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BEGIN:VEVENT
SUMMARY:Tiago Jardim Da Fonseca (Oxford)
DTSTART;VALUE=DATE-TIME:20200422T150000Z
DTEND;VALUE=DATE-TIME:20200422T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/1
DESCRIPTION:Title: On
Fourier coefficients of Poincaré series\nby Tiago Jardim Da Fonseca (
Oxford) as part of London number theory seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/LNTS/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ila Varma (Toronto)
DTSTART;VALUE=DATE-TIME:20200429T150000Z
DTEND;VALUE=DATE-TIME:20200429T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/2
DESCRIPTION:Title: Mal
le's conjecture for octic D4-fields\nby Ila Varma (Toronto) as part of
London number theory seminar\n\n\nAbstract\nWe consider the family of nor
mal octic fields with Galois group $D_4$\, ordered by their discriminants.
In forthcoming joint work with Arul Shankar\, we verify the strong form o
f Malle's conjecture for this family of number fields\, obtaining the orde
r of growth as well as the constant of proportionality. In this talk\, we
will discuss and review the combination of techniques from analytic number
theory and geometry-of-numbers methods used to prove this and related res
ults.\n
LOCATION:https://researchseminars.org/talk/LNTS/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chris Lazda (Warwick)
DTSTART;VALUE=DATE-TIME:20200506T150000Z
DTEND;VALUE=DATE-TIME:20200506T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/3
DESCRIPTION:Title: A N
éron-Ogg-Shafarevich criterion for K3 surfaces\nby Chris Lazda (Warwi
ck) as part of London number theory seminar\n\n\nAbstract\nThe naive analo
gue of the Néron–Ogg–Shafarevich criterion fails for K3 surfaces\, th
at is\, there exist K3 surfaces over Henselian\, discretely valued fields
K\, with unramified etale cohomology groups\, but which do not admit good
reduction over K. Assuming potential semi-stable reduction\, I will show h
ow to correct this by proving that a K3 surface has good reduction if and
only if its second cohomology is unramified\, and the associated Galois re
presentation over the residue field coincides with the second cohomology o
f a certain “canonical reduction” of X. This is joint work with B. Chi
arellotto and C. Liedtke.\n
LOCATION:https://researchseminars.org/talk/LNTS/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chantal David (Concordia)
DTSTART;VALUE=DATE-TIME:20200513T150000Z
DTEND;VALUE=DATE-TIME:20200513T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/4
DESCRIPTION:Title: Non
-vanishing cubic Dirichlet L-functions at s = 1/2\nby Chantal David (C
oncordia) as part of London number theory seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/LNTS/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rong Zhou (Imperial)
DTSTART;VALUE=DATE-TIME:20200520T150000Z
DTEND;VALUE=DATE-TIME:20200520T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/5
DESCRIPTION:Title: Ind
ependence of $l$ for Frobenius conjugacy classes attached to abelian varie
ties\nby Rong Zhou (Imperial) as part of London number theory seminar\
n\n\nAbstract\nLet $A$ be an abelian variety over a number field $E\\subse
t \\mathbb{C}$ and let $v$ be a place of good reduction lying over a prime
$p$. For a prime $l\\neq p$\, a theorem of Deligne implies that upon maki
ng a finite extension of $E$\, the Galois representation on the $l$-adic T
ate module factors as $\\rho_l:\\Gamma_E\\rightarrow G_A(\\mathbb{Q}_l)$\,
where $G_A$ is the Mumford-Tate group of $A$. We prove that the conjugacy
class of $\\rho_l(Frob_v)$ is defined over $\\mathbb{Q}$ and independent
of $l$. This is joint work with Mark Kisin.\n
LOCATION:https://researchseminars.org/talk/LNTS/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matteo Tamiozzo (Imperial)
DTSTART;VALUE=DATE-TIME:20200527T150000Z
DTEND;VALUE=DATE-TIME:20200527T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/6
DESCRIPTION:Title: Blo
ch-Kato special value formulas for Hilbert modular forms\nby Matteo Ta
miozzo (Imperial) as part of London number theory seminar\n\n\nAbstract\nT
he Bloch-Kato conjectures predict a relation between arithmetic invariants
of a motive and special values of the associated $L$-function. We will ou
tline a proof of (the $p$-part of) one inequality in the relevant special
value formula for Hilbert modular forms of parallel weight two\, in analyt
ic rank at most one.\n
LOCATION:https://researchseminars.org/talk/LNTS/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yunqing Tang (Paris-Saclay)
DTSTART;VALUE=DATE-TIME:20200603T150000Z
DTEND;VALUE=DATE-TIME:20200603T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/7
DESCRIPTION:Title: Pic
ard ranks of reductions of K3 surfaces over global fields\nby Yunqing
Tang (Paris-Saclay) as part of London number theory seminar\n\n\nAbstract\
nFor a K3 surface $X$ over a number field with potentially good reduction
everywhere\, we prove that there are infinitely many primes modulo which t
he reduction of $X$ has larger geometric Picard rank than that of the gene
ric fiber $X$. A similar statement still holds true for ordinary K3 surfac
es with potentially good reduction everywhere over global function fields.
