Newforms and congruence graphs

The theory of congruences of 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 Toronto), we are constructing graphs encoding congruence relations between classical newforms. These graphs have extremely interesting features: they help our understanding on 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 and generalizations.