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Talks

**Talk 17**

**Title:** Integers of prescribed arithmetic structure in residue classes

**Time:** Friday, 31th July 2020, 16:30 (IST - Indian time)

**Speaker: ** Igor Shparlinski (University of New South Wales, Sydney)

**Abstract: **We give an overview of recent results about the distribution of some special integers in residues classes modulo a large integer $q$. Questions of this type were introduced by Erdos, Odlyzko and Sarkozy (1987), who considered products of two primes as a relaxation of the classical question about the distribution of primes in residue classes. Since that time, numerous variations have appeared for different sequences of integers. The types of numbers we discuss include smooth, square-free, square-full and almost primes integers. We also expose, without going into technical details, the wealth of different techniques behind these results: sieve methods, bounds of short Kloosterman sums, bounds of short character sums and many others.

**Talk 16**

**Title: **Root number and multiplicities for Artin Twists

**Time:** Friday, 24th July 2020, 16:30 (IST - Indian time)

**Speaker: ** Somnath Jha (IIT Kanpur and IIT Goa)

**Abstract:** Given a Galois extension of number fields K/F and two elliptic curves A, B which are congruent mod p, we will discuss the relation between the p-parity conjecture of A twisted by \sigma and that of B twisted by \sigma for an irreducible, self dual, Artin representation \sigma of the Galois group of K/F. This is a joint work with Sudhanshu Shekhar and Tathagata Mandal.

**Talk 15**

**Title:** On the modular structure of Kloosterman sums after Katz

**Time:** Friday, 17th July 2020, 16:30 (IST - Indian time)

**Speaker: **Ping Xi (Xi'an Jiaotong University of Technology)

**Abstract:** It is widely believed that Kloosterman sums should behave randomly in certain suitable families, and it is particularly difficult in the horizontal sense. Motivated by deep observations on elliptic curves, Nicholas Katz proposed three problems on sign change, equidistribution and modular structure of Kloosterman sums in 1980. In this talk, we will focus on the modular structures and present some recent progresses towards this problem made by analytic number theory combining certain tools from $\ell$-adic cohomology.

**Talk 14**

**Title:** Approximations to Weyl sums

**Time:** Friday, 10th July 2020, 16:30 (IST - Indian time)

**Speaker: **Julia Brandes (Chalmers University of Technology, Gothenburg)

**Abstract:** We study two-dimensional Weyl sums involving a k-th power and a linear term. In particular, we establish asymptotics for such sums involving lower order main terms. This allows us to draw some surprising conclusions regarding the size of such exponential sums on diagonal slices of the unit torus. As an application, we improve bounds for the fractal dimension of solutions to the Schrödinger and Airy equations. This is joint work with S. T. Parsell, K. Poulias, G. Shakan and R. C. Vaughan. Link to the paper: https://arxiv.org/abs/2001.05629

**Talk 13**

**Title: **The density hypothesis for horizontal families of lattices

**Time:** Friday, 3rd July 2020, 16:30 (IST - Indian time)

**Speaker: **Gergely Harcos (Rényi Institute, Budapest)

**Abstract: **Let G be a semisimple real Lie group without compact factors and Gamma an arithmetic lattice in G. Sarnak and Xue formulated a conjecture on the multiplicity with which a given irreducible unitary representation of G occurs in the right regular representation of G on L^2(Gamma\G). It is known for the groups SL(2,R) and SL(2,C) by the work of Sarnak-Xue(1991) and Huntley-Katznelson (1993). I will report on recent joint work with Mikołaj Frączyk, Péter Maga, and Djordje Milićević, where we prove a strong, effective version of the conjecture for products of SL(2,R)'s and SL(2,C)'s. We consider congruence lattices coming from quaternion algebrasover number fields of bounded degree, and we address uniformity in all archimedean parameters.

