Trimester II

From 1st January 2022 to 30th April 2022

Seminars in January 2022

Week 1

Talk-1: "The Roles played by elliptic curves and modular forms in FLT". (Video is here and Slides are here)

Speaker: B. Sury, ISI Bangalore.

Time: 4th January, Tuesday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: S. G. Dani, CEBS, Mumbai.

Abstract: While this is an old story known to experts, in this talk we take the point of view of a mathematician not working on this topic.

We start with a brief discussion of the key notions and properties of elliptic curves and modular forms.

Following this, we describe how they combine to give a proof of FLT. Finally, more technical details are given how exactly this works.

Talk-2: "An Overview of the Taylor-Wiles Method". (Video is here and Slides are here)

Speaker: Anwesh Ray, University of British Columbia, Vancouver.

Time: 6th January, Tuesday at 16:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Dipendra Prasad, IIT Bombay.

Abstract: This talk is intended to be a gentle introduction to the Taylor-Wiles method. Along the way, we shall introduce various key ideas in the theory of Galois representations and Hecke algebras which will be subsequently revisited in greater detail later on in the semester.

Week 2

Talk-1: “Galois Representations attached to Elliptic Curves". (Video is here and Slides are here)

Speaker: Guhan Venkat, Ashoka University.

Time: 11th January, Tuesday at 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Devendra Tiwari, BP, Pune.

Abstract: In this expository talk, we will look at the Galois representations attached to elliptic curves defined over Number fields. Time permitting, we will also explore topics such as Galois cohomology, Kummer sequence and Selmer groups associated to elliptic curves. We shall primarily follow the exposition in:

[1] K. Joshi, Elliptic Curves, Serre's Conjecture and Fermat's Last Theorem, Cyclotomic fields & Related Topics, Bhaskaracharya Pratishthana (1999).

[2] K. Ribert, Galois Representations and Modular forms, Bull. Amer. Math. Soc. Vol. 32(4), 1995.

Talk-2: "Using Geometry to Understand Fermat-like Equations". (Video, Slides and some notes are here)

Speaker : Anand Deopurkar, Australian National University.

Time: 13th January, Thursday 17:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Devendra Tiwari, BP Pune.

Abstract: Geometric properties can have fascinating arithmetic consequences. As an example, I will explain a proof by Darmon-Granville that Fermat-like equations (Ax^p + By^q = Cz^r) have finitely many solutions (under some conditions on the exponents). If time permits, I will describe how geometric considerations are expected to imply uniform bounds on the number of rational points, following the work of Caporaso, Harris, and Mazur.

Week 3

Talk-1: “Modular Curves - I". (Video is here and Slides are here)

Speaker - Shaunak Deo, IISc Bangalore.

Time: 18th January, Tuesday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Somnath Jha, IIT Kanpur.

Abstract: We will begin by recalling definitions and basic properties of modular curves. Then we will introduce the modularity theorem and discuss various versions of it. Time permitting, we will explore the relationships between the different versions of modularity theorems.

Talk-2: “Modular Forms - I". (Video is here and Slides are here)

Speaker - Winnie Li, Pennsylvania State University.

Time: 20th January, Thursday 21:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Sujatha Ramdorai, UBC Vancouver.

Abstract: For over a century, modular forms for congruence subgroups of SL(2,Z) have played a central role in number theory with many earth shattering results and far-reaching consequences. In these two talks we shall survey the arithmetic of modular forms, focusing primarily on cusp forms. Topics include integrality of Fourier coefficients, Hecke operators, the theory of newforms, Galois representations, converse theorem, and connections with elliptic curves. Important conjectures such as the Ramanujan conjecture and the Sato-Tate conjecture will also be discussed.

Among the special linear groups over Z, SL(2, Z) is special in that it contains many finite index subgroups which are noncongruence. The study of modular forms for noncongruence subgroups was initiated by Atkin and Swinnerton-Dyer in 1971, followed by the monumental contributions by Scholl in 1980's. In the course of discussing various arithmetic properties, we shall also point out, when appropriate, differences between congruence and noncongruence forms.