In this talk\, I will present the proofs via the (arithmetic) intersectio
n theory on good integral models (and its special fibers) of $\\mathrm{GSp
in}$ Shimura varieties. These results are generalizations of the work of C
harles on exceptional isogenies between reductions of a pair of elliptic c
urves. This talk is based on joint work with Ananth Shankar\, Arul Shankar
\, and Salim Tayou and with Davesh Maulik and Ananth Shankar.\n
LOCATION:https://researchseminars.org/talk/LNTS/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vesselin Dimitrov (Toronto)
DTSTART;VALUE=DATE-TIME:20200612T150000Z
DTEND;VALUE=DATE-TIME:20200612T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/8
DESCRIPTION:Title: p-a
dic Eisenstein series\, arithmetic holonomicity criteria\, and irrationali
ty of the 2-adic $\\zeta(5)$\nby Vesselin Dimitrov (Toronto) as part o
f London number theory seminar\n\n\nAbstract\nIn this exposition of a join
t work in progress with Frank Calegari and Yunqing Tang\, I will explain a
new arithmetic criterion for a formal function to be holonomic\, and how
it revives an approach to the arithmetic nature of special values of L-fun
ctions. The new consequence to be proved in this talk is the irrationality
of the 2-adic version of $\\zeta(5)$ (of Kubota-Leopoldt). But I will als
o draw a parallel to a work of Zudilin\, and try to leave some additional
open ends where the holonomicity theorem could be useful. The ingredients
of the irrationality proof are Calegari's p-adic counterpart of the Apery-
Beukers method\, which is based on the theory of overconvergent p-adic mod
ular forms (IMRN\, 2005) taking its key input from Buzzard's theorem on p-
adic analytic continuation (JAMS\, 2002)\, and a Diophantine approximation
method of Andre enhanced to a power of the modular curve $X_0(2)$. The ov
erall argument\, as we shall discuss\, turns out to bear a surprising affi
nity to a recent solution of the Schinzel-Zassenhaus conjecture on the orb
its of Galois around the unit circle.\n
LOCATION:https://researchseminars.org/talk/LNTS/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yifeng Liu (Yale)
DTSTART;VALUE=DATE-TIME:20200617T130000Z
DTEND;VALUE=DATE-TIME:20200617T140000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/9
DESCRIPTION:Title: Bei
linson-Bloch conjecture and arithmetic inner product formula\nby Yifen
g Liu (Yale) as part of London number theory seminar\n\n\nAbstract\nIn thi
s talk\, we study the Chow group of the motive associated to a tempered gl
obal $L$-packet $\\pi$ of unitary groups of even rank with respect to a CM
extension\, whose global root number is $-1$. We show that\, under some r
estrictions on the ramification of $\\pi$\, if the central derivative $L'(
1/2\,\\pi)$ is nonvanishing\, then the $\\pi$-nearly isotypic localization
of the Chow group of a certain unitary Shimura variety over its reflex fi
eld does not vanish. This proves part of the Beilinson--Bloch conjecture f
or Chow groups and L-functions (which generalizes the B-SD conjecture). Mo
reover\, assuming the modularity of Kudla's generating functions of specia
l cycles\, we explicitly construct elements in a certain $\\pi$-nearly iso
typic subspace of the Chow group by arithmetic theta lifting\, and compute
their heights in terms of the central derivative $L'(1/2\,\\pi)$ and loca
l doubling zeta integrals. This confirms the conjectural arithmetic inner
product formula proposed by me a decade ago. This is a joint work with Cha
o Li.\n
LOCATION:https://researchseminars.org/talk/LNTS/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Weinstein (Boston University)
DTSTART;VALUE=DATE-TIME:20200708T150000Z
DTEND;VALUE=DATE-TIME:20200708T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/12
DESCRIPTION:Title: Pa
rtial Frobenius structures\, Tate’s conjecture\, and BSD over function f
ields.\nby Jared Weinstein (Boston University) as part of London numbe
r theory seminar\n\n\nAbstract\nTate’s conjecture predicts that Galois-i
nvariant classes in the $l$-adic cohomology of a variety are explained by
algebraic cycles. It is known to imply the conjecture of Birch and Swinne
rton-Dyer (BSD) for elliptic curves over function fields. When the variet
y\, now assumed to be in characteristic p\, admits a “partial Frobenius
