NCMW Modular Forms and Galois Representations (2019)

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Venue: IISER,  Tirupati, Andhra Pradesh
Date:  11th, Dec 2019 to 17th, Dec 2019

 

School Convener(s)

Name Prof. Eknath Ghate Dr. Shalini Bhattacharya
Mailing Address

Professor and Dean
School of Mathematics, TIFR
Homi Bhabha Road,
Mumbai- 400005

Assistant Professor
Department of Mathematics
IISER Tirupati
Tirupathi
Andhra Pradesh- 517507

 


 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 L-function and the Birch and Swinnerton-Dyer conjecture. Rational torsion points and associated congruences.Congruences between elliptic curves, visibility of elements in Shafarevich-Tate groups.

Modularity, the modular degree, the symmetric square L-function and adjoint Selmer group. Congruences and L-functions for modular forms of weight greater than two. Adjoint L-functions at non-near-central critical values. Vector-valued Siegel modular forms of genus two. Congruences between Klingen-Eisenstein series and cusp forms.

Congruence between a newform and its complex conjugate. Consequences for non-near-central critical values. Congruences involving non-parallel 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 G-ringed spaces etc. were discussed. The definition of a general rigid analytic space was given. The emphasis of the mini-course 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 p-adic Hodge theory to compute mod p reduction of semi-stable representations was disscussed. As an application deformation rings that parameterize certain semi-stable representations were constructed. Computational aspects rather than theoretical proof were focussed on.

Moreover, the conjecture about mod p local-global 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 p-adic Galois representations of the absolute Galois group of K, for a finite extension K of Qp, and its sub-categories whose objects arise from geometry were introduced. These sub-categories consist of de Rham representations, semi-stable representations, and crystalline representations. More attention was paid to semi-stable representations. By Colmez-Fontaine, it is known that the category of semi-stable 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 semi-stable representations. By Breuil and Liu, it is known that the category of Galois stable lattices in semi-stable representations with Hodge-Tate 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 p-adic 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

11:00-11:30

Tea

Tea

Tea

Tea

Tea

Tea

11:30-13:00

SV

CP

NPD

EG

CP

CP

13:00-14:30

Lunch

Lunch

Lunch

Lunch

Lunch

Lunch

14:30-15:30

DSN

SV

SD

UKA

JPS

AA

15:30-16:00

Tea

Tea

Tea

Tea

Tea

Tea

16:00-17:00

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 Post-doc 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 Post-doctoral 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 Visva-Bharati 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 BS-MS 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  

 


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