Week 4

Talk-1: "Modular Forms - II". (Video is here and Slides are here)

Speaker - Winnie Li, Pennsylvania State University.

Time: 25th January, Tuesday 21:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Siddhi Pathak, CMI Chennai.

Abstract: For over a century, modular forms for congruence subgroups of SL(2,Z) have played a central role in number theory with many earth shattering results and far-reaching consequences. In these two talks we shall survey the arithmetic of modular forms, focusing primarily on cusp forms. Topics include integrality of Fourier coefficients, Hecke operators, the theory of newforms, Galois representations, converse theorem, and connections with elliptic curves. Important conjectures such as the Ramanujan conjecture and the Sato-Tate conjecture will also be discussed.

Among the special linear groups over Z, SL(2,Z) is special in that it contains many finite index subgroups which are noncongruence. The study of modular forms for noncongruence subgroups was initiated by Atkin and Swinnerton-Dyer in 1971, followed by the monumental contributions by Scholl in 1980's. In the course of discussing various arithmetic properties, we shall also point out, when appropriate, differences between congruence and noncongruence forms.

Talk-2: "The Eichler-Shimura relation and its application to Galois representations".

Speaker - Mihir Seth, IISc Bangalore. (Video is here and Slides are here)

Time: 27th January, Tuesday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Gianluca Faraco, MPI Bonn.

Abstract: In this talk, I will talk about the classical Eichler-Shimura relation which relates the Hecke operator at a prime p to the Frobenius at p on modular curves in characteristic p. As an application, it gives the characteristic polynomial of the Frobenius acting on a 2-dimensional irreducible Galois representation attached to a modular form.

Seminars in February.

Week 5

Talk-1: "Hilbert modular forms -I: Cohomological Viewpoint". (Video is here and Slides are here)

Speaker - A. Raghuram, Fordham University, New York.

Time: 1st February, Tuesday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Ravi Raghunathan, IIT Bombay.

Abstract: I will set up the framework of the cohomology of arithmetic groups when the ambient reductive group is GL(2) over a totally real number field, and talk about the relation of Hilbert modular forms to these cohomology groups. This is in effect the Eichler-Shimura isomorphism in the Hilbert modular setting.

Reference: Raghuram, A.; Tanabe, Naomi; Notes on the arithmetic of Hilbert modular forms, Journal of Ramanujan Mathematical Society, 26 (2011), no. 3, 261–319.

Talk-2: "Galois Representations and Modular forms". (Video is here and Slides are here)

Speaker - Sudhanshu Shekhar, IIT Kanpur.

Time: 3rd February, Thursday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Sandip Singh, IIT Bombay.

Abstract: Aim of this lecture is to give an overview of the Galois representations associated to elliptic modular forms of weight at least two and discuss briefly its Geometric construction.

Prerequisite : Theory of new forms, modular curves, Hecke operators.

References :

1. Modular forms and Fermat's last theorem - Gary Cornell, Joseph H Silverman, Glenn Stevens.

2. Galois representations and modular forms, Takeshi Saito, https://www.ms.u-tokyo.ac.jp/~t-saito/talk/eepr.pdf

Week 6

Talk-1: "Hilbert modular forms -II: L-functions ". (Video is here and Slides are here)

Speaker - A. Raghuram, Fordham University, New York.

Time: 8th February, Tuesday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Guhan Venkat, Ashoka University.

Abstract: The classical Hecke L-function, given by the Mellin transform of a Hilbert modular form, will be interpreted as a Poincare pairing in cohomology. Such an interpretation lends itself to the study of arithmetic properties of the special values of L-functions.

Reference: Raghuram, A.; Tanabe, Naomi; Notes on the arithmetic of Hilbert modular forms, Journal of Ramanujan Mathematical Society, 26 (2011), no. 3, 261–319.

