Analytic Methods in Algebraic Number Theory

Convener(s)

 
Name:  Dr. Ekata Saha  Dr. Biswajyoti Saha
Mailing Address: Assistant Professor
Department of Mathematics
Indian Institute of Technology Delhi		 Assistant Professor Department of Mathematics
Indian Institute of Technology Delhi
Email:  ekata at maths.iitd.ac.in  biswajyoti at maths.iitd.ac.in

In number theory, we often see various kinds of methods employed to answer questions arising in a seemingly different context. The aim of this advanced instructional school is to introduce the participants to such an instance. To be precise, we shall discuss analytic methods in algebraic number theory, leading to the understanding of class number of imaginary quadratic fields, or more generally, CM fields, and also for almost normal number fields and their generalisations, by means of Siegel’s theorem, Brauer-Siegel theorem, and effective Brauer-Siegel theorem.

Dates: 

Monday, December 4, 2023 - 09:00 to Saturday, December 16, 2023 - 18:00

Venue: 

Venue Address: 

 

Indian Institute of Technology Delhi Hauz Khas, New Delhi 110016.

 

 

Venue State: 

Venue City: 

PIN: 

110016

Syllabus: 

Name of the Speaker with affiliation

No. of Lectures

Detailed Syllabus

Dr. Ekata Saha,
IIT Delhi, New Delhi

3

Prerequisites from algebraic number theory: Integral basis, Dirichlet unit theorem, class group and class number, discriminant, different

Dr. Sneha Chaubey,
IIIT Delhi, New Delhi

3

Prerequisites from analytic number theory: Riemann zeta function, Dirichlet L-functions, Euler product, functional equations, zero-free regions

Dr. Biswajyoti Saha,
IIT Delhi, New Delhi

2

Prerequisites from representation theory of groups: Character of a representation, regular representation, Frobenius reciprocity, Artin’s theorem, p-elementary subgroups, Brauer’s theorem.

Prof. Sanoli Gun,
IMSc, Chennai

4

Theory of Dedekind zeta function: Dedekind zeta function, functional equation, analytic class number formula, asymptotic distribution of ideals

Prof. M. Ram Murty,
Queen’s University, Canada

4

TheoryofArtinL-series:ArtinL-series,Artin’sconjecture,Aramata- Brauer theorem, Artin symbol, Artin conductor, meromorphiccontinuation

Prof. Purusottam Rath,
CMI, Chennai

4

Siegel’s theorem and Brauer’s extension: Siegel’s theorem, ineffectivity in Siegel’s theorem, Brauer’s extension of Siegel’s theorem

Prof. V. Kumar Murty,
University of Toronto, Canada

4

Effective Brauer-Siegel theorem: Explicit formulas, instances of effective Brauer-Siegel theorem andapplications

 

References:

  1. Serge Lang, Algebraic Number Theory, Springer New York,NY.

  2. Jean-Pierre Serre, Linear Representations of Finite Groups, Springer New York,NY.

  3. Harold Davenport, Multiplicative Number Theory, Springer New York,NY.

  4. Lawrence C. Washington, Introduction to Cyclotomic Fields, Springer New York,NY.

  5. Jürgen Neukirch, Algebraic Number Theory, Springer Berlin,Heidelberg.

  6. Harold Stark, Some effective cases of the Brauer-Siegel Theorem. Invent Math 23,135-152.

  7. V. Kumar Murty, Class numbers of CM-fields with solvable normal closure, Compositio Math. 127,273-287.

 

Time Table: 

 

Time-Table (with names of speakers and course associates/tutors):

Day

Date

Lecture 1

(9.30–11.00)

Tea (11.05
to
11.25)

Lecture 2

(11.30–1.00)

Lunch (1.05
to
2.25)

Tutorial (2.30–3.30)

Tea
(3.35
to
3.55)

Tutorial (4.00-5.00)

Snacks (5.05
to
5.30)

 

 

(name of the speaker)

 

(name of the speaker)

 

(name of the speaker + tutors)

 

(name of the speaker + tutors)

 

Mon

4/12/23

ES

 

SC

 

SC, BM, PSM

 

ES, AG, DK

 

Tues

5/12/23

ES

 

SC

 

SC, BM, PSM

 

ES, AG, DK

 

Wed

6/12/23

ES

 

SC

 

SC, BM, PSM

 

ES, AG, DK

 

