AIS Class Field Theory (2017)

Venue: Chennai Mathematical Institute,  Chennai, Tamil Nadu
Date:  8th, May 2017 to 27th, May 2017



School Convener(s)

Name Sanoli Gun M. Ram Murty Purusottam Rath
Mailing Address Associate Professor,
IMSc, Chennai
Queen’s University, Canada
Associate Professor,
CMI, Siruseri




Speakers and Syllabus 

speakers ,affiliation and topics covered

sr Name of speaker Affiliation Topic covered
1 Ram Murty QUEEN’S UNIVERSITY, KINGSTON, CANADA An introductory lecture on history of class field theory was done. Appealing to Wiener-Ikehara Tauberian theorem, a proof of Chebotarev was indicated. Also, Artin L-series were introduced leading to Brauer’s theorem and meromorphicity of Artin L series. Then a number of applications of Chebotarev density theorem was discussed, namely to Artin Primitive conjecture and primitive points on Eliptic curves. Finally, a sample of some of the unresolved questions in this set up was discussed.
2 Purusottam Rath CMI, CHENNAI The introductory elements of Global fields were introduced. Idea of decomposition group and ramification theory was covered. Finiteness of class number as well as a complete proof of Unit theorem was given. Finally, Adeles and Ideles were introduced, leading to the proof of strong approximation.
3 D. S. Nagaraj IMSC, CHENNAI The notion of absolute values and valuations were covered. Classification of all absolute values on a number field was done. Hensel’s lemma and its applications was done. Finally, the arithmetic of unramified extensions of local fields was discussed.
4 Anirban Mukhopadhyay IMSC, CHENNAI The notion of Dedekind zeta function was introduced. Further, the idea of L-series associated to cycles was developed leading to their analytic continuation. Finally, a detailed account of the Frobenius density theorem was given.
5 Kumar Murty UNIVERSITY OF TORONTO,CANADA The Kronecker-Weber theorem was introduced and a complete proof using higher ramification groups was done. Norm index computations were done leading to the formulation of Artin Symbol. Class field theory was introduced using the language of cycles. A complete proof of Artin Reciprocity using the language of cycles/modulus was given.
6 Dipendra Prasad TIFR, MUMBAI Building upon the idea of Idele class group, the Artin reciprocity was reformulated in the set up of the absolute pro finite (abelian) completion of number fields. The surjectivity of Artin map in the formulation was indicated. The broad strategy of the mathematics that goes into the Existence theorem was highlighted, the cyclic (Kummer) case was done in reasonable detail. Thereafter,the elements of local theory was introduced and discussed.
7 C. S. Rajan TIFR, MUMBAI The lectures started with concrete question of primes of the form x 2 + my 2 and how it leads to class field theory. Buliding upon this premise, the central role played by Quadratic reciprocity and well Cyclotomic fields was unveiled. There after, the lectures lead to the the formulation and proofs of class field theory. The emphasis was on spilitting of polynomials and primes with a down-to-earth account of the issues involved. Then, Brauer groups were introduced and their link to Class field theory was indicated. Finally, some elements of CM theory was introduced
8 D. Surya Ramana HRI, ALLAHABAD The lectures were built around the theme of analysis on number fields. The goal was to give a complete and comprehensible account of the Chebotarev density theorem, using the more involved notion of Natural density as opposed to the standard Dirichlet density. The counting of ideals in a class was done appealing to the theme of counting lattice points in domains in the Euclidean space which are Lipschitz parametriz-able. All the pre-requisites were recalled and attempt was made to make it as self-contained as possible.


  • Siddhi Pathak: Queen’s Univesity, Canada
  • Arpita Kar: Queen’s Univesity, Canada
  • Anup Dixit: University of Toronto
  • Jyothsnaa S. : IMSc, Chennai
  • Balesh Kumar: IMSc, Chennai
  • Abhishek Bhardwaj: CMI, Chennai
  • Biplab Paul: IMSc, Chennai
  • Gaurav Patil: University of Toronto

Time Table

First Week

Date 09:30 - 11:00 11:30 - 13:00 14:15 - 15:15 15:30-16:30
08.5.17 PR JS BP AB/JS
09.05.17 PR DSN BP/BK JS/SP
10.05.17 AD DSN AK BK/JS
11.05.17 SP DSN KM(V) BK/ JS
12.05.17 SP DSN KM(V) BK/JS
13.05.17 PR KM(V) AD KM(V)


