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BEGIN:VEVENT
SUMMARY:Jakub Konieczny (Hebrew University of Jerusalem)
DTSTART;VALUE=DATE-TIME:20200525T111500Z
DTEND;VALUE=DATE-TIME:20200525T121500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/1
DESCRIPTION:Title: Automatic multiplicative sequences\nby Jakub Konieczny (Hebrew Univer
sity of Jerusalem) as part of Warsaw Number Theory Seminar\n\nLecture held
in room 403 at IMPAN.\n\nAbstract\nAutomatic sequences - that is\, sequen
ces computable by finite automata - give rise to one of the most basic mod
els of computation. As such\, for any class of sequences it is natural to
ask which sequences in it are automatic. In particular\, the question of c
lassifying automatic multiplicative sequences has attracted considerable a
ttention in the recent years. In the completely multiplicative case\, such
classification was obtained independently by S. Li and O. Klurman and P.
Kurlberg. The main topic of my talk will be the resolution of the general
case\, obtained in a recent preprint with M. Lemańczyk and C. Müllner.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Youcis (IMPAN Warsaw)
DTSTART;VALUE=DATE-TIME:20200601T111500Z
DTEND;VALUE=DATE-TIME:20200601T121500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/2
DESCRIPTION:Title: An approach to the characterization of the $p$-adic local Langlands corre
spondence\nby Alex Youcis (IMPAN Warsaw) as part of Warsaw Number Theo
ry Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nIn 2013 P. S
cholze provided an alternative proof of the local Langlands correspondence
(LLC) for GLn and\, in doing so\, Scholze gave a new characterization of
the LLC via a certain trace identity. In this talk the speaker will discus
s joint work with A. Bertoloni Meli showing that a generalization of this
trace identity characterizes the LLC for much more general groups if one a
ssumes standard expected properties of such a correspondence.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anna Szumowicz (Sorbonne Université\, IMJ-PRG)
DTSTART;VALUE=DATE-TIME:20200608T111500Z
DTEND;VALUE=DATE-TIME:20200608T121500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/3
DESCRIPTION:Title: Equidistribution in number fields\nby Anna Szumowicz (Sorbonne Univer
sité\, IMJ-PRG) as part of Warsaw Number Theory Seminar\n\nLecture held i
n room 403 at IMPAN.\n\nAbstract\nThe notion of $\\mathfrak{p}$-ordering c
omes from the work of Bhargava on integer-valued polynomials. Let $k$ be a
number field and let $\\mathcal{O}_{k}$ be its ring of integers. A sequen
ce of elements in $\\mathcal{O}_{k}$ is a simultaneous $\\mathfrak{p}$-ord
ering if it is equidistributed modulo every prime ideal in $\\mathcal{O}
_{k}$ as well as possible. We prove that $\\mathbb{Q}$ the only number fie
ld $k$ such that $\\mathcal{O}_{k}$ admits a simultaneous $\\mathfrak{p}$-
ordering. It is a joint work with Mikołaj Frączyk.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jolanta Marzec (TU Darmstadt)
DTSTART;VALUE=DATE-TIME:20200622T111500Z
DTEND;VALUE=DATE-TIME:20200622T121500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/4
DESCRIPTION:Title: Maass relations for Saito-Kurokawa lifts of higher levels\nby Jolanta
Marzec (TU Darmstadt) as part of Warsaw Number Theory Seminar\n\nLecture
held in room 403 at IMPAN.\n\nAbstract\nIt is known that a Siegel modular
form is a (classical) Saito-Kurokawa\nlift of an elliptic modular form if
and only if its Fourier coefficients\nsatisfy the so-called Maass relation
s. The first construction of such a\nlift was given by Maass using corresp
ondences between various modular\nforms. However\, in order to generalize
this lift to higher levels it\nis easier to use a construction coming from
representation theory. During\nthe talk we present history of this probl
em and briefly discuss the\naforementioned constructions. We also indicate
how one can read off the\nMaass relations from the latter. This work gene
ralizes an approach\nof Pitale\, Saha and Schmidt from the classical to a
higher level case.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adam Keilthy (University of Oxford)
DTSTART;VALUE=DATE-TIME:20200629T111500Z
DTEND;VALUE=DATE-TIME:20200629T121500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/5
DESCRIPTION:Title: The block filtration and motivic multiple zeta values\nby Adam Keilth
y (University of Oxford) as part of Warsaw Number Theory Seminar\n\nLectur
e held in room 403 at IMPAN.\n\nAbstract\nMultiple zeta values are a class
of transcendental numbers\, going back to Euler in the 1700s\nand with ti
es to the Riemann zeta function. They are found arising naturally in many
areas of mathematics and physics\, from algebraic geometry to Feynman ampl
itudes. Unlike in the case of single zeta values\, we know many algebraic
relations satisfied by multiple zeta values: the double shuffle relations\
, the associator relations\, the confluence relations. However it is unkno
wn if any of these sets of relations are complete. Assuming Grothendieck's
period conjecture\, a complete set of algebraic relations are given by th
e motivic relations\, arising from a connection to P^1 minus three points.
