M.Sc. Thesis Defence Speaker: Atharva Raje, CMI Date: Thursday, 18 April 2024 Time: 9:20 AM Venue: Seminar Hall Ribet’s converse to Herbrand’s Theorem Atharva Raje Chennai Mathematical Institute. 180424 Abstract In 1850, Kummer proved that an odd prime p is irregular if and only if p divides the k^(th) Bernoulli number B_k for some even k between 2 and p3. The Herbrand Ribet Theorem is a refinement of this result, relating a Bernoulli number B_k divisible by p to a corresponding nontrivial component of the ptorsion of the class group of Q(\zeta_p). That the component corresponding to a Bernoulli number B_k which is a punit is trivial was proved by Herbrand in 1932. In this talk, we shall examine the proof of the converse result, proved by Ribet in his 1974 paper “A Modular Construction of Unramified pExtensions of Q(\mu_p)”. After reframing the problem as one of proving the existence of a modp Galois representation satisfying certain properties, we seek to find such a representation using the theory of modular forms. More specifically, by a construction due to Shimura, there is a padic Galois representation associated to every newform of weight 2 that is unramified at almost all primes. After covering some necessary background on modular forms and Hecke Operators, we shall examine the construction of a suitable normalised cusp eigenform. Thereafter, we shall prove that there is a modp reduction of the associated Galois representation that satisfies all the desired properties.
