AIS Algebraic Number Theory (2014)

Overview
The goal of this instructional school is to cover some of the salient features, both algebraic as well analytic, of algebraic number fields which every beginning researcher of number theory should be aware of. We shall also try to highlight some of the arithmetic aspects as well as indicate some open unresolved questions that can be raised even within the realms of such an introductory school.
Here is a brief description of the topics along with the speakers. The exact schedule of lectures will be announced later.

Module-1: Basic Commutative Algebra Prerequisites
(Four lectures)

1.  Integral extensions
2. Dedekind Domains
3. Discrete Valuation Rings
4. Orders
5. The notion of Picard Groups
6. Krull-Akizuki theorem.

Speakers: 

1. Krishna Hanumanthu, CMI.
2.  Clare D’Cruz, CMI.

Module-2: Introduction to algebraic number fields
(Seven Lectures)

1.  Norm, Trace, Discriminant, Different, Ramification Theory, De-composition and Inertia Groups.
2.   Geometry of numbers, Finiteness of class number, Some class number computations, Unit theorem, S-Units, Schmidt’s theorem on counting number fields   with fixed degree and bounded discriminant.

Speakers:

1. Kaneenika Sinha, IISER, Pune.
2. Sanoli Gun, IMSc.

Module-3: Arithmetic and Analysis on Quadratic fields
(Eight Lectures)

1.  Quadratic Fields, Binary Quadratic forms, Genus theory, Introduction to Gauss Class number problem.
2. Class number formula, Kronecker Limit formula, Siegel-Klingen Theorem, Brauer-Siegel theorem.

Speakers:

1. D. S. Ramana, HRI.
2. R. Balasubramanian, IMSc.

Module-4: Arithmetic and Analysis on Cyclotomic fields
(Nine Lectures)

1.  Introduction to Cyclotomic Fields, Growth of class numbers,Regular Primes.
2.  Dirichlet characters, Gauss Sums, Quadratic Reciprocity.
3. Cyclotomic Ramachandra Units, CM fields, Notion of p-adic regulator and p-adic L-functions, Leopoldt Conjecture for abelian extensions.

Speakers:

1. R. Thangadurai, HRI.
2. A. Mukhopadhyay, IMSc.
3. Purusottam Rath, CMI.

Module-5: Arithmetic and Analysis on Arbitrary Number fields
(Eight Lectures)

1.  Hecke Characters, Chebotarev Density theorem, Error terms in Chebotarev Density theorem and some arithmetic applications.
2. Introduction to Artin L-functions, Brauer’s theorem and Meromorphicity of Artin L-functions, Introduction to Stark’s conjectures.

Speakers:

1. M. Ram Murty, Queen’s University, Kingston
2. V. Kumar Murty, University of Toronto.

Tutors:

1. Anwesh Ray, CMI
2. Prem Prakash Pandey, CMI
3. Sumit Giri, IMSc
4. Ekata Saha, IMSc
5. Biswajyoti Saha, IMSc
6. A. Vatwani, Queen’s University

How to reach

• Airport: Chennai International Airport (code: MAA)
• Chennai is well-connected by rail from various cities in India.
• CMI is located in the Siruseri IT Park, off Old Mahabalipuram Road, connecting Chennai to Mahabalipuram. It is accessible by taxis from the airport and, additionally, buses from the city. Both the airport and the city are about 30km from the institute.
• Getting to CMI