structure”\, there is a natural extension of Tate’s conjecture. Ass
uming this conjecture\, we get not only BSD\, but the following result: t
he top exterior power of the Mordell-Weil group of an elliptic curve is sp
anned by a “Drinfeld-Heegner” point. This is a report on work in prog
ress.\n
LOCATION:https://researchseminars.org/talk/LNTS/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rachel Newton
DTSTART;VALUE=DATE-TIME:20201007T150000Z
DTEND;VALUE=DATE-TIME:20201007T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/13
DESCRIPTION:Title: Ev
aluating the wild Brauer group\nby Rachel Newton as part of London num
ber theory seminar\n\n\nAbstract\nThe local-global approach to the study o
f rational points on varieties over number fields begins by embedding the
set of rational points on a variety X into the set of its adelic points. T
he Brauer-Manin pairing cuts out a subset of the adelic points\, called th
e Brauer-Manin set\, that contains the rational points. If the set of adel
ic points is non-empty but the Brauer-Manin set is empty then we say there
's a Brauer-Manin obstruction to the existence of rational points on X. Co
mputing the Brauer-Manin pairing involves evaluating elements of the Braue
r group of X at local points. If an element of the Brauer group has order
coprime to p\, then its evaluation at a p-adic point factors via reduction
of the point modulo p. For p-torsion elements this is no longer the case:
in order to compute the evaluation map one must know the point to a highe
r p-adic precision. Classifying p-torsion Brauer group elements according
to the precision required to evaluate them at p-adic points gives a filtra
tion which we describe using work of Bloch and Kato. Applications of our w
ork include addressing Swinnerton-Dyer's question about which places can p
lay a role in the Brauer-Manin obstruction. This is joint work with Martin
Bright.\n
LOCATION:https://researchseminars.org/talk/LNTS/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ziyang Gao
DTSTART;VALUE=DATE-TIME:20201014T150000Z
DTEND;VALUE=DATE-TIME:20201014T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/14
DESCRIPTION:Title: Bo
unding the number of rational points on curves\nby Ziyang Gao as part
of London number theory seminar\n\n\nAbstract\nMazur conjectured\, after F
altings’s proof of the Mordell conjecture\, that the number of rational
points on a curve of genus g at least 2 defined over a number field of deg
ree d is bounded in terms of g\, d and the Mordell-Weil rank. In particula
r the height of the curve is not involved. In this talk I will explain how
to prove this conjecture and some generalizations. I will focus on how fu
nctional transcendence and unlikely intersections are applied in the proof
. If time permits\, I will talk about how the dependence on d can be furth
ermore removed if we moreover assume the relative Bogomolov conjecture. Th
is is joint work with Vesselin Dimitrov and Philipp Habegger.\n
LOCATION:https://researchseminars.org/talk/LNTS/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rob Kurinczuk (Imperial College London)
DTSTART;VALUE=DATE-TIME:20201028T160000Z
DTEND;VALUE=DATE-TIME:20201028T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/15
DESCRIPTION:Title: Mo
duli of Langlands parameters and LLIF\nby Rob Kurinczuk (Imperial Coll
ege London) as part of London number theory seminar\n\n\nAbstract\nThe con
jectural local Langlands correspondence connects representations of p-adic
groups to certain representations of Galois groups of local fields called
Langlands parameters. In recent joint work with Dat\, Helm\, and Moss\,
we have constructed moduli spaces of Langlands parameters over Z[1/p] and
studied their geometry. We expect this geometry is reflected in the repre
sentation theory of the p-adic group. In particular\, our main conjecture
"local Langlands in families" describes the GIT quotient of the moduli sp
ace of Langlands parameters in terms of the centre of the category of repr
esentations of the p-adic group generalising a theorem of Helm-Moss for GL
(n).\n
LOCATION:https://researchseminars.org/talk/LNTS/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:João Lourenço
