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BEGIN:VEVENT
SUMMARY:Santosh Nadimpalli (IIT Kanpur)
DTSTART:20200501T110000Z
DTEND:20200501T120000Z
DTSTAMP:20260422T225717Z
UID:NTdL/1
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTdL/1/">Lin
 kage principle and base change for ${\\rm GL}_2$</a>\nby Santosh Nadimpall
 i (IIT Kanpur) as part of Number theory during lockdown\n\n\nAbstract\nLet
  $l$ 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 $[E: F]=l$.\n
 Let $(\\pi\, V)$ be a cuspidal representation of ${\\rm GL}_2(F)$ with an\
 nintegral central character. Let $(\\pi_E\, W)$ be the ${\\rm GL}_2(E)$\nr
 epresentation obtained by base change of $\\pi$. The Galois group of $E/F$
 \,\ndenoted by $G$\, acts on $\\pi_E$. We show that the zeroth Tate cohomo
 logy\ngroup of $\\pi_E$\, as a $G$-module\, is isomorphic to the Frobenius
  twist of\nthe mod-$l$ reduction of $\\pi_F$. We use Kirillov model to pro
 ve this\nresult. The first half of the lecture will be a review of some pr
 eliminary\nresults on Kirillov model\, and in the latter half\, I will exp
 lain the proof\nof the above result.\n
LOCATION:https://researchseminars.org/talk/NTdL/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abhishek Saha (Queen Mary University of London)
DTSTART:20200508T110000Z
DTEND:20200508T120000Z
DTSTAMP:20260422T225717Z
UID:NTdL/2
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTdL/2/">Cri
 tical L-values and congruence primes for Siegel modular forms</a>\nby Abhi
 shek Saha (Queen Mary University of London) as part of Number theory durin
 g lockdown\n\n\nAbstract\nI will explain some recent joint work with Pital
 e and\nSchmidt where we obtain an explicit integral representation for the
 \ntwisted standard L-function on GSp_{2n} \\times GL_1 associated to a\nho
 lomorphic vector-valued Siegel cusp form of degree n and arbitrary\nlevel\
 , and a Dirichlet character. By combining this integral\nrepresentation wi
 th a detailed arithmetic study of nearly holomorphic\nSiegel cusp forms (j
 oint with Pitale\, Schmidt\, and Horinaga) we are\nable to prove an algebr
 aicity result for the critical L-values  on\nGSp_{2n} \\times GL_1. To ref
 ine this result further\, we prove that the\npullback of the nearly holomo
 rphic Eisenstein series that appears in\nour integral representation is ac
 tually cuspidal in each variable and\nhas nice p-adic arithmetic propertie
 s. This directly leads to a result\non congruences between Hecke eigenvalu
 es of two Siegel cusp forms of\ndegree 2 modulo primes dividing a certain 
 quotient of L-values.\nFinally\, I will describe a second\, more refined v
 ersion of our\ncongruence theorem\, that is obtained by looking at Arthur 
 packets and\nthe refined Gan-Gross-Prasad conjecture in this particular se
 tup.\n
LOCATION:https://researchseminars.org/talk/NTdL/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akshaa Vatwani (IIT Gandhinagar)
DTSTART:20200515T110000Z
DTEND:20200515T120000Z
DTSTAMP:20260422T225717Z
UID:NTdL/3
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTdL/3/">Log
 arithmic mean values of multiplicative functions</a>\nby Akshaa Vatwani (I
 IT Gandhinagar) as part of Number theory during lockdown\n\n\nAbstract\nA 
 general mean-value theorem for multiplicative functions taking values in t
 he unit disc was given by Wirsing (1967) and Halász (1968). We consider a
  multiplicative function f belonging to a certain class of arithmetical fu
 nctions 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. M
 ore 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 t
