ATMW Hilbert Modular forms and varieties (2013)

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.