Title: Ribet's method and beyond

Abstract: Let p be a prime number. In a famous paper from the late 1970s, Ribet was able to construct finite extensions of the pth cyclotomic field with remarkable properties, under a hypothesis on the divisibility of a particular Bernoulli number by that prime p. Interestingly, although his theorem is relatively elementary to state within the world of algebraic number theory, Ribet proved it by passing to geometry and analysis via the theory of modular forms, using remarkable congruences between such objects.

In this talk, I will review Ribet's theorem and his method as well by sketching a version of his proof. I will then describe some variations on his technique in other contexts which can be used as input for other problems in number theory, especially the Bloch--Kato conjecture.