ATMW Hilbert Modular forms and varieties (2013)

Venue: Kerela School of Mathematics in Kozhikode.
Dates: 21st to 31st Jan, 2013


Convener(s) Speakers, Syllabus and Time table Applicants/Participants


School Convener(s)

Name Eknath Ghate     Murgesan Manickam
Mailing Address Tata Institute of Fundamental Research,
School of Mathematics,
Homi Bhabha Road,
Mumbai 400005.
Kerela School of Mathematics
Kunnamangalam P.O.
Kozhikode, Kerela


Speakers and Syllabus 


Brief description of the workshop
The goal of the workshop will be to introduce researchers from scratch to some of the basic concepts in the theory of automorphic forms and varieties attached to GLn over totally real fields. We shall quickly treat some of the basic concepts over the first two or three days, and reserve the latter part of the first week for more advanced topics, some of these may include: p-adic Hilbert modular forms, Herzig’s classification of the mod p representations of GLn over a local field and Taylor’s recent construction of Galois representations for GLn over totally real fields.During the second week, there will be a short conference to collect together local experts in the area of modular forms and related areas of number theory. We will in particular concentrate on representation theoretic and p-adic aspects of the theory of automorphic forms. The conference will have several one hour research level talks every day.

Invited speakers

The following speakers have agreed to speak in the workshop/conference.

