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17th January
17th January
9.00-9.20am: Opening Ceremony
Morning session : Chair - Kaneenika Sinha
9.30 to 10.20am : N. Saradha
Title: Thue Inequalities
Abstract: In 1844, Liouville showed that algebraic numbers cannot be approximated very well by rational numbers. This led to the theory of Diophantine Approximations. In 1909, Thue showed how such approximations can be used to prove the finiteness of solutions of certain Diophantine Equations and inequalities. These are of the shape $$|F(x,y)|\leq h$$ where $F(x,y)$ is an irreducible binary form of degree $r\geq 3$ and $h\geq 1$ is an integer. These are termed as Thue inequalities in his honour. Following his seminal work, several mathematicians developed his theory and many important papers were written. In this talk, we shall give an overview of these developments.
10.25 to 11.05am : Siddhi Pathak
Title: A conjecture of Erdos about the non-vanishing of L(1,f).
Abstract: Let f be an arithmetic function on integers, periodic with period N, such that f(n) is -1 or 1 when n is not a multiple of N, and 0 otherwise. One can associate a Dirichlet series with f in the natural way by setting L(s,f) = \sum_n f(n) n^(-s). In the 1950s, Erdos conjectured that the value L(1,f) is non-zero, whenever L(s,f) is entire. This question is in the same spirit as the non-vanishing of L(1,\chi) for non-principal Dirichlet characters \chi, which implies the infinitude of primes in arithmetic progression. In this talk, we discuss a new approach to Erdos's conjecture and present recent results obtained in a joint work with Abhishek Bharadwaj and Ram Murty.
11.30 to 12.10pm: Saniya Wagh
Title: Character sheaves on certain commutative group ind-schemes.
Abstract: The idea of duality on the category of perfect connected commutative unipotent groups algebraically closed field with ch(k)=p>0 ( cpu_k) goes back to Serre. The dual of an object in cpu_k can be thought of as the “moduli space” of multiplicative Qℓ-local systems on that object. In this talk, we extend the notion of Serre dual to a larger category of certain commutative unipotent group ind-schemes. As an application, we recover the known additive self-duality for local fields. This is based on joint work with Tanmay Despande.
Afternoon Session: Chair - Sanoli Gun
2.00-2.40pm: Kaneenika Sinha
Title: Zero-free regions for modular L-functions
Abstract: “Explicit” number theory is the name given to the study of what are called zero-free regions of the Riemann zeta function and other L-functions. An explicit determination of such regions often reveals deep arithmetic properties of the underlying object attached to the concerned L-function. More generally, it could refer to the use of “explicit”, often technical methods to understand an arithmetic object. We will discuss one such method of Stechkin, and apply it to obtain zero-free regions for modular L-functions. This is ongoing joint work with Alia Hamieh, Steven Creech, Jakob Streipel, Simran Khunger and Kin-Min (Kelvin) Tsang.
2.45-3.25pm: Himanshi Chanana
Title: Sum of the GL(2) Fourier coefficients over quadratics with arithmetic weights
Abstract: In analytic number theory, it is a classical problem to study the hidden structures underlying the Fourier coefficients of automorphic forms. An effective way is to observe their summatory function over specific sequences. The problem becomes more complex when these sequences are given by polynomials of degree 2 or higher. In this talk, we study the mean behavior of GL(2) Fourier coefficients attached to holomorphic forms, Maass forms, or Eisenstein series over a quadratic form with arithmetic weights. Using the circle method, we will establish an asymptotic formula for these sums.
3.30-3.45pm : Tea Break
3.50-4.20pm: Bidisha Roy
Title: Geometric interpretation of certain moments of Hypergeometric
character sums.
Abstract: In 1987, Greene introduced Gaussian hypergeometric functions over finite fields as analogues of the classical hypergeometric functions. Not only do they share arithmetic properties with their classical counter-parts, but they also exhibit connections with various arithmetic-geometric objects. In this talk, we will present explicit formulas for certain first and second-moment sums of families of Gaussian hypergeometric functions. We will then connect these expressions with the p-th coefficients of certain new-forms by relating the traces of Hecke operators on specific cuspspaces to the symmetric-moments of trace of certain elliptic curves.
18th January
18th January
Morning Session : Chair - Debargha Banerjee.
9.30 - 10.20am : Sujatha Ramdorai
This will be an online talk
Title: Galois cohomology of elliptic curves over anticyclotomic extensions
Abstract: Iwasawa theory provides a systematic techniques for studying the arithmetic of elliptic curves over number fields. In this talk we will provide an overview of some recent results on the Galois cohomology of elliptic curves over cyclotomic extensions of imaginary quadratic number fields. This is joint work with Nguyen Tam.
10.25 -11.05 am: Shalini Bhattacharya
Title: Reduction of Galois representations modulo 2
Abstract: I will report on an ongoing work with Arathy Venugopal on the reduction of crystalline representations modulo 2.
11.10-11.30 am : Tea Break
11.30 - 12.10 pm : Moni Kumari
Title: Prime-level Ramanujan-style congruences
Abstract: In 1916, Ramanujan proved the notable congruence \tau(n) = \sigma_{11}(n) mod 691, for all natural numbers n, relating the arithmetic functions τ(n) and σ11(n). This congruence connects the cuspidal Hecke eigenform Δ(z) with the non-cuspidal eigenform E12(z) modulo the prime 691 and has played a key role in developing modern modular form theory. Various methods have been used to prove, interpret, and generalize Ramanujan’s congruence. For newforms of prime level, partial results have been established. More recently, Gaba-Popa extended these results by determining the Atkin-Lehner eigenvalue of the newform using period polynomials, under certain technical assumptions. We refine their results under a mild assumption in this talk, applying a different approach.