Talk-2: "Modular Curves - II". (Video is here and Slides are here)

Speaker: Shaunak Deo, IISc Bangalore.

Time: 10th February, Thursday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Somnath Jha, IIT Kanpur.

Abstract: Continuing our study of the modularity theorem, we will give a brief survey of various versions of the modularity theorem and explore the connections between them. If time permits, we will give a brief sketch of equivalences between them.

Week 7

Talk-1: "Uniformization of Elliptic Curves". (Video is here and Slides are here)

Speaker - Kingshook Biswas, ISI, Kolkata.

Time: 15th February, Tuesday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Krishnendu Gongopadhyay, IISER Mohali.

Abstract: We give a brief exposition of the uniformization of elliptic curves via inversion of abelian integrals. Along the way we describe how to construct the Riemann surface of a multi-valued algebraic function as a finite sheeted branched covering of the complex plane, and explain how considering the inverse of the integral of certain holomorphic differential forms on the surface leads to a parametrization of the algebraic curve by singly periodic functions when the surface is of genus zero, and by doubly periodic functions when the surface is of genus one. We will try to keep the discussion as elementary and accessible to graduate students as possible.

Talk-2: "Uniformization of Shimura Curves". (Video is here and Slides are here)

Speaker - Pilar Bayer, University of Barcelona.

Time: 17th February, Thursday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Barinder Banwait, Ruprecht-Karls-Universität Heidelberg.

Abstract: Shimura curves are a remarkable generalization of modular curves. We shall consider those defined by means of non-split rational quaternion algebras and will give their interpretation as coarse moduli spaces for fake elliptic curves. A convenient generalization of the classical complex multiplication theory, led G. Shimura to his theory of the canonical models. As in the modular case with the modular j invariant, fake elliptic curves with complex multiplication play a key role in the theoretical construction of class fields by means of special values of arithmetic automorphic functions. In joint work with M. Alsina, J. Guàrdia and A. Travesa, I shall present a method to compute the function field of some Shimura curves as well as to obtain the fake elliptic curves defined by special complex multiplication points in the canonical model.

Week 8

Talk-1: "Monodromy of the Schwarzian Equation on Riemann Surfaces".

Speaker - Subhojoy Gupta, IISc Bangalore. (Video is here and Slides are here)

Time: 22nd February, Tuesday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Sudarshan Gurjar, IIT Bombay.

Abstract: A Schwarzian equation with regular singularities on a punctured Riemann surface X is a second order linear differential equation, involving a coefficient q that is a holomorphic quadratic differential, with poles of order at most two at the punctures. The monodromy of its solutions determines a representation from the fundamental group of the surface to PSL(2,C). It has been a long-standing problem to determine the representations that thus arise, when one is also allowed the vary the complex structure on the surface, and the meromorphic quadratic differential q. This is the analogue of the Riemann-Hilbert problem in the classical theory of linear differential equations on the complex plane. I shall talk about the solution of the problem, in recent joint work with Gianluca Faraco, and its connection with the study of complex projective structures (also called Möbius structures) on surfaces.

Talk -2: "Abelian Varieties - I". (Video is here and Slides are here)

Speaker: Jaya N. Iyer, IMSc, Chennai.

Time: 24th February, Thursday 19:00 IST (GMT + 5:30).

Chair for the Talk: Swarnendu Datta, IISER Kolkata.

Abstract: We will review basics on Abelian varieties: Rigidity theorem, Cohomology of torus, Appell-Humbert theorem, Theorem of square, See-Saw principle, Dual Abelian variety, Jacobian varieties, special ones like hypergeometric.

Seminars in March 2022

Week-9

Talk-1: "Moduli Interpretation of noncongruence Modular Curves". (Video is here, and slides are here)

Speaker: Will Chen, IAS, Princeton.

Time: 1st March, Tuesday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: B. Sury, ISI Bangalore.