However these relations are inexplicit.\n\nIn this talk\, we introduce a
new filtration\, called the block filtration\, on the space of multiple ze
ta values. By considering the associated graded\, we describe several new
families of motivic relations\, that provide a complete description of rel
ations in low block degree. A generalisation of these results would thus p
rovide a complete description of relations among multiple zeta values and
aid in settling several open problems about the motivic Galois group of mi
xed Tate motives.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dorota Blinkiewicz (Adam Mickiewicz University\, Poznań)
DTSTART;VALUE=DATE-TIME:20201005T111500Z
DTEND;VALUE=DATE-TIME:20201005T121500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/6
DESCRIPTION:Title: Classes of extensions of commutative algebraic groups\nby Dorota Blin
kiewicz (Adam Mickiewicz University\, Poznań) as part of Warsaw Number Th
eory Seminar\n\n\nAbstract\nDuring the lecture I give explicit characteriz
ation of $n$-torsion elements in the group of extensions of commutative\,
smooth algebraic groups.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vesselin Dimitrov (University of Toronto)
DTSTART;VALUE=DATE-TIME:20201012T111500Z
DTEND;VALUE=DATE-TIME:20201012T121500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/7
DESCRIPTION:Title: Solution of the conjecture of Schinzel and Zassenhaus and some applicatio
ns\nby Vesselin Dimitrov (University of Toronto) as part of Warsaw Num
ber Theory Seminar\n\n\nAbstract\nWe will detail the full proof of an expl
icit form of the Schinzel-Zassenhaus conjecture: an algebraic integer of d
egree $n > 1$ is either a root of unity or else has at least one conjugate
of modulus exceeding $2^{1/(4n)}$. We furthermore obtain an extension of
the original conjecture over to the setting of holonomic functions\, with
an application to the smallest critical value for (certain) rational funct
ions.\n\nIn another application\, we would like to take the occasion to ra
ise the apparently unsolved problem of the essential irreducibility (up-to
the cyclotomic factor $X^2-X+1$ in degrees a multiple of 12) of $X^{2g} -
X^g(1+X+1/X) + 1$\, the characteristic polynomial of the integer reciproc
al Perron-Frobenius matrix of the smallest spectral radius in each given d
imension. Our explicit Schinzel-Zassenhaus bound allows for at most $10$ f
actors of each of these polynomials.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jakub Byszewski (Jagellonian University)
DTSTART;VALUE=DATE-TIME:20201019T111500Z
DTEND;VALUE=DATE-TIME:20201019T121500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/8
DESCRIPTION:Title: Finite order automorphisms of the ring of power series over a finite fiel
d\nby Jakub Byszewski (Jagellonian University) as part of Warsaw Numbe
r Theory Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nThe No
ttingham group at a prime $p$ is the group of (formal) power series $t+a_2
t^2+ a_3 t^3+ \\cdots$ in the variable $t$ with coefficients $a_i$ from t
he field with $p$ elements with the group operation given by composition o
f power series. This group is known to contain elements of order being an
arbitrary power of $p$. Elements of order $p$ have been classified by Klop
sch and have a nice description. For higher orders\, however\, only a hand
ful of examples have been known explicitly.\n\nIn the talk we will show ho
w to describe such series in closed computational form through finite auto
mata. This allows us to construct many explicit examples and formulate a n
umber of questions. The talk is based on joint work with Gunther Corneliss
en and Djurre Tijsma.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Danylo Radchenko (ETH Zürich)
DTSTART;VALUE=DATE-TIME:20201026T121500Z
DTEND;VALUE=DATE-TIME:20201026T131500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/9
DESCRIPTION:Title: Universal optimality of the E8 and Leech lattices\nby Danylo Radchenk
o (ETH Zürich) as part of Warsaw Number Theory Seminar\n\nLecture held in
room 403 at IMPAN.\n\nAbstract\nWe look at the problem of arranging point
s in Euclidean space in order to minimize the potential energy of pairwise
interactions. We show that the E8 lattice and the Leech lattice are unive
rsally optimal in the sense that they have the lowest energy for all poten
tials that are given by completely monotone potentials of squared distance
. \nThe proof uses a new kind of interpolation formula for Fourier eigenfu
nctions\, which is intimately related to the theory of modular forms.\nThe
talk is based on a joint work with Henry Cohn\, Abhinav Kumar\, Stephen D
. Miller\, and Maryna Viazovska.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julie Desjardins (University of Toronto)
DTSTART;VALUE=DATE-TIME:20201102T121500Z
DTEND;VALUE=DATE-TIME:20201102T131500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/10
DESCRIPTION:Title: Density of rational points on a family of del Pezzo surface of degree 1<
/a>\nby Julie Desjardins (University of Toronto) as part of Warsaw Number
Theory Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nLet k be
