BEGIN:VCALENDAR
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BEGIN:VEVENT
SUMMARY:Santosh Nadimpalli (IIT Kanpur)
DTSTART:20200501T110000Z
DTEND:20200501T120000Z
DTSTAMP:20260404T111213Z
UID:NTdL/1
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/NTdL/
 1/">Linkage principle and base change for ${\\rm GL}_2$</a>\nby Santosh Na
 dimpalli (IIT Kanpur) as part of Number theory during lockdown\n\n\nAbstra
 ct\nLet $l$ and $p$ be two distinct odd primes. Let $F$ be a finite extens
 ion of $Q_p$\, and let $E$ be a finite Galois extension of $F$ with $[E: F
 ]=l$.\nLet $(\\pi\, V)$ be a cuspidal representation of ${\\rm GL}_2(F)$ w
 ith an\nintegral central character. Let $(\\pi_E\, W)$ be the ${\\rm GL}_2
 (E)$\nrepresentation obtained by base change of $\\pi$. The Galois group o
 f $E/F$\,\ndenoted by $G$\, acts on $\\pi_E$. We show that the zeroth Tate
  cohomology\ngroup of $\\pi_E$\, as a $G$-module\, is isomorphic to the Fr
 obenius twist of\nthe mod-$l$ reduction of $\\pi_F$. We use Kirillov model
  to prove this\nresult. The first half of the lecture will be a review of 
 some preliminary\nresults on Kirillov model\, and in the latter half\, I w
 ill explain the proof\nof the above result.\n
LOCATION:https://stable.researchseminars.org/talk/NTdL/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abhishek Saha (Queen Mary University of London)
DTSTART:20200508T110000Z
DTEND:20200508T120000Z
DTSTAMP:20260404T111213Z
UID:NTdL/2
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/NTdL/
 2/">Critical L-values and congruence primes for Siegel modular forms</a>\n
 by Abhishek Saha (Queen Mary University of London) as part of Number theor
 y during lockdown\n\n\nAbstract\nI will explain some recent joint work wit
 h Pitale and\nSchmidt where we obtain an explicit integral representation 
 for the\ntwisted standard L-function on GSp_{2n} \\times GL_1 associated t
 o a\nholomorphic vector-valued Siegel cusp form of degree n and arbitrary\
 nlevel\, and a Dirichlet character. By combining this integral\nrepresenta
 tion with a detailed arithmetic study of nearly holomorphic\nSiegel cusp f
 orms (joint with Pitale\, Schmidt\, and Horinaga) we are\nable to prove an
  algebraicity result for the critical L-values  on\nGSp_{2n} \\times GL_1.
  To refine this result further\, we prove that the\npullback of the nearly
  holomorphic Eisenstein series that appears in\nour integral representatio
 n is actually cuspidal in each variable and\nhas nice p-adic arithmetic pr
 operties. This directly leads to a result\non congruences between Hecke ei
 genvalues of two Siegel cusp forms of\ndegree 2 modulo primes dividing a c
 ertain quotient of L-values.\nFinally\, I will describe a second\, more re
 fined version of our\ncongruence theorem\, that is obtained by looking at 
 Arthur packets and\nthe refined Gan-Gross-Prasad conjecture in this partic
 ular setup.\n
LOCATION:https://stable.researchseminars.org/talk/NTdL/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akshaa Vatwani (IIT Gandhinagar)
DTSTART:20200515T110000Z
DTEND:20200515T120000Z
DTSTAMP:20260404T111213Z
UID:NTdL/3
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/NTdL/
 3/">Logarithmic mean values of multiplicative functions</a>\nby Akshaa Vat
 wani (IIT Gandhinagar) as part of Number theory during lockdown\n\n\nAbstr
 act\nA general mean-value theorem for multiplicative functions taking valu
 es in the unit disc was given by Wirsing (1967) and Halász (1968). We con
 sider a multiplicative function f belonging to a certain class of arithmet
 ical functions and let F(s) be the associated Dirichlet series. In this se
 tting\, 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 belo
 nging to our class. \nWe also report on some recent work showing that this
  zero free region is optimal. This is joint work with Arindam Roy (UNC Cha
 rlotte).\n
