AIC Local Arithmetic (2021)
Venue:  HCRI, Prayagraj (Allahabad), Uttar Pradesh 
Date:  18th, Jan 2021 to 21st, May 2021 
Name  Chandan Singh Dalawat  Supriya Pisolkar 
Mailing Address  Professor HarishChandra Research Institute Chhatnag Road, Jhunsi Allahabad211019 INDIA 
Associate Professor, Mathematics Indian Institute of Science Education and Research (IISER) Dr. Homi Bhabha Road, Pashan, Pune 411 008 INDIA 
List of the actual speakers:
Name of the speaker 
Position 
Affiliation and address 
Chandan Singh Dalawat

Professor H+, Mathematics 
HarishChandra Research Institute Chhatnag Road, Jhunsi Allahabad211019 INDIA 
List of the actual tutors:
Name of the tutor 
Position 
Affiliation and address 
Supriya Pisolkar 
Associate Professor, Mathematics 
Indian Institute of Science Education and Research (IISER) Dr. Homi Bhabha Road, Pashan, Pune 411 008 INDIA 
Debasish Karmakar

PhD Student 
HarishChandra Research Institute Chhatnag Road, Jhunsi Allahabad211019 INDIA 
Syllabus:
Actual syllabus covered by each speaker:
Name of the Speaker 
: 
Detailed Syllabus 
Chandan Singh Dalawat


Lecture 1 : A discussion of the syllabus, inverse systems and direct systems, projective limits and inductive limits, initial objects and final objects. Lecture 2 : The ring Z_p of padic integers, padic expansions, the analogy with F_p[[t]], multiplication by p in Z_p, the ideals and quotients of the ring Z_p, invertible elements in Z_p, the padic valuation, the padic distance, dense subsets in Z_p. Lecture 3 : Hensel's lemma in Z_p, squares and quadratic characters of Z_p^*, multiplicative representatives, structure of the group Z_p^*. Lecture 4 : The field of Q_p of padic numbers, squares and quadratic characters of Q_p^*, structure of the group Q_p^*. Lecture 5 : Revision of lectures 1 to 4. Lecture 6 : The Hilbert symbol, Steinberg property, explicit formulae, the conic ax^2+by^2=1. Lecture 7 : Quadratic extensions of Q_p, their compositum, bijection with the set of quadratic characters of Q_p^*, Kummer theory, ArtinSchreier theory. Lecture 8 : Power series over a finite field F_p, analogy with Z_p, the field of fractions F_p((t)), squares in F_p((t))^*, quadratic characters of F_p((t))^*, the Hilbert symbol, Steinberg property, explicit formulae, the conic ax^2+by^2=1, quadratic extensions of F_p((t)), bijection with the set of quadratic characters. Lecture 9 : Algebraic preliminaries, Galois theory, ArtinSchreier theory, reformulation of local quadratic reciprocity. Lecture 10 : Structure of the filtered space k((t))^* modulo pth powers, structure of k((t))^+ modulo the ArtinSchreier map (k finite field), the snake lemma. Lecture 11 : pfields, rings of integers, residue fields, uniformisers, Hensel's lemma, multiplicative representatives. Lecture 12 : Structure theorem for pfields, ramification index, residual degree, raising to the power p in pfields of characteristic 0, the filtration on the quotient modulo pth powers. Lecture 13 : Valuations, equivalence of valuations, rings of integers, residue fields, discrete valuations, uniformisers, ideals in the ring of integers, norms, equivalence of norms, completions. Lecture 14 : Hensel's lemma, extension of a complete discrete valuation to a finite extension field, integral closure of the ring of integers. Lecture 15 : Krasner's lemma, ramification index and residual degree, multiplicativity in towers. Lecture 16 : Relation between the degree, the ramification index, and the residual degree, the norm map, monogenity of the ring of integers in a finite extension, unramified extensions, totally ramified extensions, tamely ramified extensions. Lecture 17 : Eisenstein polynomials, totally ramified extensions, structure of tamely ramified extensions, parametrisation of tamely ramified extensions of a given degree. Lecture 18 : Maximal tamely ramified extension, galoisian and abelian tamely ramified extensions, maximal tamely ramified abelian extension, the number of tamely ramified extensions of various kinds. Lecture 19 : The trace map, the different and the discriminant, transitivity formulae, the different and the ramification, Serre's mass formula in the tame case. Lecture 20 : Inertia subgroup and ramification subgroup, higher ramification groups, lower numbering, ramification filtration on a subgroup.
