AIS Algebraic Number Theory (2014)

Venue: Chennai Mathematical Institute , Tamil Nadu
Dates: 7th- 26st July,  2014


Convener(s) Speakers, Syllabus and Time table Applicants/Participants


School Convener(s)

Name Purusottam Rath M. Ram Murty
Mailing Address Associate Professor,
H1,Sipcot It Park, Siruseri,
Tamil Nadu – 603103
Department of Mathematics, Jeffery Hall,
99 University Avenue, Queen’s University,
Kingston, Ontario, K7L 3N6,


Speakers and Syllabus 

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.


  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.


  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.


  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.


  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.



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


  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


Selected Applicants


Serial SID Full Name Gender Affiliation City, State Position University /Institute M.Sc./ M.A. Year of Passing M.Sc. /M.A Ph.D. Degree Date
1 5134 Ms Akshaa Vatwani Female Queen's University, Kingston, Canada Kingston PhD IISER Pune 2012  
2 5143 Mr Manjil Pratim Saikia Male Tezpur University Tezpur, Assam Student Tezpur University Appeared /Awaiting Result  
3 5188 Mr Sumit Kumar Male IMSc Chennai, Tamil Nadu JRF Indian Institute of Science, Bangalore 2013  
4 5202 Mr. Abhash Kumar Jha Male NISER, Bhubaneswar Bhubaneswar, Orissa Ph.D. Banaras Hindu University, Varanasi, Uttar Pradesh 2010  
5 5402 Mr Bharath Kumar Ethamakula Male Indian Institute of Science, Bangalore Bangalore, Karnataka PhD IITKanpur 2011  
6 5492 Ms Moni Kumari Female NISER, BBSR Bhubaneswar, Orissa Ph.d BHU 2012  
7 5546 Mr. Arvind Kumar Male Harish Chandra Research Institute Allahabad, Uttar Pradesh Student BHU 2012  
8 5598 Ms. Satavisha Dey Female IIT, Bombay Mumbai, Maharashtra M.Sc Student IIT Bombay Appeared /Awaiting Result  
9 5616 Mr. Neeraj Sangwan Male IISER Mohali Mohali, Punjab Integrated MS student IISER Mohali Appeared /Awaiting Result  
10 5650 Dr. Jaban Meher Male Indian Institute of Science Bangalore, Karnataka Inspire faculty Utkal University 2006 25 Jul 2012
11 5682 Mr. Anup Kumar Singh Male Harish Chandra Research Institute Allahabad, Uttar Pradesh PhD BHU 2012  
12 5768 Mr Nabin Kumar Meher Male HRI, Allahabad Allahabad, Uttar Pradesh PhD Student IIT KANPUR 2011  
13 5871 Mr Sudhir Kumar Pujahari Male IISER-Pune Pune, Maharashtra SRF (CSIR) Utkal University 2008  
14 5873 Mr. Ankur Singh Male ISM Dhanbad Dhanbad, Jharkhand Doctoral Student IIT Madras 2013  
15 5928 Mr Kamalakshya Mahatab Male The Institute of Mathematical Sciences Chennai, Tamil Nadu PhD, SRF The Institute of Mathematical Sciences 2012  
16 5946 Ms Ritika Sharma Female Harish Chandra Research Institute Allahabad, Uttar Pradesh Ph.D Student IIT Guwahati 2012  
17 5947 Mr. Rakesh Ramlal Barai Male Guru Nanak Khalsa College , Mumbai Mumbai, Maharashtra Assistant Professor University of Mumbai 2001  
18 5948 Ms. Siddhi Sudhir Pathak Female Chennai Mathematical Institute Chennai, Tamil Nadu BSc 3rd year Student   Appeared /Awaiting Result  
19 5957 Mr. Abhishek Bharat Oswal Male Chennai Mathematical Institute Chennai , Tamil Nadu B.Sc. Student   Appeared /Awaiting Result  
20 5984 Mr Sai Mali Ananthanarayanan Male IIT Madras Chennai, Tamil Nadu B Tech & M Tech student IIT Madras M Tech Appeared /Awaiting Result  
21 5996 Mr Anand Parameswaran Male Vivekananda College/Madras University Chennai, Tamil Nadu Msc student Vivekananda College/ Madras University Appeared /Awaiting Result  
22 6078 Mr Mallesham K Male Harish Chandra Research Institute Allahabad, Uttar Pradesh Phd student IIT Madras 2011  
23 6105 Mr Adithya Murali Male BITS Pilani, Hyderabad Campus Hyderabad, Andhra Pradesh (Int) MSc (Hons) Student   Appeared /Awaiting Result  
24 6107 Mr. Mihir Sheth Male IISER Pune Pune, Maharashtra 5th year BS - MS Student IISER Pune Appeared /Awaiting Result  
25 6153 Mr Manish Kumar Pandey Male Harish Chandra Research Institute Allahabad, Uttar Pradesh PhD ISI 2011  
26 6168 Mr. Abhishek Thyagarajan Bharadwaj Male Vijaya Degree College Bangalore, Karnataka Student Vijaya Degree College, Bangalore University 2013  
27 6188 Mr. Soumyarup Banerjee Male Harish-Chandra Research Institute Allahabad, Uttar Pradesh PhD University of Hyderabad 2013  
28 6203 Mr. Pranabesh Das Male Indian Statistical Institute, New Delhi New Delhi, Research Scholar Ramakrishna Mission Vivekananda University 2011  
29 6204 Mr Sameer R Kulkarni Male IISER Pune Poona, Maharashtra Undergraduate student IISER Pune Appeared /Awaiting Result  
30 6222 Mr Sagar Balasaheb Kalane Male IIT Madras Chennai, Tamil Nadu M.Sc IIT MADRAS Appeared /Awaiting Result  


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