a number field and X an algebraic variety over k. We want to study the se
t of k-rational points X(k). For example\, is X(k) empty? If not\, is it d
ense with respect to the Zariski topology? Del Pezzo surfaces are classifi
ed by their degrees d (an integer between 1 and 9). Manin and various auth
ors proved that for all del Pezzo surfaces of degree >1 it is dense provid
ed that the surface has a k-rational point (that lies outside a specific s
ubset of the surface for d=2). For d=1\, the del Pezzo surface always has
a rational point. However\, we don't know if the set of rational points is
Zariski-dense. In this talk\, I present a result that is joint with Rosa
Winter in which we prove the density of rational points for a specific fam
ily of del Pezzo surfaces of degree 1 over k.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Masha Vlasenko (IMPAN)
DTSTART;VALUE=DATE-TIME:20201109T121500Z
DTEND;VALUE=DATE-TIME:20201109T131500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/11
DESCRIPTION:Title: Dwork crystals\nby Masha Vlasenko (IMPAN) as part of Warsaw Number T
heory Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nIn his wo
rk on rationality of zeta functions of algebraic varieties Bernard Dwork d
iscovered a number of remarkable p-adic congruences. In this talk I will d
emonstrate some of these congruences and overview our recent work with Fri
ts Beukers which explains their underlying mechanism.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matija Kazalicki (University of Zagreb)
DTSTART;VALUE=DATE-TIME:20201116T121500Z
DTEND;VALUE=DATE-TIME:20201116T131500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/12
DESCRIPTION:Title: Congruences for sporadic sequences\, three fold covers of the elliptic m
odular surfaces and modular forms for non-congruence subgroups\nby Mat
ija Kazalicki (University of Zagreb) as part of Warsaw Number Theory Semin
ar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nIn 1979\, in the cou
rse of the proof of the irrationality of $\\zeta(2)$\n Ap\\'ery introduced
numbers $b_n = \\sum_{k=0}^n {n \\choose k}^2{n+k\n\\choose k}$ that are\
, surprisingly\, integral solutions of recursive\nrelations $$(n+1)^2 u_{
n+1} - (11n^2+11n+3)u_n-n^2u_{n-1} = 0.$$\nZagier performed a computer sea
rch on first 100 million triples\n$(A\,B\,C)\\in \\mathbb{Z}^3$ and found
that the recursive relation\ngeneralizing $b_n$\n$$(n+1)u_{n+1} - (An^2+An
+B)u_n + C n ^2 u_{n-1}=0\,$$\nwith the initial conditions $u_{-1}=0$ and
$u_0=1$ has (non-degenerate\ni.e. $C(A^2-4C)\\ne 0$) integral solution for
only six more triples\n(whose solutions are so called sporadic sequences)
.\n\nStienstra and Beukers showed that the for prime $p\\ge 5$\n\\begin{e
quation*}\nb_{(p-1)/2} \\equiv \\begin{cases} 4a^2-2p \\pmod{p} \\textrm{
if } p =\na^2+b^2\,\\textrm{ a odd}\\\\ 0 \\pmod{p} \\textrm{ if } p\\equi
v 3\n\\pmod{4}.\\end{cases}\n\\end{equation*}\n\nRecently\, Osburn and Str
aub proved similar congruences for all but one\nof the six Zagier's sporad
ic sequences (three cases were already known\nto be true by the work of St
ienstra and Beukers) and we proved the\ncongruence for the sixth sequence.
\n\nIn this talk we describe congruences for the Ap\\'ery numbers\n$b_{(p-
1)/3}$ (and also for the other sporadic sequences).\nFor that we study Atk
in and Swinnerton-Dyer type of congruences\nbetween Fourier coefficients o
f cusp forms for non-congruence\nsubgroups\, $L$-functions of three covers
of elliptic modular surfaces\nand Galois representations attached to thes
e covers.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wadim Zudilin (Radboud University\, Nijmegen)
DTSTART;VALUE=DATE-TIME:20201123T121500Z
DTEND;VALUE=DATE-TIME:20201123T131500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/13
DESCRIPTION:Title: Dwork-type ($q$-)(super)congruences\nby Wadim Zudilin (Radboud Unive
rsity\, Nijmegen) as part of Warsaw Number Theory Seminar\n\nLecture held
in room 403 at IMPAN.\n\nAbstract\nThe "microscope" principle is this: If
a rational function A(q) of variable q vanishes at every p-th root of unit
y (for p prime)\, then A(q) == 0 modulo Φ_p(q)\, the p-th cyclotomic poly
nomial\; assuming that A(1) is a well-defined rational number with a p-fre
e denominator and specialising the congruence at q=1 we conclude with A(1)
== 0 modulo p.\nIn other words\, behaviour of rational functions at p-th
roots of unity may be instructive for gaining information about their valu
es at 1 modulo p. With some "creative" extras\, we can further consider di
visibility by higher powers of primes (and we can even deal with not neces
sarily primes).\nIn my talk\, partly based on recent joint work with Victo
r Guo\, I plan to highlight some novel outcomes of this "creative microsco
pe" methodology -- examples of Dwork-type supercongruences for truncated h
ypergeometric sums.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Osburn (University College Dublin)
DTSTART;VALUE=DATE-TIME:20201130T121500Z
DTEND;VALUE=DATE-TIME:20201130T131500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/14