**Talk 12**

**Title:** The Julia line of a Riemann-type functional equation

**Time:** Friday, 26th June 2020, 16:30 (IST - Indian time)

**Speaker: **Ade Irma Suriajaya (Kyushu University at Fukuoka, Japan)

**Abstract:** The notion of a Julia line is a concept introduced by Gaston Julia about one hundred years ago in his improvement upon Picard's Great Theorem. In this talk we apply this idea to Dirichlet series satisfying a Riemann-type functional equation (more precisely, Dirichlet series with periodic coefficients and, if there is enough time, elements of the extended Selberg class) and discuss aspects of their value-distribution. This is joint work in progress with Jörn Steuding (University of Würzburg) and Thanasis Sourmelidis (Graz University of Technology), and it extends previous joint work of Jörn Steuding with Justas Kalpokas and Maxim Korolev (Steklov Mathematical Institute of Russian Academy of Sciences).

**Talk 11**

**Title:** Schinzel Hypothesis with probability 1 and rational points

**Time:** Friday, 19th June 2020, 16:30 (IST - Indian time)

**Speaker:** Efthymios Sophos (University of Glasgow)

**Abstract:** Schinzel's Hypothesis states that there are infinitely many primes represented by any integer polynomial satisfying the necessary congruence assumptions. Equivalently, there exists at least one prime represented by any such polynomial. The problem is completely open, except in the very special case of polynomials of degree 1. We shall describe our recent proof of the existence version of Schinzel's Hypothesis for almost all polynomials, preprint: https://arxiv.org/abs/2005.02998.We apply our result to showing that generalised Châtelet surfaces have a rational point with positive probability. These surfaces play an important role in the Brauer-Manin obstruction in arithmetic geometry, however, very little is known about their arithmetic.The talk is based on joint work with Alexei Skorobogatov.**Title:** Delta methods and subconvexity

**Talk 10**

**Title: **Delta methods and subconvexity

**Time:** Friday, 12th June 2020, 16:30 (IST - Indian time)

**Speaker: **Ritabrata Munshi (Indian Statistical Institute Kolkata)

**Abstract: **We will discuss some recent progress in the subconvexity problem, with a focus on the results obtained via the delta method.

**Talk 9**

**Title:** Beyond the Selberg Class: $0\le d_F\le 2$

**Time:** Friday, 5th June 2020, 16:30 (IST - Indian time)

**Speaker: **Ravi Raghunathan (IIT Bombay)

**Abstract: **I will define a class of Dirichlet series $\mathfrak^{#}$ which strictly contains the extended Selberg class as well as several $L$-functions (including the tensor product, symmetric square and exterior square $L$-functions of automorphic representations of $GL_n$. I will describe a number of classification results which generalise the work of Kaczorowski and Perelli and provide simpler proofs in many cases. Time permitting, I will discuss some applications concerning the zero sets of $L$ functions. Some of the results have been obtained in collaboration with R. Balasubramanian.

**Talk 8**

**Title:** Metric results on summatory arithmetic functions on Beatty sets and beyond

**Time:** Friday, 29th May 2020, 16:30 (IST)

**Speaker: **Marc Technau (Graz University of Technology)

**Abstract: **The \emph $\mathcal(\alpha) = \lbrace\, \lfloor n\alpha \rfloor : n\in\mathbb \,\rbrace$ associated to a real number $\alpha>1$ may be viewed as a generalised arithmetic progression (consecutive elements differ by either $\lfloor \alpha \rfloor$ or $\lfloor \alpha \rfloor+1$) and there are numerous results in the literature on averages of arithmetically interesting function $f\colon\mathbb\to\mathbb$ along such Beatty sets. (Here $\lfloor\xi\rfloor$ denotes the integer part of a real number $\xi$.) For fixed $\alpha$, the quality of such results is usually intricately linked to Diophantine properties of $\alpha$. However, it turns out that the metric theory is much cleaner: in this talk I will discuss recent joint work with A.\ Zafeiropoulos showing that

\[ \Bigl\lvert \sum_{\substack{ 1\leq m\leq x \\ m\in \mathcal(\alpha) }} f(m) - \frac{\alpha} \sum_ f(m) \Bigr\rvert^2 \ll_ (\log x) (\log\log x)^ \sum_ \lvert f(m) \rvert^2 \]