Thu

7/12/23

BS

 

SG

 

BS, AG, BM

 

SG, DK, DS

 

Fri

8/12/23

BS

 

SG

 

BS, AG, BM

 

SG, DK, DS

 

Sat

9/12/23

RM

 

PR

 

RM, RL, SH

 

PR, P, PSM

 

SUNDAY: OFF

Mon

11/12/23

RM

 

SG

 

RM, RL, SH

 

SG, DK, DS

 

Tues

12/12/23

PR

 

SG

 

PR, P, PSM

 

SG, DK, DS

 

Wed

13/12/23

PR

 

KM

 

PR, P, SH

 

KM, BS, RL

 

Thu

14/12/23

RM

 

KM

 

RM, DS, SH

 

KM, BS, RL

 

Fri

15/12/23

RM

 

KM

 

RM, DS, SH

 

KM, BS, RL

 

Sat

16/12/23

PR

 

KM

 

PR, P, SH

 

KM, BS, RL

 

 

Tutorial Assistants:

S. No.

Name

Affiliation

1

Akanksha Gupta

IIT Delhi, New Delhi

2

Bikram Misra

IIT Delhi, New Delhi

3

Divyanshu Kala

IIT Delhi, New Delhi

4

Dhananjaya Sahu

IMSc, Chennai

5

Pawan Singh Mehta

IIT Delhi, New Delhi

6

Preeti

CMI, Chennai

7

Rashi Lunia

IMSc, Chennai

8

Suhita Hazra

CMI, Chennai

Full forms for the abbreviations of speakers and tutors:

BS:

Biswajyoti Saha

AG:

Akanksha Gupta

ES:

Ekata Saha

BM:

Bikram Misra

KM:

V. Kumar Murty

DK:

Divyanshu Kala

PR:

Purusottam Rath

DS:

Dhananjaya Sahu

RM:

M. Ram Murty

PSM:

Pawan Singh Mehta

SC:

Sneha Chaubey

P:

Preeti

SG:

Sanoli Gun

RL:

Rashi Lunia

 

 

SH:

Suhita Hazra

Selected Applicants: 

 