DTSTART;VALUE=DATE-TIME:20201111T160000Z
DTEND;VALUE=DATE-TIME:20201111T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/16
DESCRIPTION:Title: Th
e Scholze-Weinstein conjecture on local models\nby João Lourenço as
part of London number theory seminar\n\n\nAbstract\nInspired by the theory
of local models of Shimura varieties\, Scholze-Weinstein proposed a conje
cture predicting representability of certain minuscule closed sub-v-sheave
s of their p-adic de Rham affine Grassmannian by a projective flat and geo
metrically reduced normal scheme.\n\nIn my talk\, I'll explain the motivat
ion behind the problem stemming from Shimura varieties\, review the necess
ary technical background and ultimately sketch a proof for pseudo-tame gro
ups without exceptional factors. To achieve this\, I'll determine the Pica
rd group of the Witt vectors affine Grassmannian as conjectured by Bhatt-S
cholze. Time permitting\, I might outline a (very much incomplete) strateg
y for handling exceptional groups.\n
LOCATION:https://researchseminars.org/talk/LNTS/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Annette Huber (Universität Freiburg)
DTSTART;VALUE=DATE-TIME:20201118T160000Z
DTEND;VALUE=DATE-TIME:20201118T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/17
DESCRIPTION:Title: Ex
ponential periods and o-minimality\nby Annette Huber (Universität Fre
iburg) as part of London number theory seminar\n\n\nAbstract\n(joint work
with Johan Commelin and Philipp Habegger)\nRoughly\, period numbers are de
fined by integrals of the form\n$\\int_\\sigma\\omega$ with $\\omega$ and
$\\sigma$ of algebraic nature.\nThis can be made precise in very different
languages: as values of\nthe period pairing between de Rham cohomology an
d singular homology\nof algebraic varieties or motives defined over number
fields\, or more\nnaively as\nvolumes of semi-algebraic sets.\n\nMore rec
ently\, exponential periods have come into focus. Roughly\, they\nare of t
he form $\\int_\\sigma e^{-f}\\omega$ with $\\sigma\,\\omega$ and now\nals
o $f$ of algebraic nature. They appear are periods for the Rham complex\no
f an irregular connection. We want to explain how the "naiv" side of\nthe
story can be formulated in this case.\n
LOCATION:https://researchseminars.org/talk/LNTS/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Garcia (UCl)
DTSTART;VALUE=DATE-TIME:20201209T160000Z
DTEND;VALUE=DATE-TIME:20201209T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/20
DESCRIPTION:Title: Ei
senstein classes and hyperplane complements\nby Luis Garcia (UCl) as p
art of London number theory seminar\n\n\nAbstract\nIn recent years several
authors (Sczech\, Nori\, Hill\, Charollois-Dasgupta-Greenberg\, Beilinson
-Kings-Levin) have defined and studied certain group cocycles ("Eisenstein
cocycles") in the cohomology of arithmetic groups. I will discuss how the
se constructions can be understood in terms of equivariant cohomology and
characteristic classes. This point of view relates the cocycles to the the
ta correspondence and leads to generalisations relating the homology of ar
ithmetic groups to algebraic objects such as meromorphic differentials on
hyperplane complements. I will discuss these generalisations and an applic
ation to critical values of L-functions. \n\nThe talk is based on joint wo
rk-in-progress with Nicolas Bergeron\, Pierre Charollois and Akshay Venkat
esh.\n
LOCATION:https://researchseminars.org/talk/LNTS/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pedro Lemos (University College London)
DTSTART;VALUE=DATE-TIME:20201216T160000Z
DTEND;VALUE=DATE-TIME:20201216T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/21
DESCRIPTION:Title: Re
sidual Galois representations of elliptic curves with image in the normali
ser of a non-split Cartan\nby Pedro Lemos (University College London)
as part of London number theory seminar\n\n\nAbstract\nIt is known that if
$p$ is a prime $>37$\, then the image of the residual Galois representati
on $\\bar{\\rho}_{E\,p}: G_{\\mathbb{Q}}\\rightarrow {\\rm GL}_2(\\mathbb{
F}_p)$ attached to an elliptic curve $E/\\mathbb{Q}$ without complex multi
plication is either ${\\rm GL}_2(\\mathbb{F}_p)$\, or is contained in the
normaliser of a non-split Cartan subgroup of ${\\rm GL}_2(\\mathbb{F}_p)$.