holds for almost all $\alpha>1$ with respect to the Lebesgue measure. This significantly improves a beautiful earlier result due to Abercrombie, Banks, and Shparlinski. The proof uses a recent Fourier-analytic result of Lewko and based on the classical Carleson--Hunt inequality. Moreover, it can be shown that the above result is optimal (up to logarithmic factors) in a suitable sense. If time permits, I shall also discuss ongoing work on Piatetski-Shaprio sequences $\lbrace\, \lfloor n^c \rfloor : n\in\mathbb \,\rbrace$ ($c>1$) of a related spirit.

**Talk 7**

**Title:**Optimality of the logarithmic upper-bound sieve, with explicit estimates

(joint with Emanuel Carneiro, Andrés Chirre and Julian Mejía-Corde)

**Time:** Friday, 22nd May 2020, 16:30 (IST)

**Speaker:**Harald Helfgott (University of Göttingen / CNRS)

**Abstract:**At the simplest level, an upper bound sieve of Selberg type is a choice of rho(d), d<=D, with rho(1)=1, such that S = \sum_ \left(\sum_ \mu(d) \rho(d)\right)^2 is as small as possible. The optimal choice of rho(d) for given D was found by Selberg. However, for several applications, it is better to work with functions rho(d) that are scalings of a given continuous or monotonic function eta. The question is then what is the best function eta, and how does S for given eta and D compares to S for Selberg's choice. The most common choice of eta is that of Barban-Vehov (1968), which gives an S with the same main term as Selberg's S. We show that Barban and Vehov's choice is optimal among all eta, not just (as we knew) when it comes to the main term, but even when it comes to the second-order term, which is negative and which we determine explicitly.

**Talk 6**

**Title:** Logarithmic mean values of multiplicative functions

**Time:** Friday, 15th May 2020, 16:30 (IST - Indian time)

**Speaker: **Akshaa Vatwani (IIT Gandhinagar)

**Abstract: **A general mean-value theorem for multiplicative functions taking values in the unit disc was given by Wirsing (1967) and Halász (1968). We consider a multiplicative function f belonging to a certain class of arithmetical functions and let F(s) be the associated Dirichlet series. In this setting, we obtain new Halász-type results for the logarithmic mean value of f. More precisely, we give estimates in terms of the size of $|F(1+1/\log x)|$ and show that these estimates are sharp. As a consequence, we obtain a non-trivial zero-free region for partial sums of L-functions belonging to our class. We also report on some recent work showing that this zero free region is optimal. This is joint work with Arindam Roy (UNC Charlotte).

**Talk 5**

**Title:** Critical L-values and congruence primes for Siegel modular forms

**Time:** Friday, 8th May 2020, 16:30 (IST - Indian time)

**Speaker:** Abhishek Saha (Queen Mary, London)

**Abstract:**I will explain some recent joint work with Pitale and Schmidt where we obtain an explicit integral representation for the twisted standard L-function on GSp_ \times GL_1 associated to a holomorphic vector-valued Siegel cusp form of degree n and arbitrary level, and a Dirichlet character. By combining this integral representation with a detailed arithmetic study of nearly holomorphic Siegel cusp forms (joint with Pitale, Schmidt, and Horinaga) we are able to prove an algebraicity result for the critical L-values on GSp_ \times GL_1. To refine this result further, we prove that the pullback of the nearly holomorphic Eisenstein series that appears in our integral representation is actually cuspidal in each variable and has nice p-adic arithmetic properties. This directly leads to a result on congruences between Hecke eigenvalues of two Siegel cusp forms of degree 2 modulo primes dividing a certain quotient of L-values. Finally, I will describe a second, more refined version of our congruence theorem, that is obtained by looking at Arthur packets and the refined Gan-Gross-Prasad conjecture in this particular setup.