Second Week

Date 09:30 - 11:00 11:30 - 13:00 14:15 - 15:15 15:30-16:30
15.05.17 DP AM DP AM
16.05.17 DP AM DP AM
17.05.17 DP AM DP AM
18.05.17 DP KM(V) AD/SP DP
19.05.17 DP KM (V) AD KM(V)
20.05.17 RM (V) RM (V) AD SP


 Third Week

Date 09:30 - 11:00 11:30 - 13:00 14:15 - 15:15 15:30-16:30
22.05.17 SR CSR CSR SR
23.05.17 SR CSR CSR SR
24.05.17 SR CSR CSR SR
25.05.17 SR CSR CSR SR
26.05.17 SR CSR CSR SR
27.05.17 SR CSR CSR SR
  • CSR - C.S. RAJAN


Actual Participants 


Sr. SID Full name Gender Affiliation Position in College/ University University/Institute M.Sc./ M.A. Year of Passing M.Sc./ M.A Ph.D. Deg. Date
1 7988 Mr. Jaitra Chattopadhyay Male HCRI PhD Ramakrishna Mission Vidyamandira 2015  
2 8001 Dr. Satyanarayana Reddy Arikatla Male Shiv Nadar University Assistant Professor Andhra University Campus 1995 7/31/2012
3 8003 Ms Bidisha Roy Female HCRI JRE in Mathematics IIT madras 2015  
4 8014 Mr. Subha Sarkar Male HCRI PhD IIT Guwahati 2015  
5 8256 Mr Shubham Sinha Male Indian Institute of Science Student      
6 8300 Mr Agniva Dasgupta Male Indian Institute of Science Bsc , 4th year      
7 8341 Mr. Debasish Karmakar Male HCRI PhD Banaras Hindu University 2012  
8 8544 Mr. Devadatta Ganesh Hegde Male IIT Bombay Int MSc Student      
9 8546 Mr. Atharva Dilip Korde Male IIT Bombay Int MSc Student      
10 8566 Mr Shayeef Murshid Male indian statistical institute,kolkata M.Math      
11 8719 Mr. Subham Bhakta Male Chennai Mathematical Institute BSc 3rd year Student      
12 8753 Mr Plawan Das Male UM DAE Centre for Excellence in Basic Sciences, PhD UM DAE Centre for Excellence in Basic Sciences, Mumbai (CBS) 2014  
13 8892 Mr Nandagopal R Male Indian Statistical Institute Bangalore M.Math Student      
14 8933 Mr Vishwas Bhargava Male IIT kanpur Research Associate      
15 8939 Ms. Arpita Kar Female Queen's University MSc Student Queen's University    
16 8940 Mr. Rijul Saini Male Tata Institute of fundamental research PhD Integrated Ph.D. at Tata Institute of fundamental research Awaiting Result  
17 10324 Mr. Praneel Samanta Male UM-DAE CBS, Mumbai Junior Project Assistant UM-DAE CBS 2016  
18 10418 Mr Lalit Vaishya Male HCRI JRF(PHD) Anaras Hindu University 2014  
19 10419 Mr. Rakesh Ramlal Barai Male Guru Nanak Khalsa College, Assistant Professor Mumbai 2001  
20 10428 Ms Sampa Dey Female IIT MADRAS PhD IIT Madras 2014  
21 10430 Mr Sayak Chakrabarty Male Chennai Mathematical Institute BSc student      
22 10480 Mr Mainak Ghosh Male Tata Institute of Fundamental Research Ph.D Student      
23 10697 Mr. Debasish Sadhukhan Male University Of Hyderabad MSC STUDENT University Of Hyderabad Appeared  
24 10786 Ms Ashvni Narayanan Female Indian Statistical Institute, Bangalore BMath 3rd year student      
25 10816 Mr. Kumarmanas Nethil Male Indian Institute of Technology Kanpur B.S. Student      
26 10821 Ms. Sanjana Agarwal Female Indiana university PhD BhopalBhopal 2016  
27 10822 Mr Aranya Lahiri Male Indiana University, Bloomington PhD IISER Bhopal 2014  
28    Aritriya Mukhapadhyay   CMI        
29   Arnab Kundu   CMI        
30   Shravan Patankar   CMI        
31   Soumyadeep Sahu   CMI        
32   Azizul Haque   HRI        
33   Karthik   IMSc        
34   Mr. Damanvir Singh   IISER Mohali        
35   Ms Siddhi Pathak   Kingston, Canada        
36   Anup Dixit   Toronto, Canada        
37   Biplab Paul   IMSc, Chennai        
38   Jyothsnaa S   IMSc Chennai        
39   Pranabendu Darbar   IMSc Chennai        
40   Abhishek Bharadwaj   CMI Chennai        


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