Abstract: In this talk a "modular curve" will denote the quotient of the upper half plane by a finite index but possibly noncongruence subgroup of SL(2,Z). We will explain how all modular curves can be obtained as moduli spaces for elliptic curves with certain "G-structures", where G is a finite group. When G is abelian, G-structures are equivalent to classical congruence "level n" structures. When G is nonabelian, the moduli spaces are often (but not always!) noncongruence. By a theorem of Asada, every modular curve can be given a moduli interpretation in this way. Moreover, using the Galois correspondence, the geometric structure of the moduli spaces can be described entirely in terms of group theory, and is amenable to computation. In this talk we will describe these moduli interpretations, and discuss some examples, questions that arise, and applications as time allows.

Talk-2: Modularity over Totally Real Fields. (Video is here and slides are here)

Speaker - Patrick Allen, McGill University.

Time: 3rd March, Thursday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Shaunak Deo, IISc Bangalore.

Abstract: We will introduce what it means for an elliptic curve over a general totally real field to be modular. Over the rational numbers, there are many equivalent formulations of being modular, but not all of these generalize easily. We will discuss which generalize easily and which are more mysterious.

Week 10

Talk: "The Unbounded Denominator Conjecture". (Video is here and Slides are here)

Speaker: Vesselin Dimitrov, University of Toronto.

Time: 10th March, Thursday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: B. Sury, ISI Bangalore.

Abstract: Atkin and Swinnerton-Dyer conjectured a modularity characterization for Belyi maps in terms of the arithmetic properties of their x = 0 Puiseux expansion. In this talk, I will explain what the theorem precisely says and how it is proved, using ideas from arithmetic algebraization, Nevanlinna theory, and the congruence subgroup property of the Ihara modular group SL_2(Z[1/p]).

Week 11

Talk: "Abelian Varieties - II". (Video is here and Slides are here)

Speaker - Jaya N. Iyer, IMSc, Chennai.

Time: 15th March, Tuesday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Devendra Tiwari, BP Pune.

Abstract: We will review basics on Abelian varieties: Rigidity theorem, Cohomology of torus, Appell-Humbert theorem, Theorem of square, See-Saw principle, Dual Abelian variety, Jacobian varieties, special ones like hypergeometric.

Week 12

Talk-1: "Modularity over C implies modularity over Q". (Video is here and Sides are here)

Speaker: Barinder Banwait, Heidelberg Germany.

Time: 22nd March, Tuesday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Dipendra Prasad, IIT Bombay.

Abstract: We give an overview of Mazur's proof that, for an elliptic curve over Q, if it admits a non-constant mapping from X(N) defined over the complex numbers C, for some N, then it admits a non-constant mapping from X_0(N) defined over the rational numbers Q as well, possibly for a different N. We will also briefly discuss some more refined open questions of Khare concerning uniformizations of elliptic curves by non-congruence modular curves.

Talk-2: "Modularity and Effective Mordell". (Video is here and Slides are here)

Speaker - Levent Alpoge, Harvard University, USA.

Time: 24th March, Thursday 19:00 Hrs IST (GMT +5:30).

Chair for the Talk: Soumya Sankar, Ohio State USA.

Abstract: I will give a finite-time algorithm that, on input (g,K,S) with g > 0, K a totally real number field of odd degree, and S a finite set of places of K, outputs the finitely many g-dimensional abelian varieties A/K which are of GL_2-type over K and have good reduction outside S.

The point of this is to effectively compute the S-integral K-points on a Hilbert modular variety, and the point of that is to be able to compute all K-rational points on complete curves inside such varieties.

This gives effective height bounds for rational points on infinitely many curves and (for each curve) over infinitely many number fields. For example one gets effective height bounds for odd-degree totally real points on x^6 + 4y^3 = 1, by using the hypergeometric family associated to the arithmetic triangle group Δ (3,6,6).

Week 13

Talk-1: Cyclic Coverings of the Projective line and Arithmetic Groups -I.