LOCATION:https://stable.researchseminars.org/talk/NTdL/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Harald Helfgott (Univ. Gottingen/CNRS/Inst. Math. Jussieu)
DTSTART:20200522T110000Z
DTEND:20200522T120000Z
DTSTAMP:20260404T111213Z
UID:NTdL/4
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/NTdL/
 4/">Optimality of the logarithmic upper-bound sieve\, with explicit estima
 tes (joint with Emanuel Carneiro\, Andrés Chirre and Julian Mejía-Corder
 o)</a>\nby Harald Helfgott (Univ. Gottingen/CNRS/Inst. Math. Jussieu) as p
 art of Number theory during lockdown\n\n\nAbstract\nAt the simplest level\
 , an upper bound sieve of Selberg type is a choice of rho(d)\, d<=D\, with
  rho(1)=1\, such that\n\nS = \\sum_{n\\leq N} \\left(\\sum_{d|n} \\mu(d) \
 \rho(d)\\right)^2\n\nis as small as possible.\n\nThe 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 c
 ontinuous 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 Selbe
 rg's choice.\n\nThe most common choice of eta is that of Barban-Vehov (196
 8)\, 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-or
 der term\, which is negative and which we determine explicitly.\n
LOCATION:https://stable.researchseminars.org/talk/NTdL/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marc Technau (Graz University of Technology)
DTSTART:20200529T110000Z
DTEND:20200529T120000Z
DTSTAMP:20260404T111213Z
UID:NTdL/5
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/NTdL/
 5/">Metric results on summatory arithmetic functions on Beatty sets and be
 yond</a>\nby Marc Technau (Graz University of Technology) as part of Numbe
 r theory during lockdown\n\n\nAbstract\nThe \\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 ge
 neralised 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 interesti
 ng function $f\\colon\\mathbb{N}\\to\\mathbb{C}$ along such Beatty sets. (
 Here $\\lfloor\\xi\\rfloor$ denotes the integer part of a real number $\\x
 i$.) For fixed $\\alpha$\, the quality of such results is usually intricat
 ely 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\n\n\\[ \\Bigl\\lvert \\su
 m_{\\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 \\]\n\nholds for almost all $\\alpha>
 1$ with respect to the Lebesgue measure. This significantly improves a bea
 utiful earlier result due to Abercrombie\, Banks\, and Shparlinski. The pr
 oof uses a recent Fourier-analytic result of Lewko and {Radziwi\\l\\l} bas
 ed 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-S
 haprio sequences $\\lbrace\\\, \\lfloor n^c \\rfloor : n\\in\\mathbb{N} \\
 \,\\rbrace$ ($c>1$) of a related spirit.\n
LOCATION:https://stable.researchseminars.org/talk/NTdL/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ravi Raghunathan (IIT Bombay)
DTSTART:20200605T110000Z
DTEND:20200605T120000Z
DTSTAMP:20260404T111213Z
UID:NTdL/6
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/NTdL/
 6/">Beyond the Selberg Class: $0\\le d_F\\le 2$</a>\nby Ravi Raghunathan (
 IIT Bombay) as part of Number theory during lockdown\n\n\nAbstract\nI will
  define a class of Dirichlet series $\\mathfrak{A}^{#}$ which strictly con
 tains the extended Selberg class as well as several $L$-functions (includi
 ng the tensor product\, symmetric square and exterior square $L$-functions
  of\nautomorphic representations of $GL_n$. I will describe a number of cl
 assification results which generalise the work of\nKaczorowski and Perelli
  and provide simpler proofs in many cases. Time permitting\, I will discus
 s some applications concerning the zero sets of $L$ functions. Some of the
  results have been obtained in collaboration with R. Balasubramanian.\n
LOCATION:https://stable.researchseminars.org/talk/NTdL/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ritabrata Munshi (Indian Statistical Institute\, Kolkata)
DTSTART:20200612T110000Z
DTEND:20200612T120000Z
DTSTAMP:20260404T111213Z
UID:NTdL/7
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/NTdL/
 7/">Delta methods and subconvexity</a>\nby Ritabrata Munshi (Indian Statis
