NCMW Modular Forms and Galois Representations (2019)

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