 o our class. \nWe also report on some recent work showing that this zero f
 ree region is optimal. This is joint work with Arindam Roy (UNC Charlotte)
 .\n
LOCATION:https://researchseminars.org/talk/NTdL/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Harald Helfgott (Univ. Gottingen/CNRS/Inst. Math. Jussieu)
DTSTART:20200522T110000Z
DTEND:20200522T120000Z
DTSTAMP:20260422T225717Z
UID:NTdL/4
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTdL/4/">Opt
 imality of the logarithmic upper-bound sieve\, with explicit estimates (jo
 int with Emanuel Carneiro\, Andrés Chirre and Julian Mejía-Cordero)</a>\
 nby Harald Helfgott (Univ. Gottingen/CNRS/Inst. Math. Jussieu) as part of 
 Number theory during lockdown\n\n\nAbstract\nAt the simplest level\, an up
 per 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 continuo
 us or monotonic function eta. The question is then what is the best functi
 on eta\, and how does S for given eta and D compares to S for Selberg's ch
 oice.\n\nThe most common choice of eta is that of Barban-Vehov (1968)\, wh
 ich 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 ter
 m\, which is negative and which we determine explicitly.\n
LOCATION:https://researchseminars.org/talk/NTdL/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marc Technau (Graz University of Technology)
DTSTART:20200529T110000Z
DTEND:20200529T120000Z
DTSTAMP:20260422T225717Z
UID:NTdL/5
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTdL/5/">Met
 ric results on summatory arithmetic functions on Beatty sets and beyond</a
 >\nby Marc Technau (Graz University of Technology) as part of Number theor
 y during lockdown\n\n\nAbstract\nThe \\emph{Beatty set} $\\mathcal{B}(\\al
 pha) = \\lbrace\\\, \\lfloor n\\alpha \\rfloor : n\\in\\mathbb{N} \\\,\\rb
 race$ associated to a real number $\\alpha>1$ may be viewed as a generalis
 ed arithmetic progression (consecutive elements differ by either $\\lfloor
  \\alpha \\rfloor$ or $\\lfloor \\alpha \\rfloor+1$) and there are numerou
 s results in the literature on averages of arithmetically interesting func
 tion $f\\colon\\mathbb{N}\\to\\mathbb{C}$ along such Beatty sets. (Here $\
 \lfloor\\xi\\rfloor$ denotes the integer part of a real number $\\xi$.) Fo
 r fixed $\\alpha$\, the quality of such results is usually intricately lin
 ked to Diophantine properties of $\\alpha$. However\, it turns out that th
 e metric theory is much cleaner: in this talk I will discuss recent joint 
 work with A.\\ Zafeiropoulos showing that\n\n\\[ \\Bigl\\lvert \\sum_{\\su
 bstack{ 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 beautiful 
 earlier result due to Abercrombie\, Banks\, and Shparlinski. The proof use
 s a recent Fourier-analytic result of Lewko and {Radziwi\\l\\l} based on t
 he classical Carleson--Hunt inequality. Moreover\, it can be shown that th
 e 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 $\\lbrace\\\, \\lfloor n^c \\rfloor : n\\in\\mathbb{N} \\\,\\rbr
 ace$ ($c>1$) of a related spirit.\n
LOCATION:https://researchseminars.org/talk/NTdL/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ravi Raghunathan (IIT Bombay)
DTSTART:20200605T110000Z
DTEND:20200605T120000Z
DTSTAMP:20260422T225717Z
UID:NTdL/6
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTdL/6/">Bey
 ond the Selberg Class: $0\\le d_F\\le 2$</a>\nby Ravi Raghunathan (IIT Bom
 bay) as part of Number theory during lockdown\n\n\nAbstract\nI will define
  a class of Dirichlet series $\\mathfrak{A}^{#}$ which strictly contains t
 he extended Selberg class as well as several $L$-functions (including the 