Lecture 21 : The ramification filtration on a quotient group, HasseHerbrandt functions, transitivity, roots on unity. Lecture 22 : Ramification filtration on cyclotomic extensions, ramification jumps, cyclic extensions, abelian extensions of prime exponent, HasseArf. Lecture 23 : cyclic extensions of pfields of characteristic 0, raising to the exponent p, filtration on the units modulo pth powers, primitive roots of unity, the unramified F_pline, ramification breaks of cyclic extensions. Lecture 24 : ramification breaks of cyclic extensions in the absence of a primitive pth root of unity, contribution of cyclic extensions to Serre's mass formula. Lecture 25 : Parametrisation of solvable extensions of prime degree over an arbitrary field. Lecture 26 : Specialisation to the case of pfields, the maximal abelian extension of exponent dividing p1, the galois module structure of units modulo pth powers, Serre's mass formula in prime degree. Lecture 27 : The norm map and the ramification filtration. Lecture 28 : The local KroneckerWeber theorem. Lecture 29 & 30 : Primitive solvable extensions of an arbitrary field. Lectures 31 & 32 : Irreducible F_prepresentations of the absolute Galois group of a pfield. Lectures 33 & 34 : The Galois module structure of units modulo pth powers in a tamely ramified split extension. Lecture 35 & 36 : Primitive extensions of pfields. 
Time Table
Actual timetable that was followed during the programme:

Day
Date
Lecture
(15:0016:30)
Tutorial
(15:0016:00)
(Name of the speaker in
abbreviated form)
(Names of the speaker and tutor in abbreviated forms)
Monday
18/01/2021
CSD
Wednesday
20/01/2021
CSD
Friday
22/01/2021
SP & CSD
Monday
25/01/2021
CSD
Wednesday
27/01/2021
CSD
Friday
29/01/2021
SP & CSD
Monday
01/02/2021
CSD
Wednesday
03/02/201
CSD
Friday
05/02/2021
SP & CSD
Monday
08/02/2021
CSD
Wednesday
10/02/2021
CSD
Friday
12/02/2021
SP & CSD
Monday
15/02/2021
CSD
Wednesday
17/02/2021
CSD
Friday
19/02/2021
SP & CSD
Monday
22/02/2021
CSD
Wednesday
24/02/2021
CSD
Friday
26/02/2021
SP & CSD
Monday
01/03/2021
CSD
Wednesday
03/03/2021
CSD
Friday
05/03/2021
SP & CSD
Monday
08/03/2021
CSD
Wednesday
10/03/2021
CSD
Friday
12/03/2021
SP & CSD
Monday
15/03/2021
CSD
Wednesday
17/03/2021
CSD
Friday
19/03/2021
SP & CSD
Monday
22/03/2021
CSD
Wednesday
24/03/2021
CSD
Friday
26/03/2021
DK & CSD
Monday
29/03/2021
CSD
Wednesday
31/03/2021
CSD
Friday
02/04/2021
DK & CSD
Monday
05/04/2021
CSD
Wednesday
07/04/2021
CSD
Friday
09/04/2021
DK & CSD
Monday
12/04/2021
CSD
Wednesday
14/04/2021
CSD
Friday
16/04/2021
DK & CSD
Monday
19/04/2021
CSD
Wednesday
21/04/2021
CSD
Friday
23/04/2021
DK & CSD
Monday
26/04/2021
CSD
Wednesday
28/04/2021
CSD
Friday
30/04/2021
DK & CSD
Monday
03/05/2021
CSD
Wednesday
05/05/2021
CSD
Friday
07/05/2021
DK & CSD
Monday
10/05/2021
CSD
Wednesday
12/05/2021
CSD
Friday
14/05/2021
DK & CSD
Monday
17/05/2021
CSD
Wednesday
19/05/2021
CSD
Friday
21/05/2021
DK & CSD
Full forms for the abbreviations of speakers and tutors:
CSD : Prof. Chandan Singh Dalawat
SP :Dr. Supriya Pisolkar
DK : Debasish Karmakar
List of actual Participants
Serial  SID  Full Name  Gender  Affiliation  Position in College/ University  Year of Passing M.Sc./M.A  Ph.D. Degree Date 
1  35019  Mr. Gopal Jhakharait  Male  Lalit Narayan Mithila University Darbhanga, Bihar  MSC STUDENT  Awaiting Result  
2  35055  Mr. Mahesh Kumar Ram  Male  IISER Berhampur  PhD  2016  
3  35073  Mr. Abhishek Koparde  Male  IISER Pune  MSc Student  
4  35085  Ms. Yashi Jain  Female  IISER Pune  BS MS student  Awaiting Result  
5  35119  Mr Abhishek Thyagarajan Bharadwaj  Male  Chennai Mathematical Institute  PostDoctoral Fellow  2013  22/06/2020 
6  35140  Mr. Aniruddha S  Male  IISER  Integrated PhD Student  2021  
7  35148  Mr. Deepak Madhukar Bakal  Male  Department of Mathematics, Savitribai Phule Pune University, Pune 411007. 