DESCRIPTION:Title: Generalized Fishburn numbers\, torus knots and quantum modularity\nb
y Robert Osburn (University College Dublin) as part of Warsaw Number Theor
y Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nThe Fishburn
numbers are a sequence of positive integers with numerous combinatorial in
terpretations and interesting asymptotic properties. In 2016\, Andrews and
Sellers initiated the study of arithmetic properties of these numbers. In
this talk\, we discuss a generalization of this sequence using knot theor
y and the quantum modularity of the associated Kontsevich-Zagier series.\n
\nThe first part is joint work with Colin Bijaoui (McMaster)\, Hans Boden
(McMaster)\, Beckham Myers (Harvard)\, Will Rushworth (McMaster)\, Aaron T
ronsgard (Toronto) and Shaoyang Zhou (Vanderbilt) while the second part is
joint work with Ankush Goswami (RISC).\n
LOCATION:https://researchseminars.org/talk/WarsawNT/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gunther Cornelissen (Utrecht University)
DTSTART;VALUE=DATE-TIME:20201207T121500Z
DTEND;VALUE=DATE-TIME:20201207T131500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/15
DESCRIPTION:Title: An analogy between number theory and spectral geometry\nby Gunther C
ornelissen (Utrecht University) as part of Warsaw Number Theory Seminar\n\
n\nAbstract\nSunada’s construction of non-isometric\, isospectral manifo
lds proceeds in the same way as Gassmann’s construction of non-isomorphi
c number fields with the same zeta function\, using a group G with two non
-conjugate subgroups H and K such that the permutation representations giv
en by G acting on their cosets are isomorphic. In Gassmann’s example\, G
was the permutation group on 6 letters and H and K the groups generated b
y (12)(34) and (13)(24)\, and (12)(34) and (12)(56)\, respectively. These
can be realized as covering groups of a compact Riemann surface of genus 2
. Recently\, the speaker and collaborators showed that isomophism of numbe
r fields can be detected by equality of suitable L-series. This talk is ab
out the finding the analogous result for manifolds. The result says that i
f two manifolds are finite Riemannian covers of a developable orbifold\, a
nd such that a certain homological condition is satisfied\, then the manif
olds are isometric if and only if the spectra of finitely many Laplacians
twisted by suitable unitary representations of the fundamental group are e
qual. The result is explicit: in the above example\, one needs 56 spectral
equalities corresponding to 180-dimensional representations. (Joint work
with Norbert Peyerimhoff.)\n
LOCATION:https://researchseminars.org/talk/WarsawNT/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tim Dokchitser (University of Bristol)
DTSTART;VALUE=DATE-TIME:20201214T121500Z
DTEND;VALUE=DATE-TIME:20201214T131500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/16
DESCRIPTION:Title: Curves with tame torsion\nby Tim Dokchitser (University of Bristol)
as part of Warsaw Number Theory Seminar\n\nLecture held in currently onlin
e.\n\nAbstract\nIn this talk I will sketch the proof of the fact that in e
very\ngenus and for every prime p there are curves over Q with tame p-tors
ion.\nIn genus 1\, this is something this is quite easy to deduce from the
\ntheory of the Tate curve\, and I will explain how an explicit version of
\nthe theory of hyperelliptic Mumford curves gives this in arbitrary\ngenu
s. This is joint work with Matthew Bisatt.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bidisha Roy (IMPAN\, Warsaw)
DTSTART;VALUE=DATE-TIME:20201221T121500Z
DTEND;VALUE=DATE-TIME:20201221T131500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/17
DESCRIPTION:Title: Torsion groups of elliptic curves over number fields\nby Bidisha Roy
(IMPAN\, Warsaw) as part of Warsaw Number Theory Seminar\n\nLecture held
in currently online.\n\nAbstract\nComputing torsion groups of elliptic cur
ves defined over number fields is a classical topic and it has a vast lite
rature in algebraic number theory. Any elliptic curve of the form y^2 = x^
3 + c is called a Mordell curve. Mordell curves are well studied elliptic
curves with complex multiplication.\n\nIn this talk\, for Mordell curves
defined over the field of rational numbers we will discuss the classificat
ion of torsion groups over cubic and sextic fields. Also\, we present the
classification of torsion groups of Mordell curves defined over cubic fiel
ds. For Mordell curves over sextic fields\, we provide all possible torsio
n groups. In the second part\, we briefly discuss torsion groups of Mordel
l curves over higher degree number fields.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wojtek Wawrów (London School of Geometry and Number Theory)
DTSTART;VALUE=DATE-TIME:20210111T121500Z
DTEND;VALUE=DATE-TIME:20210111T131500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/18
DESCRIPTION:Title: Ranks of Jacobians over rational numbers\nby Wojtek Wawrów (London
School of Geometry and Number Theory) as part of Warsaw Number Theory Semi
nar\n\nLecture held in currently online.\n\nAbstract\nThe problem of findi
ng out the possible Mordell-Weil ranks of abelian varieties of given dimen
