Description：

In the 1976 Ribet proved converse to Herbrand’s theorem. The result states that numerator of a Bernoulli number is divisible by a prime number p if, and only if, a certain eigencomponent of the class group of Q(mu_p) is nontrivial. The methods developed by Ribet have been very influential in number theory in the last 50 years. In this course we will start with the study Ribet’s paper. An important part of the Ribet’s paper is a construct of a nontrivial extension of two one dimensional representations. This is often referred to as Ribet’s lemma. Ribet’s lemma has been vastly generalised. Second part of the course will concentrate on the generalisation due to Bellaiche—Chenevier. The results of Bellaiche—Chenevier only construct extension of residually distinct representations. In a recent joint work with Dasgupta—Silliman—Wang, we removed this restriction in some important cases, leading to the proof of the Equivariant Tamagawa Number Conjecture. The last part of the course will concentrate on this more general result. If time permits we will explore possible generalisations of our result.  

Prerequisite： Algebraic number theory, class field theory, representation theory.

Reference：

1.A modular construction of unramified p-extensions of Q(mu_p), Kenneth Ribet, Invent. Math. Vl 34, (1976).

2.Families of Galois representations and Selmer groups, J. Bellaiche, G. Chenevier, Asterisque, Vol 324, (2009)

3.On constructing extension of residually isomorphic characters, S Dasgupta, Preprint. https://arxiv.org/abs/2310.16631

4.The residually indistinguishable case of Ribet’s methods for GL_2, S Dasgupta, M Kakde, J Silliman, J Wang, Preprint https://arxiv.org/abs/2310.16396

Target Audience：Undergraduate students, Graduate students

Teaching Language: English

Bio: Mahesh Kakde is a professor of mathematics at Indian Institute of Science, Bangalore. He is a visiting faculty at YMSC, Tsinghua University.

Registration：https://v.wjx.cn/vm/Q4kxhAM.aspx