PhD Student  2019  
8  35200  Ms. Neha Maurya  Female  IIT Madras  MSc  2020  
9  35220  Mr. S Sundara Narasimhan  Male  Indian Institute of Science Education and Research Mohali (IISER M)  BSMS (Mathematics) 4th year student  
10  35230  Mr. Prem Dagar  Male  IIT ROORKEE  Ph.D STUDENT  2017  
11  35231  Ms . Mridusmita Hazarika  Female  North Eastern Hill University  MSc Student  2019  
12  35235  Mr. Hariom Sharma  Male  IIT ROORKEE  PhD  2019  
13  35236  Mr. Mohit Mishra  Male  Harish Chandra Research Institute  PhD student  2017  
14  35237  Mr. Rishabh Agnihotri  Male  Harish chandra Research Institute  PhD Student  2017  
15  35244  Mr. Anupam Datta  Male  Chennai Mathematical Institute  BSc Student  
16  35248  Mr. Vaibhav Kalia  Male  IIT ROPAR  PhD  2018  
17  35252  Mr. Shivansh Pandey  Male  National Institute of Science Education and Research, Bhubaneswar  PhD  2019  
18  35272  Mr Gorekh Prasad Sena  Male  National Institute of Science Education and Research.  PhD  2015  
19  35276  Mr Pronay Kumar Karmakar  Male  IISER Mohali  PhD  2017  
20  35284  Mr. Sudip Pandit  Male  Sudip Pandit  PhD  2018  
21  35296  Ms. Pallavi Agrahari  Female  university of Allahabad  MSc student  Appeared / Awaiting Result  
22  35297  Mr. Naveen Kumar  Male  Indian institute of science education and research  PhD  2019  
23  35302  Mr. Arpan Das  Male  HarishChandra Research Institute  PhD student  2019  
24  35311  Mr. Hargun Preet Singh Bhatia  Male  Indian Institute of Technology Kanpur  BS Student  
25  35313  Mr Rajat Kumar Mishra  Male  IIT Gandinagar  PhD  2019  
26  35322  Mr Mohamed Harif B  Male  Rajah Serfoji Govt. College(Autonomous), Thanjavur5  Assistant Professor  2006  23/02/2019 
27  35330  Mr Siddharth Ramakrishnan  Male  Kerala School of Mathematics  Visiting Student  2020  
28  35333  Mr, Aditya Bhan Ojha  Male  MSCM Govt college Thural  Assistant Professor  2009  
29  35336  Mr Pasupulati Sunil Kumar  Male  IISER TVM  PhD student  2018  
30  35337  Mr. Ashutosh Suresh Jangle  Male  IISER Pune  MSc Student  2021  
31  35338  Mr. Ajay Dabral  Male  University of Delhi  MPhil  2017  
32  35349  Mr. Sampurna Pal  Male  ISI, Kolkata  PhD  2019  
33  35355  Mr Vignesh Perumal  Male  Patrician college of arts and science  Asst Professor  2009  
34  35358  Ms. Jaskaran Kaur  Female  Panjab University  MSc Student  2020  
35  35359  Mr. Surendra Kumar Mishra  Male  University of Allahabad  PhD  2014  
36  35372  Mr. Avinash Roy  Male  IISER Pune  BSMS Student  Awaiting Result  
37  35374  Ms Pratiksha Satish Shingavekar  Female  IIT Madras  PhD student  2017  
38  35376  Ms Gunileima Thokchom  Female  DMU Manipur  MSc student  
39  35377  Mr. Himanshu Mittal  Male  GD Goenka University, Gurugram  PhD  2016  
40  35380  Mr. Rohit Pokhrel  Male  Sikkim University  MPhil  2019  
41  Mr. Abhishek Sarma  Tezpur University(Alumni  MSc Student  
42  Dr. M. Subramani  Indian Institute of Information Technology DM, Kancheepuram  Asst Professor  
43  Mr. Shrinit Singh  Shiv Nadar University  PhD student 
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