sion over a fixed number field has a long history. In this talk we shall s
urvey various results which have appeared over the years bounding from bel
ow the maximal rank that those varieties can have\, with particular emphas
is on Jacobians of special families of curves over the field of rational n
umbers.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Netan Dogra (King's College London)
DTSTART;VALUE=DATE-TIME:20210118T121500Z
DTEND;VALUE=DATE-TIME:20210118T131500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/19
DESCRIPTION:Title: Some new results in the nonabelian Chabauty method\nby Netan Dogra (
King's College London) as part of Warsaw Number Theory Seminar\n\nLecture
held in currently online.\n\nAbstract\nIn this talk I will discuss the non
abelian Chabauty method\, which seeks to use p-adic analytic functions to
determine the finite sets of rational points on higher genus curves\, and
some new cases where it can be used to determine the solutions to Diophant
ine equations.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleksiy Klurman (University of Bristol)
DTSTART;VALUE=DATE-TIME:20210125T121500Z
DTEND;VALUE=DATE-TIME:20210125T131500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/20
DESCRIPTION:Title: Monotone chains of Hecke cusp forms\nby Oleksiy Klurman (University
of Bristol) as part of Warsaw Number Theory Seminar\n\nLecture held in cur
rently online.\n\nAbstract\nWe discuss a general joint equidistribution re
sult for the Fourier coefficients of Hecke cusp forms. One simple conseque
nce of such a result is that there exist infinitely many integers n (in fa
ct an upper density of this set is positive) such that \n$\\tau (n)<\\tau
(n+1)<\\tau(n+2)$ where $\\tau$ is a Ramanujan $\\tau$-function. This is b
ased on a joint work with A. Mangerel (CRM).\n
LOCATION:https://researchseminars.org/talk/WarsawNT/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikołaj Frączyk (University of Chicago)
DTSTART;VALUE=DATE-TIME:20210301T160000Z
DTEND;VALUE=DATE-TIME:20210301T170000Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/22
DESCRIPTION:Title: Sarnak’s density hypothesis in horizontal families\nby Mikołaj Fr
ączyk (University of Chicago) as part of Warsaw Number Theory Seminar\n\n
Lecture held in currently online.\n\nAbstract\nLet $G$ be a real semi simp
le Lie group with an irreducible unitary representation $\\pi$. The non-te
mperedness of $\\pi$ is measured by the real parameter $p(\\pi)$ which is
defined as the infimum of $p$ such that $\\pi$ has non-zero matrix coeffic
ients in $L^p(G)$. Sarnak and Xue conjectured that for any arithmetic latt
ice $\\Gamma\\subset G$ and principal congruence subgroup $\\Gamma(q)\\sub
set \\Gamma$\, the multiplicity of $\\pi$ in $L^2(G/\\Gamma(q))$ is at mos
t $O(V(q)^{2/p(\\pi) +\\varepsilon})$\, where $V(q)$ is the covolume of $\
\Gamma(q)$. Sarnak and Xue proved this conjecture for $G=SL(2\,\\mathbb R)
\,SL(2\,\\mathbb C)$. I will talk about the joint work with Gergely Harcos
\, Peter Maga and Djordje Milicevic where we prove bounds of the same qual
ity that hold uniformly for families of pairwise non-commensurable lattice
s in $G=SL(2\,\\mathbb R)^a\\times SL(2\,\\mathbb C)^b$. These families of
lattices\, which we call horizontal\, are given as unit groups of maximal
orders of quaternion algebras over number fields.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sudhir Pujahari (University of Warsaw)
DTSTART;VALUE=DATE-TIME:20210308T121500Z
DTEND;VALUE=DATE-TIME:20210308T131500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/23
DESCRIPTION:Title: Arithmetic and statistics of sums of eigenvalues of Hecke operators\
nby Sudhir Pujahari (University of Warsaw) as part of Warsaw Number Theory
Seminar\n\nLecture held in currently online.\n\nAbstract\nIn the first pa
rt of the talk\, we will study about the distribution of gaps between eige
nvalues of Hecke operators in both horizontal and vertical settings. As an
application of this we will obtain a strong multiplicity one theorem and
evidence towards Maeda conjecture. The horizontal setting is a joint work
with M. Ram Murty. In the second part of the talk\, using recent developme
nts in the theory of l-adic Galois representations we will study the norma
l number of prime factors\nof sums of Fourier coefficients of eigenforms.
Moreover\, we will see the distribution of distinct prime factors of sums
of Fourier coefficients of eigenforms. The final part is a joint work with
M. Ram Murty and V. Kumar Murty.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Verschoor (University of Utrecht)
DTSTART;VALUE=DATE-TIME:20210315T121500Z
DTEND;VALUE=DATE-TIME:20210315T131500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/24
DESCRIPTION:Title: Bailey Type Factorizations of Horn Functions\nby Carlo Verschoor (Un
iversity of Utrecht) as part of Warsaw Number Theory Seminar\n\nLecture he
ld in currently online.\n\nAbstract\nA well-known identity by Bailey state
s that Appell’s F4 function can be written as the product of two Gauss h
ypergeometric functions under a suitable specialization of its parameters.