 tical Institute\, Kolkata) as part of Number theory during lockdown\n\n\nA
 bstract\nWe will discuss some recent progress in the subconvexity problem\
 , with a focus on the results obtained via the delta method.\n
LOCATION:https://stable.researchseminars.org/talk/NTdL/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Efthymios Sophos (University of Glasgow)
DTSTART:20200619T110000Z
DTEND:20200619T120000Z
DTSTAMP:20260404T111213Z
UID:NTdL/8
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/NTdL/
 8/">Schinzel Hypothesis with probability 1 and rational points</a>\nby Eft
 hymios Sophos (University of Glasgow) as part of Number theory during lock
 down\n\n\nAbstract\nSchinzel's Hypothesis states that there are infinitely
  many primes represented by any integer polynomial satisfying the necessar
 y congruence assumptions. Equivalently\, there exists at least one prime r
 epresented by any such polynomial. The problem is completely open\, except
  in the very special case of polynomials of degree 1. We shall describe ou
 r recent proof of the existence version of Schinzel's Hypothesis for almos
 t all polynomials\, preprint: https://arxiv.org/abs/2005.02998.\nWe 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 littl
 e is known about their arithmetic.\nThe talk is based on joint work with A
 lexei Skorobogatov.\n
LOCATION:https://stable.researchseminars.org/talk/NTdL/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ade Irma Suriajaya (Kyushu University at Fukuoka\, Japan)
DTSTART:20200626T110000Z
DTEND:20200626T120000Z
DTSTAMP:20260404T111213Z
UID:NTdL/9
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/NTdL/
 9/">The Julia line of a Riemann-type functional equation</a>\nby Ade Irma 
 Suriajaya (Kyushu University at Fukuoka\, Japan) as part of Number theory 
 during lockdown\n\n\nAbstract\nThe notion of a Julia line is a concept int
 roduced by Gaston Julia about one hundred years ago in his improvement upo
 n Picard's Great Theorem. In this talk we apply this idea to Dirichlet ser
 ies satisfying a Riemann-type functional equation (more precisely\, Dirich
 let series with periodic coefficients and\, if there is enough time\, elem
 ents of theextended Selberg class) and discuss aspects of their value-dist
 ribution. This is joint work in progress with Jörn Steuding  (University 
 of Würzburg) and Thanasis Sourmelidis (Graz University of Technology)\, a
 nd it extends previous joint work of Jörn Steuding with Justas Kalpokas a
 nd Maxim Korolev (Steklov Mathematical Institute of Russian Academy of Sci
 ences).\n
LOCATION:https://stable.researchseminars.org/talk/NTdL/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gergely Harcos (Rényi Institute\, Budapest\, Hungary)
DTSTART:20200703T110000Z
DTEND:20200703T120000Z
DTSTAMP:20260404T111213Z
UID:NTdL/10
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/NTdL/
 10/">The density hypothesis for horizontal families of lattices</a>\nby Ge
 rgely Harcos (Rényi Institute\, Budapest\, Hungary) as part of Number the
 ory during lockdown\n\n\nAbstract\nLet G be a semisimple real Lie group wi
 thout compact factors and\nGamma an arithmetic lattice in G. Sarnak and Xu
 e formulated a conjecture\non the multiplicity with which a given irreduci
 ble unitary representation\nof G occurs in the right regular representatio
 n of G on L^2(Gamma\\G). It\nis known for the groups SL(2\,R) and SL(2\,C)
  by the work of Sarnak-Xue\n(1991) and Huntley-Katznelson (1993). I will r
 eport on recent joint work\nwith Mikołaj Frączyk\, Péter Maga\, and Djo
 rdje Milićević\, where we prove a\nstrong\, effective version of the con
 jecture for products of SL(2\,R)'s and\nSL(2\,C)'s. We consider congruence
  lattices coming from quaternion algebras\nover number fields of bounded d
 egree\, and we address uniformity in all\narchimedean parameters.\n
LOCATION:https://stable.researchseminars.org/talk/NTdL/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julia Brandes (Chalmers University of Technology\, Gothenburg\, Sw
