NCMW Modular Forms and Galois Representations (2019)
Venue:  IISER, Tirupati, Andhra Pradesh 
Date:  11th, Dec 2019 to 17th, Dec 2019 
Name  Prof. Eknath Ghate  Dr. Shalini Bhattacharya 
Mailing Address 
Professor and Dean 
Assistant Professor 
Speakers with their affiliations:
Name of the speaker 
Position 
Affiliation 
Neil Dummigan 
Professor 
University of Sheffield 
Eknath Ghate 
Professor 
TIFR, Mumbai 
Chol Park 
Assistant Professor 
Ulsan National Institute of Science and Technology, South Korea 
Sandeep Varma 
Reader 
TIFR, Mumbai 
List of Tutors
Name of the tutor 
Position 
Affiliation 
Anand Chitrao 
Research Scholar 
TIFR, Mumbai 
Arvind Kumar 
Postdoctoral fellow 
TIFR, Mumbai 
Vivak Rai 
Postdoctoral fellow 
IISER, Pune 
Ravitheja Vangala 
Postdoctoral fellow 
TIFR, Mumbai 
Syllabus:
Name of the Speaker 
Detailed Syllabus 
Neil Dummigan 
Elliptic curves over the rationals. Rational points and Selmer groups. The Lfunction and the Birch and SwinnertonDyer conjecture. Rational torsion points and associated congruences.Congruences between elliptic curves, visibility of elements in ShafarevichTate groups. Modularity, the modular degree, the symmetric square Lfunction and adjoint Selmer group. Congruences and Lfunctions for modular forms of weight greater than two. Adjoint Lfunctions at nonnearcentral critical values. Vectorvalued Siegel modular forms of genus two. Congruences between KlingenEisenstein series and cusp forms. Congruence between a newform and its complex conjugate. Consequences for nonnearcentral critical values. Congruences involving nonparallel weight Hilbert modular forms. 
Eknath Ghate

An introduction to rigid analytic geometry. Basic notions that go into defining a rigid analytic variety like Tate algebras, Gauss norms, affinoid algebras and spaces, Weirstrass, Laurent, rational subdomains, affinoid subdomains, weak and strong Grothendieck topologies, locally Gringed spaces etc. were discussed. The definition of a general rigid analytic space was given. The emphasis of the minicourse was on definitions, examples and statements of results rather than on their proofs. In particular, open and closed disks, annuli, rigid affine spaces, and Tate elliptic curves were discussed. 
Chol Park