Other identities of this type are known for Appell’s F2 and F3\, which
are closely related to Bailey’s identity. The aim of this talk is to sho
w that the same can be done for Horn’s H1\, H4 and H5 functions.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Enis Kaya (University of Groningen)
DTSTART;VALUE=DATE-TIME:20210322T121500Z
DTEND;VALUE=DATE-TIME:20210322T131500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/25
DESCRIPTION:Title: Explicit Vologodsky Integration for Hyperelliptic Curves\nby Enis Ka
ya (University of Groningen) as part of Warsaw Number Theory Seminar\n\nLe
cture held in currently online.\n\nAbstract\nLet $X$ be a curve over a $p$
-adic field with semi-stable reduction and let $\\omega$ be a meromorphic
$1$-form on $X$. There are two notions of $p$-adic integration one may ass
ociate to this data: the Berkovich–Coleman integral which can be perform
ed locally\; and the Vologodsky integral with desirable number-theoretic p
roperties. In this talk\, we present a theorem comparing the two\, and des
cribe algorithms for computing these integrals in the case that $X$ is a h
yperelliptic curve. We also illustrate our algorithm with a numerical exam
ple computed in Sage. This talk is partly based on joint work with Eric Ka
tz.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Adamczewski (Institut Camille Jordan & CNRS)
DTSTART;VALUE=DATE-TIME:20210329T111500Z
DTEND;VALUE=DATE-TIME:20210329T121500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/26
DESCRIPTION:Title: Furstenberg's conjecture\, Mahler's method\, and finite automata\nby
Boris Adamczewski (Institut Camille Jordan & CNRS) as part of Warsaw Numb
er Theory Seminar\n\nLecture held in currently online.\n\nAbstract\nIt is
commonly expected that expansions of numbers in multiplicatively independe
nt bases\, such as 2 and 10\, should have no common structure. However\, i
t seems extraordinarily difficult to confirm this naive heuristic principl
e in some way or another. In the late 1960s\, Furstenberg suggested a seri
es of conjectures\, which became famous and aim to capture this heuristic.
The work I will discuss in this talk is motivated by one of these conject
ures. Despite recent remarkable progress by Shmerkin and Wu\, it remains t
otally out of reach of the current methods. While Furstenberg’s conjectu
res take place in a dynamical setting\, I will use instead the language of
automata theory to formulate some related problems that formalize and exp
ress in a different way the same general heuristic. I will explain how the
latter can be solved thanks to some recent advances in Mahler’s method\
; a method in transcendental number theory initiated by Mahler at the end
of the 1920s. This a joint work with Colin Faverjon.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anna Szumowicz (Caltech)
DTSTART;VALUE=DATE-TIME:20210412T140000Z
DTEND;VALUE=DATE-TIME:20210412T150000Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/27
DESCRIPTION:Title: Cuspidal types on GL_p(O)\nby Anna Szumowicz (Caltech) as part of Wa
rsaw Number Theory Seminar\n\nLecture held in currently online.\n\nAbstrac
t\nLet $F$ be a non-Archimedean local field and let $O$ be its ring of\nin
tegers. We describe the cupidal types on $\\mathrm{GL}_p(O)$ (where $p$ is
a prime\nnumber) using Clifford theory. This gives some information and\n
invariants attached to cuspidal types called orbits. We give an\nexample w
hich shows that the orbit of a representation does not give\nenough inform
ation to determine whether a representation is a cuspidal\ntype or not.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rusen Li (Shandong University)
DTSTART;VALUE=DATE-TIME:20210419T111500Z
DTEND;VALUE=DATE-TIME:20210419T121500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/28
DESCRIPTION:Title: Summation formulas of q-hyperharmonic numbers\nby Rusen Li (Shandong
University) as part of Warsaw Number Theory Seminar\n\nLecture held in cu
rrently online.\n\nAbstract\nIn 1990\, Spieß gave some identities includi
ng the types of $\\sum_{\\ell=1}^n\\ell^k H_\\ell$\, $\\sum_{\\ell=1}^n\\e
ll^k H_{n-\\ell}$ and $\\sum_{\\ell=1}^n\\ell^k H_\\ell H_{n-\\ell}$. In t
his talk\, based upon a certain type of $q$-harmonic numbers $H_n^{(r)}(q)
$\, several formulas of $q$-hyperharmonic numbers are derived as $q$-gener
alizations. The main tools used in the talk are Abel’s identity and a q-
version of the relation by Spieß.\nThis is based on a joint work with Tak
ao Komatsu.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentijn Karemaker (Universiteit Utrecht)
DTSTART;VALUE=DATE-TIME:20210426T111500Z
DTEND;VALUE=DATE-TIME:20210426T121500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/29
DESCRIPTION:Title: Polarisations of abelian varieties over finite fields via canonical lift
ings\nby Valentijn Karemaker (Universiteit Utrecht) as part of Warsaw
Number Theory Seminar\n\nLecture held in currently online.\n\nAbstract\nIn