**Talk 4**

**Title:** Linkage principle and Base change for ${\rm GL}_2$.

**Time:** Friday, 1st May 2020, 16:30 (IST)

**Speaker:** Santosh Nadimpalli (IIT Kanpur)

**Abstract:**Let $l$ and $p$ be two distinct odd primes. Let $F$ be a finite extension of $Q_p$, and let $E$ be a finite Galois extension of $F$ with $[E: F]=l$.Let $(\pi, V)$ be a cuspidal representation of ${\rm GL}_2(F)$ with an integral central character. Let $(\pi_E, W)$ be the ${\rm GL}_2(E)$ representation obtained by base change of $\pi$. The Galois group of $E/F$, denoted by $G$, acts on $\pi_E$. We show that the zeroth Tate cohomology group of $\pi_E$, as a $G$-module, is isomorphic to the Frobenius twist of the mod-$l$ reduction of $\pi_F$. We use Kirillov model to prove this result. The first half of the lecture will be a review of some preliminary results on Kirillov model, and in the latter half, I will explain the proof of the above result.

**Talk 3**

**Title:** Values of inhomogeneous quadratic forms at integer points

**Time:** Friday, 24th April 2020, 16:30 (IST)

**Speaker:** Anish Ghosh (TIFR Mumbai)

**Abstract:**An inhomogeneous quadratic form is a quadratic form along with a shift. These forms arise in integrable systems, in number theory as well as in dynamics. I will describe two recent results on the values taken by such forms at integer points. The results, which complement each other, are proved using very different techniques. The first uses moment formulas of incomplete Eisenstein series, while the second uses asymptotic representation theory, namely quantitative versions of Kazhdan's property T. This work is joint with Dubi Kelmer (Boston College) and Shucheng Yu (Technion, Israel) and is based on the two papers https://arxiv.org/abs/1911.04739 and https://arxiv.org/abs/2001.10990.

**Talk 2**

**Title: **Products of primes in arithmetic progressions (and generalisations)

**Time:** Friday, 17th April 2020, 16:30 (IST)

**Speaker:** Olivier Ramaré (CNRS / Institut de Mathématiques de Marseille)

(work in common at different stages with Aled Walked, Priamvad Srivasta, Oriol Serra, R. Balasubramanian, Sanoli Gun and Jyothsnaa Sivaraman)

**Abstract:** We shall describe a path that enables us to show by a mix of Analytic Number Theory and Additive Combinatorics that every invertible class modulo q contains a product of exactly three primes, all of them below Cq^ for some large constant C. Notice that 15/8 < 2 ! This exponent is currently being decreased. The same path shows that there exists a product of four primes in this class, all of them of size at most Cq(\log q)^6. An analogous statement may be made with products of three completely split ideals in a narrow ray class group class modulo some integral ideal q. The current state of our work enables us to take the norm of these primes less than C x the norm of q to the 6, improving markedly on what is known about the bound for corresponding Linnik's constant (which is about 40). There are numerous questions concerning the dependencies on the parameter of the field.

**Talk 1**

**Title: **Restricted Diophantine approximation in quadratic number fields

**Time:** Friday, 10th April 2020 at 16:30 (IST)

**Speaker:** Stephan Baier (RKMVERI)

**Abstract:** Dirichlet's approximation theorem tells us that, given any irrational $\alpha$, the inequality $|\alpha-a/q| \le q^{-2}$ is satisfied for infinitely many fractions $a/q$ of coprime integers $a$ and $q$. The more difficult problem of approximating irrationals by fractions $a/p$ with denominator $p$ restricted to primes has a long history starting with Vinogradov, who managed to handle linear exponential sums over primes unconditionally, and culminating in Matomaki's work making use of bounds for averages of Kloosterman sums which marks the limit of the currently available technology. We report about recent progress on the same problem in quadratic number fields of class number 1. This is joint work with Marc Technau (University of Graz) in the case of imaginary quadratic fields and Dwaipayan Mazumder (RKMVERI) in the case of real quadratic fields. ** **