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.
18-04-24

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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 p-3. The Herbrand Ribet Theorem is a refinement of this result, relating a Bernoulli number B_k divisible by p to a corresponding non-trivial component of the p-torsion of the class group of Q(\zeta_p). That the component corresponding to a Bernoulli number B_k which is a p-unit 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 p-Extensions of Q(\mu_p)”. After reframing the problem as one of proving the existence of a mod-p 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 p-adic 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 mod-p reduction of the associated Galois representation that satisfies all the desired properties.