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
51344 Ms. Himanshi Chanana Female IIT Kanpur PhD IIT Ropar 2019 14 Feb 2022
51388 Ms Aparna Tripathi Female Harish-Chandra Research Institute. PhD University of Delhi 2018  
51407 Mrs. Raveena Ganash Female National Institute of Science, Education and Research Research Scholar (JRF) Jawaharlal Nehru UNiversity 2022  
51442 Mr. Kushal Bhowmick Male Harish Chandra Research Institute PhD Student Indian Statistical Institute, Kolkata 2016 3 Jul 2022
51673 Mr Oais Ahmad Bhat Male Harish-Chandra Research Institute Prayagraj Ph.D Univerity of Kashmir 2020  
51737 Ms. Sudipa Das Female Harish-Chandra Research Institute PhD RKMVERI 2021  
51795 Mr. Aviral Srivastava Male IIT Gandhinagar Phd University of Hyderabad,M.Sc. 2022  
51797 Mr Archit Agarwal Male IIT Indore PhD IIT Gandhinagar 2018  
51798 Ms. Meghali Garg Female IIT Indore Ph.D. IIT Gandhinagar 2019  
51820 Ms Meenu Meenu Female Kerala School of Mathematics PhD Student Central University of Rajasthan, Rajasthan 2021  
51882 Ms Srijonee Shabnam Chaudhury Female Harish-Chandra Research Institute, Allahabad PhD Student Jadavpur University    
51904 Mr Jagannath Sahoo Male IIT Gandhinagar PhD Scholar University of Hyderabad 2020 15 May 2023
51906 Mr. Guru Sharan N Male Indian Institute of Technology PhD University of Hyderabad 2020  
52148 Ms H Laxmi Female Indian Institute of Technology Guwahati Phd Scholar Indian Institue of Technology Guwahati 2020  
52291 Ms Amrutha C Female SRM University AP PhD Central University of Kerala 2020  
52301 Mr. Sushant Kala Male The Institute of Mathematical Sciences, Chennai SRF IIT Ropar 2020  
52351 Ms. Diksha Rani Female Indian Institute of Technology Indore PhD Indian Institute of Technology Indore 2022  
52434 Mr. Vinodkumar Rajlingappa Ghale Male BITS-Pilani, Hyderabad Campus Ph.D. student University Of Hyderabad 2016  
52510 Mr Jay Ram Thakur Male IISER BHOPAL PH.D University of Delhi 2020  
52595 Mr. Jewel Mahajan Male IISER Pune PhD Student University of Calcutta (Pure Mathematics) 2018  
52596 Mr. Jishu Das Male IISER Pune PhD Student IISER Pune 2018  
53045 Mr. Vaibhav Kalia Male INDIAN INSTITUTE OF TECHNOLOGY ROPAR PhD student Motilal Nehru National Institute of Technology Allahabad 2018  
53072 Ms. Amrinder Kaur Female University of Hyderabad PhD Thapar University 2017  
53195 Mr. Sandeep E M Male NISER Bhubaneswar Post Doctoral Fellow IISER Thiruvananthapuram 2015 14 Feb 2022
53325 Mr. Suraj Panigrahy Male Indian Institute of Technology Bombay PhD Indian Institute of Technology Guwahati 2020  
53364 Mr Ravi Kalwaniya Male Indian institute of Technology Madras PhD IIT KGP 2022  
53435 Mr. Prashant Tiwari Male HRI Prayagraj Post Doctoral Fellow M.S. 2018 3 Jul 2022
52422 Mr. Subham Kundu Male Indian Institute of Science Education and Research Bhopal Integrated PhD Student Indian Institute of Science Education and Research Bhopal Appeared / Awaiting Result 14 Oct 2019
51755 Mr. Subham Garai Male IIT Kanpur PhD IIT Kanpur 2023  
52406 Mr. Ansh Agarwal Male IIT Guwahati PhD IIT Hyderabad 2022  
Local participants
51479 Mr. Santak Panda Male Indian Institute of Technology Delhi Dual Degree Student MTech Appeared / Awaiting Result  
51816 Ms. Shivani Goel Female Indraprastha Institute of Information Technology, Delhi PhD Chaudhary Charan Singh University, Meerut 2018  
51917 Dr. Anubhav Sharma Male IIT Bombay Post doctoral fellow University of Delhi 2018 15 May 2023
51945 Mr. Sarthak Gupta Male Ashoka University Teaching Fellow ALGANT Masters: Leiden University and University of Bordeaux 2022  
51991 Ms Khyati Female Indraprastha Institute of Information and Technology, Delhi Research scholar Banaras Hindu University 2018  
52042 Ms. Kritika Aggarwal Female Indraprastha Institute of Information Technology, Delhi PhD student M.Sc. Mathematics, Hindu College, University of Delhi 2021  
52548 Mr. Prabhakar Ratipal Yadav Male Indian Statistical Institute (ISI), Delhi Centre PhD Banaras Hindu University 2019  
52614 Mr. Bittu Male IIIT Delhi PhD Indian Institute of Technology Roorkee 2019  
52635 Mrs. Bina Jha Female Harish-Chandra Reasearch Institute PhD National institute of technology Rourkela 2020  
52668 Mr. Himanshu Rewri Male IIT Delhi PhD MSc 2019  
53087 Ms. Saheli Dutta Female Indian Statistical Institute, Delhi Centre. PhD IIT Bombay 2020  
Waiting List of Candidates (as per order)
52760 Mr. Adhip Ganguly Male IIT Guwahati PhD student University of Delhi M.Sc 2021  
52569 Mr. Sanskar Agrawal Male IISER Tirupati BSMS student IISER Tirupati Appeared / Awaiting Result  
52740 Mr. Ayanava Mandal Male National Institute of Science Education and Research Integrated MSc Student HBNI Appeared / Awaiting Result  
52465 Mr. Saunak Bhattacharjee Male IISER Tirupati MSc Student IISER Tirupati    
51701 Mr. Sourajyoti Maiti Male Ramakrishna Mission Vivekananda Educational and Research Institute (passed M.Sc in 2023) M.Sc Graduate Ramakrishna Mission Vivekananda Educational and Research Institute 2023  
53047 Dr. Devendra Tiwari Male Bhaskaracharya Pratishthana Postdoctoral Fellow Banaras Hindu University 2010 14 Oct 2019

How to Reach: 

TBA

School Short Name: 

amant

Last Date Application: 

Tuesday, May 30, 2023

School Type: 

AIS

Separate faculty form: 

0