In this short lecture series, Breuil's integral padic Hodge theory to compute mod p reduction of semistable representations was disscussed. As an application deformation rings that parameterize certain semistable representations were constructed. Computational aspects rather than theoretical proof were focussed on. Moreover, the conjecture about mod p localglobal compatibility was discussed. This roughly states that a certain space of mod p algebraic automorphic forms on a unitary group determines a given Galois representation. This gives evidence for the mod p Langlands correspondence. In the first lecture, the category of padic Galois representations of the absolute Galois group of K, for a finite extension K of Qp, and its subcategories whose objects arise from geometry were introduced. These subcategories consist of de Rham representations, semistable representations, and crystalline representations. More attention was paid to semistable representations. By ColmezFontaine, it is known that the category of semistable representations is equivalent to the category of admissible filtered (phi, N)modules. In the second lecture, it was shown how to compute the mod p reductions of semistable representations. By Breuil and Liu, it is known that the category of Galois stable lattices in semistable representations with HodgeTate weights in [0, r] is equivalent to the category of strongly divisible modules, provided that r < p  1. To study the mod p reduction, Breuil modules, which correspond to the mod p reductions of strongly divisible modules were studied. In the third lecture, Galois deformation theory was quickly reviwed, and then by making use of the parameterization of the families of strongly divisible modules discussed in the second lecture, the irreducible components of those deformation rings were constructed. The last lecture was more expository. For a given mod p representation of the absolute Galois group of a padic field K, a mod p automoprhic representation of GLn (K) was defined, using a certain space of mod p algebraic automorphic forms on a unitary group. It was hoped that this latter representation corresponds to the former under a mod p Langlands correspondence, but the structure of the latter is still quite mysterious. It is natural to ask if the latter determines the former and this question was asnwered in certain cases. 
Sandeep Varma 
A brief introduction to the theory of representations of a finite group of order divisible by a prime p, on vector spaces over an algebraically closed field of characteristic p was given. This representation theory works very differently from the theory of complex representations of finite groups. The notion of blocks and defects were introduced and some important examples were discussed. 
U.K. Anandavardhanan 
Mod p representations of GL_2(F) 
Anilatmaja Aryasomayajula 
On a conjecture by Sarnak 
Shaunak Deo 
Universal deformations of dihedral representations 
D. S. Nagaraj 
Galois representations attached to an elliptic curve over a number field 
Jyoti Prakash Saha 
Purity for families of Galois representations 
Sandeep Varma 
Some Bernstein projectors for SL_2 
Time Table
Period / Day 
Wed 11 Dec 
Thu 12 Dec 
Fri 13 Dec 
Sat 14 Dec 
Mon 16 Dec 
Tue 17 Dec 
9:30—11:00 
NPD 
EG 
SV 
CP 
NPD 
NPD 
11:0011:30 
Tea 
Tea 
Tea 
Tea 
Tea 
Tea 
11:3013:00 
SV 
CP 
NPD 
EG 
CP 
CP 
13:0014:30 
Lunch 
Lunch 
Lunch 
Lunch 
Lunch 
Lunch 
14:3015:30 
DSN 
SV 
SD 
UKA 
JPS 
AA 
15:3016:00 
Tea 
Tea 
Tea 
Tea 
Tea 
Tea 
16:0017:00 
RV and AC 
AK and VR 
AC and RV 
VR and AK 
RV and VR 
AK and AC 
Full forms for the abbreviations of speakers and tutors :
 NPD : Neil Dummigan
 EG : Eknath Ghate
 CP : Chol Park
 SV : Sandeep Varma
 DSN : D. S. Nagraj
 SD : Shaunak Deo
 UKA : U K Anandavardhanan
 JPS : Jyoti Prakash Saha
 AA : Anilatmaja Aryasomayajula
 AC : Anand Chitrao
 AK : Arvind Kumar
 VR : Vivek Rai
 RV : Ravitheja Vangala
List of actual Participants
Sr.  SID  Full Name  Gender  Affiliation  Position in College /University  Univ./Inst. M.Sc./M.A.  Year of Passing M.Sc / M.A  Ph.D. Deg. Date 
1  29575  Mr. Abhishek Juyal  Male  Institute of Mathematical Sciences  Post Doctoral Fellow  Banaras Hindu University  2011  15/12/2018 
2  29579  Mr Sumit Kumar  Male  ISI Kolkata  PhD  IIT KANPUR  2016  
3  29603  Mr. Murtuza Ibrahim Nullwala  Male  Ramrao Adik Institute of Technology  Assistant Professor  University of Mumbai  2014  
4  29617  Mr Pasupulati Sunil Kumar  Male  Indian Institute of Science Education and Research Thiruvananthapuram  PhD student  Indian Institute of Science Education and Research Thiruvananthapuram  2018  
5  29692  Mr Lalit Vaishya  Male  HARISH CHANDRA RESEARCH INSTITUTE PRAYAGRAJ (ALLAHABAD) INDIA 
Ph D STUDENT  BHU VARANASI  2014  
6  29912  Mr Manish Kumar Pandey  Male  IISER BHopal  Postdoc  Indian Statistical Institute  2011  01/08/2019 
7  30141  Mr. Satyabrat Sahoo  Male  IIT HYDERABAD  Ph.D. Student  M.Sc  2016  
8  30327  Mrs. Moni Kumari  Female  TIFR, Mumbai  Visiting fellow  BHU, Baranas  2012  05/01/2019 
9  Sameer Kulkarni 
TIFR Mumbai 
RS  
10  30457  Mr Sreejith M M  Male  Kerala School of Mathematics  PhD Student  IISER Thiruvanathapuram  2016  
11  30477  Dr. Anuj Jakhar  Male  The Institute of Mathematical Sciences, Chennai  Postdoctoral fellow  Indian Institute of Science Education and Research Mohali  2014  25/05/2018 
12  30527  Mr Pronay Kumar Karmakar  Male  IISER Mohali  PhD  Tezpur University  2017  
13  30626  Mr Subhasis Panda  Male  IIT Madras.  PhD  University of Hyderabad  2015  
14  30627  Mr. Adit Vishnu Pm  Male  Indian Institute of Science  Integrated MSc Student  Indian Institute of Science  
15  30640  Mr. Suneel Kumar  Male  IISER Mohali  PhD student  IIT Kanpur  2017  
16  30667  Mr Sohan Ghosh  Male  Indian Institute of Technology, Kanpur  Research Scholar  Indian Institute of Technology, Guwahati  2017  
17  30670  Ms Pratiksha Satish Shingavekar  Female  IIT Madras  Research scholar  University of Pune  2017  
18  30691  Dr. Tathagata Mandal  Male  IIT Kanpur  Postdoc  VisvaBharati University  2012  23/11/2018 
19  30692  Dr. Neeraj Neeraj  Male  IIT Bombay  Post Doctoral fellow  IISER Mohali  2014  20/05/2018 
20  30697  Mr Siddharth Ramakrishnan  Male  IISER PUNE  BSMS Student  IISER PUNE  App / Awt Res. 

21  30708  Mr Sumukha S  Male  NITK Suratkal  Research Scholar  Mangalore University  2017  
22  29639  Nazim Khan  Male  Aligarh Muslim University  PhD  ALIGARH MUSLIM UNIVERSITY  2015  
23  29724  Krishnarjun K.  Male  Harish Chandra Research Institute  PhD  Central University of Tamil Nadu  2018  
24  30277  Mabud Ali Sarkar  Male  THE UNIVERSITY OF BURDWAN  PhD  UNIVERSITY OF NORTH BENGAL  2015  
25  29584  Mr. Rishabh Agnihotri  Male  Harish Chandra Research Institute  PhD  IIT Kanpur  2017  
26  30567  Ms. Dyuti Roy  Female  IISER TIRUPATI  PhD Student  2015  
27  30623  Mr. Debasish Sadhukhan  Male  IISER TIRUPATI  PhD  UNIVERSITY OF HYDERABAD  2017  
28  30650  Ms. Poornima B  Female  IISER Tirupati  Project assistant  IISER Pune  2019  
29  30669  Mr. Harinarayanan G  Male  IISER  PhD Student  Cochin University of Science and Technology  2018  
30  30671  Ms Harshitha C  Female  IISER  PhD Student  IIT Gandhinagar  2018 
How to reach