researchseminars.org

Santosh Nadimpalli (IIT Kanpur)
2020-05-01 11:00 - 12:00 UTC | https://researchseminars.org/talk/NTdL/1/
Title: Linkage principle and base change for ${\rm GL}_2$ (https://researchseminars.org/talk/NTdL/1/)
by Santosh Nadimpalli (IIT Kanpur) as part of Number theory during lockdown

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.

Abhishek Saha (Queen Mary University of London)
2020-05-08 11:00 - 12:00 | https://researchseminars.org/talk/NTdL/2/
Title: Critical L-values and congruence primes for Siegel modular forms (https://researchseminars.org/talk/NTdL/2/)
by Abhishek Saha (Queen Mary University of London) as part of Number theory during lockdown

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_{2n} \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_{2n} \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.

Akshaa Vatwani (IIT Gandhinagar)
2020-05-15 11:00 - 12:00 | https://researchseminars.org/talk/NTdL/3/
Title: Logarithmic mean values of multiplicative functions (https://researchseminars.org/talk/NTdL/3/)
by Akshaa Vatwani (IIT Gandhinagar) as part of Number theory during lockdown

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).

Harald Helfgott (Univ. Gottingen/CNRS/Inst. Math. Jussieu)
2020-05-22 11:00 - 12:00 | https://researchseminars.org/talk/NTdL/4/
Title: Optimality of the logarithmic upper-bound sieve, with explicit estimates (joint with Emanuel Carneiro, Andrés Chirre and Julian Mejía-Cordero) (https://researchseminars.org/talk/NTdL/4/)
by Harald Helfgott (Univ. Gottingen/CNRS/Inst. Math. Jussieu) as part of Number theory during lockdown

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_{n\leq N} \left(\sum_{d|n} \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.

Marc Technau (Graz University of Technology)
2020-05-29 11:00 - 12:00 | https://researchseminars.org/talk/NTdL/5/
Title: Metric results on summatory arithmetic functions on Beatty sets and beyond (https://researchseminars.org/talk/NTdL/5/)
by Marc Technau (Graz University of Technology) as part of Number theory during lockdown

Abstract
The \emph{Beatty set} $\mathcal{B}(\alpha) = \lbrace\, \lfloor n\alpha \rfloor : n\in\mathbb{N} \,\rbrace$ 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 $f\colon\mathbb{N}\to\mathbb{C}$ 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

\[ \Bigl\lvert \sum_{\substack{ 1\leq m\leq x \\ m\in \mathcal{B}(\alpha) }} f(m) - \frac{1}{\alpha} \sum_{1\leq m\leq x} f(m) \Bigr\rvert^2 \ll_{f,\alpha,\varepsilon} (\log x) (\log\log x)^{3+\varepsilon} \sum_{1\leq m\leq x} \lvert f(m) \rvert^2 \]

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 {Radziwi\l\l} 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 $\lbrace\, \lfloor n^c \rfloor : n\in\mathbb{N} \,\rbrace$ ($c>1$) of a related spirit.

Ravi Raghunathan (IIT Bombay)
2020-06-05 11:00 - 12:00 | https://researchseminars.org/talk/NTdL/6/
Title: Beyond the Selberg Class: $0\le d_F\le 2$ (https://researchseminars.org/talk/NTdL/6/)
by Ravi Raghunathan (IIT Bombay) as part of Number theory during lockdown

Abstract
I will define a class of Dirichlet series $\mathfrak{A}^{#}$ 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.

Ritabrata Munshi (Indian Statistical Institute, Kolkata)
2020-06-12 11:00 - 12:00 | https://researchseminars.org/talk/NTdL/7/
Title: Delta methods and subconvexity (https://researchseminars.org/talk/NTdL/7/)
by Ritabrata Munshi (Indian Statistical Institute, Kolkata) as part of Number theory during lockdown

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

Efthymios Sophos (University of Glasgow)
2020-06-19 11:00 - 12:00 | https://researchseminars.org/talk/NTdL/8/
Title: Schinzel Hypothesis with probability 1 and rational points (https://researchseminars.org/talk/NTdL/8/)
by Efthymios Sophos (University of Glasgow) as part of Number theory during lockdown

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.

Ade Irma Suriajaya (Kyushu University at Fukuoka, Japan)
2020-06-26 11:00 - 12:00 | https://researchseminars.org/talk/NTdL/9/
Title: The Julia line of a Riemann-type functional equation (https://researchseminars.org/talk/NTdL/9/)
by Ade Irma Suriajaya (Kyushu University at Fukuoka, Japan) as part of Number theory during lockdown

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 theextended 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).

Gergely Harcos (Rényi Institute, Budapest, Hungary)
2020-07-03 11:00 - 12:00 | https://researchseminars.org/talk/NTdL/10/
Title: The density hypothesis for horizontal families of lattices (https://researchseminars.org/talk/NTdL/10/)
by Gergely Harcos (Rényi Institute, Budapest, Hungary) as part of Number theory during lockdown

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 algebras
over number fields of bounded degree, and we address uniformity in all
archimedean parameters.

Julia Brandes (Chalmers University of Technology, Gothenburg, Sweeden)
2020-07-10 11:00 - 12:00 | https://researchseminars.org/talk/NTdL/11/
Title: Approximations to Weyl sums (https://researchseminars.org/talk/NTdL/11/)
by Julia Brandes (Chalmers University of Technology, Gothenburg, Sweeden) as part of Number theory during lockdown

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

Ping Xi (Xi'an Jiaotong University, Xi'an, China)
2020-07-17 11:00 - 12:00 | https://researchseminars.org/talk/NTdL/12/
Title: On the modular structure of Kloosterman sums after Katz (https://researchseminars.org/talk/NTdL/12/)
by Ping Xi (Xi'an Jiaotong University, Xi'an, China) as part of Number theory during lockdown

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.

Somnath Jha (IIT Kanpur & IIT Goa, India)
2020-07-24 11:00 - 12:00 | https://researchseminars.org/talk/NTdL/13/
Title: Root number and multiplicities for Artin twists (https://researchseminars.org/talk/NTdL/13/)
by Somnath Jha (IIT Kanpur & IIT Goa, India) as part of Number theory during lockdown

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.

Igor Shparlinski (The University of New South Wales, Sydney, Australia)
2020-07-31 11:00 - 12:00 | https://researchseminars.org/talk/NTdL/14/
Title: Integers of prescribed arithmetic structure in residue classes (https://researchseminars.org/talk/NTdL/14/)
by Igor Shparlinski (The University of New South Wales, Sydney, Australia) as part of Number theory during lockdown

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.
