### Modular form congruences and the reducibility of residual modular representations

Linfoot Number Theory Seminar

17th February 2021, 11:00 am – 12:00 pm

Virtual Seminar, https://bristol-ac-uk.zoom.us/j/93536951034

During the late sixties, Serre linked some congruences satisfied by the Fourier coefficients of the Ramanujan Delta function to the existence of some system of mod l-Galois representations. After Deligne and Serre constructed these residual Galois representations for any modular newform, Ribet proved in 1985 that all but finitely many of them were irreducible.

In this talk, I will present my recent work on the reducibility of these residual modular representations and how it is related to congruences between modular newforms and Eisenstein series.

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