Afternoon Session : Chair - Srimathy Srinivasan
2.00 - 2.40 pm: Preeti Raman
Title: Hermitian forms, Classical groups and Galois cohomology
Abstract: We discuss Galois cohomological invariants of Hermitian forms over central simple algebras with an involution over a field F. When the cohomological dimension of F is less than or equal to 3 there are classification results for such Hermitian forms in terms of these invariants. We also mention some applications of these classification results, especially in connection with R-equivalence classes of suitable adjoint groups.
2.45 - 3.25 pm: Sonam Garg
Title: Generalizations and Arithmetic Properties of the q-Riemann Zeta Function.
Abstract: Kurokawa and Wakayama (2003) initially introduced a q-analogue of Euler’s constant, exploring the irrationality of certain numbers related to a q-Euler constant. In this presentation, we extend their results and explore the linear independence properties for certain numbers involving a q-analogue of Euler’s constant. Additionally, we derive the closed-form expression for a q-analogue of the k-th Stieltjes constant, denoted as γk(q). Further, using Nesterenko’s result, we address a question raised by Erd˝ os in 1948 and using an answer to his question, we discuss the arithmetic nature of some infinite series involving γ1(2). Continuing further, we delve into q-analogues of multiple zeta functions, exploring their behaviour and interrelations. In particular, we calculate a mathematical expression for γ0,0(q), which serves as a “q” version of Euler’s constant with a height of 2. Our exploration then advances to introducing several q-variants of the double zeta function, examining their algebraic identities, and uncovering connections among them. Furthermore, we also investigate the coefficients of the Laurent series expansion of the q-analogue of the Hurwitz zeta function, which was introduced by Kurokawa and Wakayama in 2003
3.30 - 3.45pm : Tea Break
3.50 - 4.30pm : Jyothsnaa Sivaraman
Title: Counting ideals and related questions.
Abstract: The number of positive integers less than or equal to a positive real number x is given by [x], the integral part of x. The number of positive integers less than or equal to a positive real number x, in a given arithmetic progression modulo q, is given by x/q upto an error which is bounded above by 1. In the analogous setup of rings of integers of a number field, a result due to Weber gives an asymptotic with an error term for the number of integral ideals with absolute norm less than or equal to x. The implied constant in the O error term in Weber's theorem is effective but inexplicit. In this talk we will first talk about improving the constant given by Weber's method. We further extend these ideas towards counting ideals in ray classes (bounding the implied constant in the error term in a theorem of Tatuzawa in terms of the modulus and number field invariants). The work on these results is motivated by a myriad of applications. One important application is towards proving a number field analogue of a theorem of Ramar{\'e}, Srivastava and Serra on Linnik's problem of finding the least prime in an arithmetic progression. In addition, these ideas can be used to prove new variants of the Brun-Titchmarsh theorem and finding upper bounds for the average value of Ihara's Euler-Kronecker constants for certain families of ray class fields of imaginary quadratic fields. This talk will range over several papers which have been written jointly with various authors.
4.35- 5.15pm: Neelam Saikia
Title: Distribution of the Hessian values of Gaussian hypergeometric functions.
Abstract: In this talk we consider a special family of Gaussian hypergeometric functions whose entries are cubic and trivial characters over finite fields. The special values of these functions are known to give the Frobenius traces of families of Hessian elliptic curves. Using the theory of harmonic Maass forms and mock modular forms, we will discuss that the limiting distribution of these values is semi-circular, confirming the usual Sato-Tate distribution in this setting. This is a joint work with Ken Ono, Sudhir Pujahari and Hasan Saad.
19th January
19th January
Morning session : Chair - Chandrasheel Bhagwat
9.30 - 10.20am : Jaya Iyer
This will be an online talk
Title: Period index questions.
Abstract: Let k be a field and Br(k) the Brauer group of k. There are two numerical invariants attached to a Brauer class α ∈ Br(k); index(α) = sqrt[D : k] if α is represented by a central division algebra D over k and period(α) = order of α in Br(k). When the field is a function field of a curve over k, we expect these two invariants are the same under some assumption of local triviality of the Brauer element. We show this is true when genus of the curve is two and in higher genus the same holds in a quadratic extension of k. Joint work with R. Parimala
10.25am-11.05am: Ekata Saha
Title: On extensions of a question of Chowla on polynomial Pell equations.
Abstract: S. Chowla posed the problem of determining all quadratic integer polynomials $D(x)$ of the form $x^2+d$ such that the polynomial Pell equation $P(x)^2 - D(x) Q(x)^2=1$ has non-trivial solutions $P(x),Q(x)$ over integers. This is part of the open question of determining all polynomial $D(x)$ for which such polynomial Pell equations have non-trivial solutions. M.B. Nathanson answered Chowla's question to show that only admissible values of $d$ are $\pm 1, \pm 2$. Certain extensions of Chowla's question turn out to be rather delicate and exhibit some interesting features. In this talk, we will discuss some of them and report on joint works with Akanksha Gupta.
11.10 - 11.30am: Tea Break
11.30 - 12.10pm: Shivani Goel
Title: On the Pair correlation of real-valued vector sequences
Abstract: In this talk, we generalize the work of Boca and Zaharescu on local statistics of integer valued vector sequences and investigate the pair correlation function for real valued vector sequences. We show Poissonian behavior of the pair correlation function for certain classes of real valued vector sequences.
12.15pm - Concluding Remarks
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