  1. Profesor U K Anandavardhanan, IIT, Mumbai.
    Title: Classification of Irreducible Admissible mod p Representations of GL(n) (3 hours)
    Abstract: In these lectures, we'll sketch an outline of the results of F.Herzig on the classification of irreducible admissible mod p representations of a p-adic GL(n). We'll first introduce the weights and the relevant Hecke algebras and then come to the classification.
  2. Professor Baskar Balasubramanyam, IISER, Pune.
    Title: Discrete subgroups of SL_2(R)^n
    Abstract: Introductory lectures, Ref: Freitag, Chapters 1.1, 1.2
  3. Ms. Shalini Bhattacharya, TIFR, Mumbai
    Title: Hilbert modular forms
    Abstract: Introductory lecture, Ref: Freitag, Chapter 1.4
  4. Dr. Abhik Ganguli, TIFR, Mumbai.
    Title: Finiteness of dimensions of space of Hilbert modular forms
    Abstract: Introductory lectures, Ref: Freitag, Chapter 1.6
  5. Professor Eknath Ghate, TIFR, Mumbai.
    Title: Congruences for Hilbert modular forms
    Abstract: The first lecture will be an overview lecture on Hilbert modular forms and varieties, and their L-functions. As motivation for the workshop,  We will explain several results which both directly and indirectly use Hilbert modular forms. Time permitting,  We will also give some more detailed lectures on congruences for Hilbert modular forms towards the end of the workshop, including work done by the author in the area and some open problems that remain.
  6. Dr. Srilaxmi Krishnamoorthy, IMSc, Chennai
    Title : Doi-Naganuma Lifting and Base Change for Real quadratic fields.
    Abstract: Introductory lecture, Ref: van der Geer, Chapter 6.4.  We will define the Doi-Naganuma Lifting from classical modular forms to Hilbert modular forms for a real quadratic field. We will present a proof of a conjecture by Hirzebruch and Zagier. We will discuss about L-series associated to Hilbert modular cusp forms.
  7. Professor M. Manickam, KSOM, Kozhikode
    Title: Construction of Hilbert modular forms
    Abstract: Introductory lecture, Ref: Freitag, 1.5
  8. Dr. Amrita Muralitharan, TIFR, Mumbai.
    Title: An algebraic geometric method
    Abstract: Introductory lecture, Ref: Freitag, 2.4
  9. Professor Ravi Raghunathan, IIT, Mumbai.
    Title: Contribution of cusps to the trace forumla
    Abstract: Introductory lecture, Ref: Freitag, 2.3
  10. Professor A. Raghuram, IISER, Pune
    Title: Automorphic Cohomology and Hilbert modular form (3 hours)
    Abstract: We  will give a short course of three lectures discussing the connections between holomorphic Hilbert modular forms and the cohomology of arithmetic subgroups of the algebraic group GL(2) over a totally real field F. The topics to be covered:
    1. A long exact sequence.
    We will define certain sheaves E on a locally symmetric space S_G attached to G = GL(2)/F and consider the sheaf cohomology group H*(S_G,E). The space S is not compact, and we will consider the Borel-Serre compactification and from this arises a long exact sequence in sheaf cohomology. This sequence is one of the fundamental tools to analyze the automorphic cohomology of G.
    2. Cuspidal cohomology.
    This is a very interesting transcendentally defined subspace of H*(S_G,E) which is built out of holomorphic Hilbert modular cusp forms of a given weight which can be read off from the sheaf E. I will talk about Hilbert modular forms in the representation theoretic language where it is easy to see the dictionary between the weight and sheaf.
    3. Rationality questions.
    I will make sense of the statement "Cuspidal cohomology admits a rational structure." This is a cohomological version of the statement that the space of cusp forms of a given weight and level admits a basis of containing cusp forms with rational Fourier coefficients. The rational structure on cuspidal cohomology is the one of the most important ingredients in defining periods which appear in the special values of L-functions attached to Hilbert modular forms.
  11. Mr. Sudanshu Shekhar, TIFR, Mumbai
    Title: The Tate Conjecture for Hilbert modular surfaces
    Abstract: Introductory lecture, Ref: van der Geer, Chapter 11.
  12. Ms. Devika Sharma, TIFR, Mumbai
    Title: The Hilbert modular group
    Abstract: Introductory lecture, Ref: Freitag, Chapter 1.3
  13. Professor  Ramesh Sreekantan, ISI, Bangalore
    Title:  Resolution of cusp singularities of Hilbert modular surfaces (2hours)
    Abstract: Introductory lectures, Ref: van der Geer, Chapter 2
  14. Professor Sandeep Varma, TIFR, Mumbai
    Title: Selberg trace forumla and dimension formula
    Abstract: Introductory lectures, Ref: Freitag, Chapters 2.1, 2.2
  15. Professor Michael Schein, Ramat Gan.
    Title: 1. Galois representations, modularity, and Serre's conjecture (3 hours)
    Abstact: This series of lectures will review how Hilbert modular forms over a totally real field F give rise to Galois representations, i.e., representations of the absolute Galois group of F over a p-adic or characteristic p field. One can ask the reverse question: given a Galois representation, when does it come from a Hilbert modular form? We will discuss conjectures and theorems in this subject, including Serre's classical modularity conjecture and its beautiful proof by Khare and Wintenberger.
    Title 2. The p-adic and mod p local Langlands correspondence (3 hours)
    Abstract: Let K be a p-adic field, and let E be a coefficient field .The Langlands philosophy predicts, very roughly, a correspondence between representations of the absolute Galois group of K on an n-dimensional E-vector space and E-representations of GL(n,K). If E is an l-adic field, for l a prime different from p, the correspondence was established by Harris and Taylor. We will discuss this question in the cases where E is a p-adic or characteristic p field. In both cases, everything is known for GL(2, Qp) and almost nothing is known otherwise. We will discuss known results, their proofs, and the close connections to the modularity conjectures from the first series of lectures.
  16. Professor Payman Kassaei, London.
    Title:  Hilbert modular forms: mod-p and p-adic Methods (6 hours)
    Abstract: Recently, analytic continuation of overconvergent p-adic Hilbert modular forms has provided applications to problems in the classical Langlands program. In these talks, we will present a geometric approach to Hilbert modular forms employing the p-adic and mod-p geometry of Hilbert modular varieties.  We will give an introduction to the theory of  overconvergent p-adic Hilbert modular forms, and show how geometry can be used to shed light on the extent of overconvergence of an eigenform. We will then present some applications of these methods.
  17. Professor Victor Rotger, Barcelona.
    Title: Modular forms, cycles and rational points on elliptic curves (6 hours)
    Abstract: The Birch and Swinnerton-Dyer conjecture predicts that an elliptic curve should have as many independent rational points over a number field as the order of vanishing of its L-series at the central critical point. In this series of lectures I will explain several methods, some of them still subject to unproven conjectures, to construct such points. All them use heavily the theory of (classical and Hilbert) forms, both in its  complex and p-adic manifestations. And many of them, at least conjecturally, can be explained algebraically by means of cycles on auxiliary modular higher-dimensional varieties. The object of this course is describing some of these constructions and explain the state of the art of the question.