Speaker - T. N. Venkataramana, TIFR, Mumbai. (Video is here and Slides are here)

Time: 29th March, Tuesday 19:00 Hrs IST (GMT +5:30).

Chair for the Talk:: S. G. Dani, CEBS Mumbai.

Abstract: A degree d cyclic covering of the projective line is obtained by taking the d-th root of a polynomail f in one variable with n distinct roots, say; if the multiplicities of the roots of f are fixed, and n,d are fixed, we get a family of cyclic coverings of the projective line ramified at varying points of the line.

In these two lectures, we prove that if the multiplicities are coprime to d, and if the number n of distinct roots of f exceeds 2d, then the monodromy of this family is an arithmetic group (in a suitable product of unitary groups).

To do this, we first relate the monodromy representation to certain well known representations of the braid group (or the pure braid group), namely the Burau-Gassner representations evaluated at d-th roots of unity.

We then prove the arithmeticity of the image of the latter representations in the range n>2d by using old results on arithmeticity of certain integral groups generated by unipotent elements.

Talk-2: Cyclic Coverings of the Projective Line and Arithmetic Groups -II.

Speaker: T. N. Venkataramana, TIFR, Mumbai. (Video is here and Slides are here)

Time: 31st March, Tuesday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Sandip Singh, IIT Bombay.

Abstract: Same as talk-1 above.

Seminars in April

Week 14

Talk-1: "Introduction to the Modular Method". (Video is here and Slides are here)

Speaker - Nicolas Billerey, University Clermont Auvergne, France

Time: 5th April, Tuesday 19:00 Hrs IST (GMT +5:30).

Chair for the Talk: Manami Roy, Fordham University New York.

Abstract: The modular method is a powerful approach to Diophantine equations which was first introduced by Frey and Serre for the case of Fermat's Last Theorem. In its original version, it is based on the modularity of semistable elliptic curves over Q proved by Wiles and relies on deep results on Galois representations due to Mazur and Ribet.

In this lecture I will explain the main steps of the modular method following first the classical FLT proof. Then I will discuss some of the issues that appear when applying extensions of this method to other Fermat type equations.

Talk-2: "Rational Isogenies of Prime Degree". (Video is here and Slides are here)

Speaker - Philippe Michaud-Jacobs, University of Warwick.

Time: 7th April, Thursday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Mihir Seth, IISc Bangalore.

Abstract: Let E be an elliptic curve over Q. For which primes p does E admit a rational p-isogeny? This question was answered by Mazur in 1978, who proved that the only such primes are p = 2,3,5,7,11,13,17,19,37,43,67, and 163. This is a fundamental result in number theory and one of the key components of the proof of Fermat's Last Theorem. The aim of this talk is to give an accessible overview of Mazur's proof of this result. We will see how modular curves and their Jacobians, as well as Galois representations of elliptic curves, play a key role in the proof.

Week 15

Talk-1: "Diophantine Applications of Modularity". (Video is here and Slides are here)

Speaker: Samir Siksek, University of Warwick.

Time: 12th April, Tuesday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Shaunak Deo, IISc Bangalore.

Abstract: The machinery of Galois representations and modular forms allows us to relate solutions of certain Diophantine equations to modular forms of weight 2 and small level. We study variants of this argument that allow us to relate solutions of certain Diophantine equations with large exponents to solutions to S-unit equations.

This for example can be used to prove an asymptotic version of Fermat's Last Theorem over some infinite families of number fields (following in the footsteps of Serre, Mazur, Kraus, Freitas, Ozman, Kara, Isik, Mocanu and many others).

Talk-2: "Overview of Modularity of Elliptic Curves over Totally Real Fields".

Speaker: Nuno Freitas, ICMAT Madrid. (Video is here and Slides are here)

Time: 14th April, Thursday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Narasimha Kumar, IIT Hyderabad.