 tensor product\, symmetric square and exterior square $L$-functions of\nau
 tomorphic representations of $GL_n$. I will describe a number of classific
 ation results which generalise the work of\nKaczorowski and Perelli and pr
 ovide simpler proofs in many cases. Time permitting\, I will discuss some 
 applications concerning the zero sets of $L$ functions. Some of the result
 s have been obtained in collaboration with R. Balasubramanian.\n
LOCATION:https://researchseminars.org/talk/NTdL/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ritabrata Munshi (Indian Statistical Institute\, Kolkata)
DTSTART:20200612T110000Z
DTEND:20200612T120000Z
DTSTAMP:20260422T225717Z
UID:NTdL/7
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTdL/7/">Del
 ta methods and subconvexity</a>\nby Ritabrata Munshi (Indian Statistical I
 nstitute\, Kolkata) as part of Number theory during lockdown\n\n\nAbstract
 \nWe will discuss some recent progress in the subconvexity problem\, with 
 a focus on the results obtained via the delta method.\n
LOCATION:https://researchseminars.org/talk/NTdL/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Efthymios Sophos (University of Glasgow)
DTSTART:20200619T110000Z
DTEND:20200619T120000Z
DTSTAMP:20260422T225717Z
UID:NTdL/8
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTdL/8/">Sch
 inzel Hypothesis with probability 1 and rational points</a>\nby Efthymios 
 Sophos (University of Glasgow) as part of Number theory during lockdown\n\
 n\nAbstract\nSchinzel's Hypothesis states that there are infinitely many p
 rimes represented by any integer polynomial satisfying the necessary congr
 uence assumptions. Equivalently\, there exists at least one prime represen
 ted by any such polynomial. The problem is completely open\, except in the
  very special case of polynomials of degree 1. We shall describe our recen
 t proof of the existence version of Schinzel's Hypothesis for almost all p
 olynomials\, preprint: https://arxiv.org/abs/2005.02998.\nWe apply our res
 ult to showing that generalised Châtelet surfaces have a rational point w
 ith positive probability. These surfaces play an important role in the Bra
 uer-Manin obstruction in arithmetic geometry\, however\, very little is kn
 own about their arithmetic.\nThe talk is based on joint work with Alexei S
 korobogatov.\n
LOCATION:https://researchseminars.org/talk/NTdL/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ade Irma Suriajaya (Kyushu University at Fukuoka\, Japan)
DTSTART:20200626T110000Z
DTEND:20200626T120000Z
DTSTAMP:20260422T225717Z
UID:NTdL/9
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTdL/9/">The
  Julia line of a Riemann-type functional equation</a>\nby Ade Irma Suriaja
 ya (Kyushu University at Fukuoka\, Japan) as part of Number theory during 
 lockdown\n\n\nAbstract\nThe notion of a Julia line is a concept introduced
  by Gaston Julia about one hundred years ago in his improvement upon Picar
 d's Great Theorem. In this talk we apply this idea to Dirichlet series sat
 isfying a Riemann-type functional equation (more precisely\, Dirichlet ser
 ies with periodic coefficients and\, if there is enough time\, elements of
  theextended Selberg class) and discuss aspects of their value-distributio
 n. This is joint work in progress with Jörn Steuding  (University of Wür
 zburg) and Thanasis Sourmelidis (Graz University of Technology)\, and it e
 xtends previous joint work of Jörn Steuding with Justas Kalpokas and Maxi
 m Korolev (Steklov Mathematical Institute of Russian Academy of Sciences).