Selected Applicants

Sr. SID Full Name Gender Affiliation State Position in College/ University University/ Institute M.Sc./M.A. Year of Passing M.Sc./M.A Ph.D. Degree Date
1 2197 Mr Sudhansu Sekhar Rout Male National Institute of Technology Rourkela Orissa Ph.D. Scholar Utkal University Bhubaneswar 2007 --
2 2517 Mr. Sampath Lonka Male University of Hyderabad, Hyderabad Andhra Pradesh PhD University of Hyderabad 2009 --
3 2553 Mr. Suman Ahmed Male University of Burdwan, Burdwan West Bengal Phd Calcutta University 2008 --
4 2571 Mr Subham Sarkar Male Indian Statistical Institute, Bangalore Karnataka student R.K.M. Vivekananda University, Belur math 2010 --
5 2592 Mr. Amiya Kumar Mondal Male IIT Bombay Maharashtra Ph.D. Jadavpur University 2007 --
6 2633 Mr Kasi Viswanadham G Male Harishchandra Research Institute, Allahabad Uttar Pradesh phD Andhra University 2009 --
7 2644 Mr. Subair Kuniyil Male Aligarh Muslim University, Aligarh Uttar Pradesh PhD student Aligarh Muslim University, Aligarh, UP 2008 --
8 2661 Mr Mahendra Kumar Verma Male IIT Bombay Maharashtra Research Scholar University of Allahabad 2005 --
9 2663 Ms Shalini Bhattacharya Female TIFR Mumbai Maharashtra PhD student -- 2010 --
10 2682 Ms Devika Sharma Female TIFR Mumbai Maharashtra Research Scholar IIT, Bombay 2009 --
11 2704 Mr. Saurabh Kumar Singh Male TIFR Mumbai Maharashtra PhD IIT Kanpur 2010 --
12 2705 Mr. Abhash Kumar Jha Male NISER, Bhubaneswar Orissa Ph.D. Banaras Hindu University 2010 --
13 2710 Mr. Sudhanshu Shekhar Male TIFR Mumbai Maharashtra PhD University of Hyderabad 2008 --
Second list
1 2677 Mr. A Manivel Male Pachaiyappa's college Tamil Nadu M.Phil University of Madras 2012 --
2 2751 Mr. Keshav Aggarwal Male IISER Mohali Punjab MS student IISER Mohali Result Awaiting  --
3 2754 Dr.Srilakshmi Krishnamoorthy Female The Institute of Mathematical Sciences Tamil Nadu Post Doctoral Fellow  -- 2004  --
4 2762 Dr. Ravinder Singh Male Institute of Mathematics and Applications,Bhubaneswar Orissa Visiting Faculty Punjab University Chandigarh 2003 --
5 2769 Dr. Amrita Muralidharan Female Tata Institute of Fundamental Research Maharashtra Visiting fellow University of Durham 2007 --
6 2790 Mr. Srivatsa Vasudevan Male Kerals School Of Mathematics Kerala Student Ramakrishna mission Vivekananda College 2011 --
7 2793 Mr. Anwesh R Male Chennai Mathematical Institute Tamil Nadu Student  -- --  --
8 2887 Mr. Abhik Ganguli Male TIFR, Mumbai Maharashtra Postdoctoral fellow University of Oxford, Oxford, UK 2004  --
9 2963 Mr.Ngairangbam Sudhir Singh Male Manipur University Manipur Research Scholar Manipur University 2008  --