this talk we will give a widely applicable and computable description of
polarisations of abelian varieties over finite field. More precisely\, we
will describe all polarisations of all abelian varieties over a finite fie
ld in a fixed isogeny class corresponding to a squarefree Weil polynomial\
, when one variety in the isogeny class admits a canonical lifting to char
acteristic zero. The computability of the description relies on applying c
ategorical equivalences between abelian varieties over finite fields and f
ractional ideals in étale algebras. \nThis is joint work with Jonas Bergs
tröm and Stefano Marseglia.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Slob (Ulm University)
DTSTART;VALUE=DATE-TIME:20210510T111500Z
DTEND;VALUE=DATE-TIME:20210510T121500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/30
DESCRIPTION:Title: Primitive divisors of sequences associated to elliptic curves over funct
ion fields\nby Robert Slob (Ulm University) as part of Warsaw Number T
heory Seminar\n\nLecture held in currently online.\n\nAbstract\nIn the fir
st part of the talk\, we give a gentle introduction into the subject of di
visibility sequences over the rational numbers and discuss the notion of a
primitive divisor/Zsigmondy bound. We then explain how these notions can
be extended to number fields and function fields\, and how to obtain a div
isibility sequence from a non-torsion point on an elliptic curve over any
of these fields. There will also be plenty of nice examples.\n\nIn the sec
ond part of the talk\, we discuss the typical methods that are used to pro
ve the existence of a Zsigmondy bound for a divisibility sequence obtained
from a non-torsion point on an elliptic curve $E$ over a number or functi
on field $K$. Let $P$ be this non-torsion point in $E(K)$\, and suppose Q
is a torsion point in $E(K)$. We can also associate a sequence of divisors
$\\{D_{nP+Q}\\}$ on $K$ to the sequence of points $\\{nP+Q\\}$. In my pre
print\, we proved the existence of a Zsigmondy bound for this sequence $\\
{D_{nP+Q}\\}$ for $K$ a function field (under some minor conditions)\, ext
ending the analogous result of Verzobio over number fields. I will provide
the crucial ideas to apply the existing methods of the case $\\{nP\\}$ to
my case $\\{nP+Q\\}$. Additionally\, I will highlight the differences wit
h the number field case.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marco Streng (Universiteit Leiden)
DTSTART;VALUE=DATE-TIME:20210517T111500Z
DTEND;VALUE=DATE-TIME:20210517T121500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/31
DESCRIPTION:Title: Obtaining modular units via a recurrence relation\nby Marco Streng (
Universiteit Leiden) as part of Warsaw Number Theory Seminar\n\nLecture he
ld in currently online.\n\nAbstract\nThe modular curve $Y^1(N)$ parametris
es pairs $(E\,P)$\, where $E$ is an elliptic curve and $P$ is a point of o
rder $N$ on $E$. One tool for studying this curve is the group of modular
units on it\, that is\, the group of algebraic functions with no poles or
zeroes.\n\nWe first review how a recurrence relation (related to elliptic
divisibility sequences) gives rise to defining equations for the curves $Y
^1(N)$. We then show that the same recurrence relation also gives explicit
algebraic formulae for a basis of the group of units on $Y^1(N)$.\n\nThis
proves a conjecture of Maarten Derickx and Mark van Hoeij.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eric Delaygue (University Lyon 1)
DTSTART;VALUE=DATE-TIME:20210524T111500Z
DTEND;VALUE=DATE-TIME:20210524T121500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/32
DESCRIPTION:Title: On primary pseudo-polynomials and around Ruzsa's Conjecture\nby Eric
Delaygue (University Lyon 1) as part of Warsaw Number Theory Seminar\n\nL
ecture held in currently online.\n\nAbstract\nEvery polynomial $P(X)$ with
integer coefficients satisfies the congruences $P(n+m)=P(n) \\mod m$ for
all integers $n$ and $m$. An integer valued sequence is called a pseudo-po
lynomial when it satisfies these congruences. Hall characterized pseudo-po
lynomials and proved that they are not necessarily polynomials. A long sta
nding conjecture of Ruzsa says that a pseudo-polynomial $a(n)$ is a polyno
mial as soon as $\\limsup |a_n|^{1/n} < e$. A primary pseudo-polynomial is
an integer valued sequence $a(n)$ such that $a(n+p)=a(n) \\mod p$ for all
integers $n ≥ 0$ and all prime numbers $p$. The same conjecture has bee
n formulated for them\, which implies Ruzsa’s\, and this talk will revol
ve around this conjecture.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Vargas-Montoya (University Lyon 1)
DTSTART;VALUE=DATE-TIME:20210531T111500Z
DTEND;VALUE=DATE-TIME:20210531T121500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/33
DESCRIPTION:Title: Algebraicity modulo $p$ of G functions\, hypergeometric series and stron
g Frobenius structure\nby Daniel Vargas-Montoya (University Lyon 1) as
part of Warsaw Number Theory Seminar\n\nLecture held in currently online.