Abstract: The modularity of all elliptic curves over Q has been known since the work of Wiles and Breuil, Conrad, Diamond, and Taylor. Essential to this proof are modularity lifting theorems and a 3-5 modularity switching argument.

In this talk, we will overview recent attempts to carry over this approach to totally real fields. More precisely, we will discuss how recent progress on modularity lifting theorems together with a 3-5-7 modularity switching argument allows us to reduce the proof of modularity of elliptic curves over a totally real field K to a calculation of K-points on certain modular curves. This final calculation is often impractical but it has been successfully completed for all quadratic and cubic totally real fields, yielding modularity of all elliptic curves over such fields.

Week 16

Talk-1: "Modularity of Rigid Galois Representations." (Video is here and Slides are here)

Speaker - Anwesh Ray, UBC Vancouver.

Time: 19th April, Tuesday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Devendra Tiwari, BP Pune.

Abstract: A hypergeometric abelian variety is an abelian scheme over $\mathbb{P}^1-\{0,1,\infty\}$. Such abelian varieties are of natural interest when studying generalizations of Fermat's equation. Associated to a hypergeometric abelian variety with multiplications by a totally real field, we consider the associated rigid Galois representation. This Galois representation is fibred over $\mathbb{P}^1-\{0,1,\infty\}$.

We study the properties of such Galois representations and explain what it means for such Galois representations to be modular. Finally, we explain a result of Darmon, which establishes the modularity of rigid Galois representations, assuming a modularity lifting conjecture over totally real fields.

Talk-2: "Darmon's Program - I". (Video is here and Slides are here)

Speaker - Angelos Koutsianas, Aristotle University of Thessaloniki, Greece.

Time: 21st April, Thursday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Narasimha Kumar, IIT Hyderabad.

Abstract: The proof of Fermat's Last Theorem by Wiles and Taylor-Wiles ushered in a new era in the study of Galois representations and applications to Diophantine problems. Motivated by the study of Beal's conjecture and generalized Fermat equations, Darmon proposed a program to tackle these problems using modularity of abelian varieties of GL_2 type over totally real fields. In this talk, we will give an introduction to his program, with an emphasis on the case of signature (p,p,r), and illustrating it with results which are currently possible for r = 5 (joint work with Imin Chen).

Week 17

Talk-1: "Torsion points of elliptic curves from Hellegouarch to the present day".

Speaker - Loic Merel, University of Paris. (Video is here and Slides are here)

Time: 26th April, Tuesday 19:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Dipendra Prasad, IIT Bombay.

Abstract: Yves Hellegouarch passed away on February 6 2022. He seems to have been the first, or perhaps the first with Demjanenko, as soon as 1969, to consider what is often called the «Frey elliptic curve», attached to an hypothetical solution to Fermat's Last Theorem. That led him to consideration on torsion points of elliptic curves, a few years before Mazur determined the possible torsion points on elliptic curve over Q. To continue an account of research to the present day, we will focus on what is now known and what is still conjectured about the asymptotic of the boundedness of torsion of elliptic curves over number fields.

Talk-2: "Darmon's Program-II". (Video is here and Slides are here)

Speaker - Imin Chen, Simon Fraser University.

Time: 28th April, Thursday 21:00 Hrs IST (GMT + 5:30).

Chair for the Talk: Guhan Venkat, Ashoka University.

Abstract: Darmon's program gives a general framework to resolve generalized Fermat equations of signature (p,q,r) where q, r are fixed and p is varying. In this talk, we will look at the case of signature (r, r, p). We review the known Frey varieties available in this signature and introduce a new one due to Kraus which is a hyperelliptic realization of the Frey abelian varieties constructed by Darmon for signature (r, r, p). We illustrate a number of new results which are possible using these explicit hyperelliptic curves for r = 7, 11 and ideas from Darmon's program (joint work with Nicolas Billerey, Luis Dieulefait, and Nuno Freitas).

Here concludes the second trimester of the Year-long series of virtual seminars.