 eeden)
DTSTART:20200710T110000Z
DTEND:20200710T120000Z
DTSTAMP:20260404T111213Z
UID:NTdL/11
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/NTdL/
 11/">Approximations to Weyl sums</a>\nby Julia Brandes (Chalmers Universit
 y of Technology\, Gothenburg\, Sweeden) as part of Number theory during lo
 ckdown\n\n\nAbstract\nWe 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 surpri
 sing conclusions regarding the size of such exponential sums on diagonal s
 lices of the unit torus. As an application\, we improve bounds for the fra
 ctal dimension of solutions to the Schrödinger and Airy equations. This i
 s joint work with S. T. Parsell\, K. Poulias\, G. Shakan and R. C. Vaughan
 . Link to the paper: https://arxiv.org/abs/2001.05629\n
LOCATION:https://stable.researchseminars.org/talk/NTdL/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ping Xi (Xi'an Jiaotong University\, Xi'an\, China)
DTSTART:20200717T110000Z
DTEND:20200717T120000Z
DTSTAMP:20260404T111213Z
UID:NTdL/12
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/NTdL/
 12/">On the modular structure of Kloosterman sums after Katz</a>\nby Ping 
 Xi (Xi'an Jiaotong University\, Xi'an\, China) as part of Number theory du
 ring lockdown\n\n\nAbstract\nIt is widely believed that Kloosterman sums s
 hould behave randomly in certain suitable families\, and it is particularl
 y difficult in the horizontal sense. Motivated by deep observations on ell
 iptic curves\, Nicholas Katz proposed three problems on sign change\, equi
 distribution and modular structure of Kloosterman sums in 1980. In this ta
 lk\, we will focus on the modular structures and present some recent progr
 esses towards this problem made by analytic number theory combining certai
 n tools from $\\ell$-adic cohomology.\n
LOCATION:https://stable.researchseminars.org/talk/NTdL/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Somnath Jha (IIT Kanpur & IIT Goa\, India)
DTSTART:20200724T110000Z
DTEND:20200724T120000Z
DTSTAMP:20260404T111213Z
UID:NTdL/13
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/NTdL/
 13/">Root number and multiplicities for Artin twists</a>\nby Somnath Jha (
 IIT Kanpur & IIT Goa\, India) as part of Number theory during lockdown\n\n
 \nAbstract\nGiven a Galois extension of number fields K/F and two elliptic
  curves A\, B  which are congruent mod p\, we will discuss the relation be
 tween the p-parity conjecture of A twisted by \\sigma and that of B twiste
 d by \\sigma for an irreducible\, self dual\, Artin representation \\sigma
  of the Galois group of K/F. \n\nThis is a joint work with Sudhanshu Shekh
 ar and Tathagata Mandal.\n
LOCATION:https://stable.researchseminars.org/talk/NTdL/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Shparlinski (The University of New South Wales\, Sydney\, Aus
 tralia)
DTSTART:20200731T110000Z
DTEND:20200731T120000Z
DTSTAMP:20260404T111213Z
UID:NTdL/14
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/NTdL/
 14/">Integers of prescribed arithmetic structure in residue classes</a>\nb
 y Igor Shparlinski (The University of New South Wales\, Sydney\, Australia
 ) as part of Number theory during lockdown\n\n\nAbstract\nWe give an overv
 iew of recent results about the distribution of some special integers in r
 esidues classes modulo a large integer $q$. Questions of this type were in
 troduced by Erdos\, Odlyzko and Sarkozy (1987)\, who considered products o
 f two primes as a relaxation of the classical question about the distribut
 ion of primes in residue classes. Since that time\, numerous variations ha
 ve appeared for different sequences of integers. The types of numbers we d
 iscuss include smooth\, square-free\, square-full and almost primes intege
 rs. 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.\n
LOCATION:https://stable.researchseminars.org/talk/NTdL/14/
END:VEVENT
END:VCALENDAR