\n\nAbstract\nB. Dwork in his work about zeta function of a hypersurface o
ver finite fields introduced the notion of strong Frobenius structure. In
this talk we are going to take up this notion for the study of algebraicit
y modulo $p$ of Siegel G functions\, where $p$ is a prime number. \nFirst
ly\, we are going to see that if $f(t)$ is a power series (or Siegel G fun
ction) with coefficients in the ring of integers $\\mathbb{Z}$ and if $f(t
)$ is solution of a differential operator $L$ having strong Frobenius stru
cture for $p$ of period $h$\, then the reduction of $f$ modulo $p$ is alge
braic over $\\mathbb{F}_p(t)$ and its algebraicity degree is bounded by $p
^{n^2h}$\, where $n$ is the order of L and $\\mathbb{F}_p$ is the field of
$p$ elements. Secondly\, we are going to show that\, under reasonable hyp
otheses\, rigid differential operators have a strong Frobenius structure
for almost every prime number $p$.\nFinally\, we are going to illustrate
our results with several examples coming of hypergeometric series of type
${}_nF_n-1$.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael J. Schlosser (Unviersity of Vienna)
DTSTART;VALUE=DATE-TIME:20210607T140000Z
DTEND;VALUE=DATE-TIME:20210607T150000Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/34
DESCRIPTION:Title: On the infinite Borwein product raised to a real power\nby Michael J
. Schlosser (Unviersity of Vienna) as part of Warsaw Number Theory Seminar
\n\nLecture held in currently online.\n\nAbstract\nWe study the $q$-series
coefficients appearing in the expansion of $\\prod_{n\\ge 1}[(1-q^n)/(1-q
^{pn})]^\\delta$\, the infinite Borwein product for an arbitrary prime $p$
\, raised to an arbitrary positive real power $\\delta$. Application of th
e Hardy-Ramanujan-Rademacher circle method gives an asymptotic formula for
the coefficients. For $p=3$ we give an estimate of their growth which ena
bles us to partially confirm an earlier conjecture we made concerning an o
bserved sign pattern of the coefficients when the exponent $\\delta$ is wi
thin a specified range of positive real numbers. We then take a closer loo
k at the cube of the infinite Borwein product\, for arbitrary $p$ (now a p
ositive integer)\, and establish some vanishing and divisibility propertie
s of the respective coefficients.\nThis is joint work with Nian Hong Zhou.
\n
LOCATION:https://researchseminars.org/talk/WarsawNT/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jason Bell (University of Waterloo)
DTSTART;VALUE=DATE-TIME:20210614T111500Z
DTEND;VALUE=DATE-TIME:20210614T121500Z
DTSTAMP;VALUE=DATE-TIME:20210612T235246Z
UID:WarsawNT/35
DESCRIPTION:Title: Effective isotrivial Mordell-Lang in positive characteristic\nby Jas
on Bell (University of Waterloo) as part of Warsaw Number Theory Seminar\n
\nLecture held in currently online.\n\nAbstract\nThe Mordell-Lang conjectu
re (now a theorem\, proved by Faltings\, Vojta\, McQuillan\, …) asserts
that if $G$ is a semiabelian variety $G$ defined over an algebraically clo
sed field of characteristic zero\, $X$ is a subvariety of $G$\, and $\\Gam
ma$ is a finite rank subgroup of $G$\, then $X\\cap \\Gamma$ is a finite u
nion of cosets of $\\Gamma$. In positive characteristic\, the naive trans
lation of this theorem does not hold\, however Hrushovski\, using model th
eoretic techniques\, showed that in some sense all counterexamples arise f
rom semiabelian varieties defined over finite fields (the isotrivial case)
. This was later refined by Moosa and Scanlon\, who showed in the isotriv
ial case that the intersection of a subvariety of a semiabelian variety $G
$ with a finitely generated subgroup $\\Gamma$ of $G$ that is invariant un
der the Frobenius endomorphism $F:G\\to G$ is a finite union of sets of th
e form $S+A$\, where $A$ is a subgroup of $\\Gamma$ and $S$ is a sum of or
bits under the map $F$. We show how how one can use finite-state automat
a to give a concrete description of these intersections $\\Gamma\\cap X$ i
n the isotrivial setting\, by constructing a finite machine that identifie
s all points in the intersection. In particular\, this allows us to give d
ecision procedures for answering questions such as: is $X\\cap \\Gamma$ em
pty? finite? does it contain a coset of an infinite subgroup? In addition\
, we are able to read off coarse asymptotic estimates for the number of po
ints of height $\\le H$ in the intersection from the machine. This is joi
nt work with Dragos Ghioca and Rahim Moosa.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/35/
END:VEVENT
END:VCALENDAR