study of numerous conjectures. In this talk I will describe these new objects, show examples and explain some of the possible applications.

Joint work with Vandita Patel (University of Warwick).

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