 \n
LOCATION:https://researchseminars.org/talk/NTdL/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gergely Harcos (Rényi Institute\, Budapest\, Hungary)
DTSTART:20200703T110000Z
DTEND:20200703T120000Z
DTSTAMP:20260422T225717Z
UID:NTdL/10
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTdL/10/">Th
 e density hypothesis for horizontal families of lattices</a>\nby Gergely H
 arcos (Rényi Institute\, Budapest\, Hungary) as part of Number theory dur
 ing lockdown\n\n\nAbstract\nLet G be a semisimple real Lie group without c
 ompact factors and\nGamma an arithmetic lattice in G. Sarnak and Xue formu
 lated a conjecture\non the multiplicity with which a given irreducible uni
 tary representation\nof G occurs in the right regular representation 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 report o
 n recent joint work\nwith Mikołaj Frączyk\, Péter Maga\, and Djordje Mi
 lićević\, where we prove a\nstrong\, effective version of the conjecture
  for products of SL(2\,R)'s and\nSL(2\,C)'s. We consider congruence lattic
 es coming from quaternion algebras\nover number fields of bounded degree\,
  and we address uniformity in all\narchimedean parameters.\n
LOCATION:https://researchseminars.org/talk/NTdL/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julia Brandes (Chalmers University of Technology\, Gothenburg\, Sw
 eeden)
DTSTART:20200710T110000Z
DTEND:20200710T120000Z
DTSTAMP:20260422T225717Z
UID:NTdL/11
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTdL/11/">Ap
 proximations to Weyl sums</a>\nby Julia Brandes (Chalmers University of Te
 chnology\, Gothenburg\, Sweeden) as part of Number theory during lockdown\
 n\n\nAbstract\nWe study two-dimensional Weyl sums involving a k-th power a
 nd a linear term. In particular\, we establish asymptotics for such sums i
 nvolving lower order main terms. This allows us to draw some surprising co
 nclusions regarding the size of such exponential sums on diagonal slices o
 f the unit torus. As an application\, we improve bounds for the fractal di
 mension 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\n
LOCATION:https://researchseminars.org/talk/NTdL/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ping Xi (Xi'an Jiaotong University\, Xi'an\, China)
DTSTART:20200717T110000Z
DTEND:20200717T120000Z
DTSTAMP:20260422T225717Z
UID:NTdL/12
DESCRIPTION:Title: <a href="https://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 during lo
 ckdown\n\n\nAbstract\nIt is widely believed that Kloosterman sums should b
 ehave randomly in certain suitable families\, and it is particularly diffi
 cult in the horizontal sense. Motivated by deep observations on elliptic c
 urves\, Nicholas Katz proposed three problems on sign change\, equidistrib
 ution and modular structure of Kloosterman sums in 1980. In this talk\, we
  will focus on the modular structures and present some recent progresses t
 owards this problem made by analytic number theory combining certain tools
  from $\\ell$-adic cohomology.\n
LOCATION:https://researchseminars.org/talk/NTdL/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Somnath Jha (IIT Kanpur & IIT Goa\, India)
DTSTART:20200724T110000Z
DTEND:20200724T120000Z
DTSTAMP:20260422T225717Z
UID:NTdL/13
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTdL/13/">Ro
 ot number and multiplicities for Artin twists</a>\nby Somnath Jha (IIT Kan
 pur & IIT Goa\, India) as part of Number theory during lockdown\n\n\nAbstr
 act\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 between t
 he 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. \n\nThis is a joint work with Sudhanshu Shekhar and 
 Tathagata Mandal.\n
LOCATION:https://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:20260422T225717Z
UID:NTdL/14
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTdL/14/">In
 tegers of prescribed arithmetic structure in residue classes</a>\nby Igor 
 Shparlinski (The University of New South Wales\, Sydney\, Australia) as pa
 rt of Number theory during lockdown\n\n\nAbstract\nWe 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 introduce
 d by Erdos\, Odlyzko and Sarkozy (1987)\, who considered products of two p
 rimes as a relaxation of the classical question about the distribution of 
 primes in residue classes. Since that time\, numerous variations have appe
 ared 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 differe
 nt techniques behind these results: sieve methods\, bounds of short Kloost
 erman sums\, bounds of short character sums and many others.\n
LOCATION:https://researchseminars.org/talk/NTdL/14/
END:VEVENT
END:VCALENDAR