I will report on a recent joint work with Samuel Le Fourn where we improv
e this result by showing that if $p>1.4\\times 10^7$\, then $\\bar{\\rho}_
{E\,p}$ is either surjective\, or its image is the normaliser of a non-spl
it Cartan subgroup of ${\\rm GL}_2(\\mathbb{F}_p)$.\n
LOCATION:https://researchseminars.org/talk/LNTS/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lennart Gehrmann (University of Duisburg-Essen / McGill University
)
DTSTART;VALUE=DATE-TIME:20201021T150000Z
DTEND;VALUE=DATE-TIME:20201021T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/22
DESCRIPTION:Title: L-
invariants\, completed cohomology and big principal series\nby Lennart
Gehrmann (University of Duisburg-Essen / McGill University) as part of Lo
ndon number theory seminar\n\n\nAbstract\nLet $f$ be a newform of weight $
2$ that is Steinberg at $p$. Darmon showed that the Fontaine-Mazur $L$-inv
ariant of the associated local $p$-adic Galois representation can be compu
ted in terms of the cohomology of $p$-arithmetic subgroups of the group $P
GL_2(\\mathbb{Q})$.\nOn the other hand Breuil showed that one can compute
the $f$-isoyptical part of completed cohomology of the modular curve in te
rms of the cohomology of $p$-arithmetic groups.\nIn this talk I will give
generalizations of both results to higher rank reductive groups. This is p
artly joint work with Giovanni Rosso.\n
LOCATION:https://researchseminars.org/talk/LNTS/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sally Gilles
DTSTART;VALUE=DATE-TIME:20201104T160000Z
DTEND;VALUE=DATE-TIME:20201104T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/23
DESCRIPTION:Title: Pe
riod morphisms and syntomic cohomology\nby Sally Gilles as part of Lon
don number theory seminar\n\n\nAbstract\nIn 2017\, Colmez and Nizioł prov
ed a comparison theorem between arithmetic p-adic nearby cycles and syntom
ic cohomology sheaves. To prove it\, they gave a local construction using
$(\\varphi\,\\Gamma)$-modules theory which allows to reduce the period iso
morphism to a comparison theorem between Lie algebras. In this talk\, I wi
ll first give the geometric version of this construction before explaining
how to globalize it. This period morphism can be used to describe the é
tale cohomology of rigid analytic spaces. In particular\, we deduce the se
mi-stable conjecture of Fontaine-Jannsen\, which relates the étale cohom
ology of the rigid analytic variety associated to a formal proper semi-sta
ble scheme to its Hyodo-Kato cohomology. This result was also proved by (a
mong others) Tsuji\, via the Fontaine-Messing map\, and by Česnavičius
and Koshikawa\, which generalized the proof of the crystalline conjecture
by Bhatt\, Morrow and Scholze. In the second part of the talk\, I will ex
plain how we can use the previous map to show that the period morphism of
Tsuji and the one of Česnavičius-Koshikawa are the same.\n
LOCATION:https://researchseminars.org/talk/LNTS/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Graham (Imperial College London)
DTSTART;VALUE=DATE-TIME:20210113T160000Z
DTEND;VALUE=DATE-TIME:20210113T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/24
DESCRIPTION:Title: An
ticyclotomic Euler systems for conjugate self-dual representations of $GL(
2n)$\nby Andrew Graham (Imperial College London) as part of London num
ber theory seminar\n\n\nAbstract\nAn Euler system is a collection of Galoi
s cohomology classes which satisfy certain compatibility relations under c
orestriction\, and by constructing an Euler system and relating the classe
s to $L$-values\, one can establish instances of the Bloch--Kato conjectur
e. In this talk\, I will describe a construction of an anticyclotomic Eule
r system for a certain class of conjugate self-dual automorphic representa
tions\, which can be seen as a generalisation of the Heegner point constru
ction. The classes arise from special cycles on unitary Shimura varieties
and are closely related to the branching law associated with the spherical
pair $(GL(n) \\times GL(n)\, GL(2n))$. This is joint work with S.W.A. Sha
h.\n
LOCATION:https://researchseminars.org/talk/LNTS/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raphael Steiner (ETH Zürich)
DTSTART;VALUE=DATE-TIME:20210120T160000Z
DTEND;VALUE=DATE-TIME:20210120T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/25
DESCRIPTION:Title: Fo
urth moments and sup-norms with the aid of theta functions\nby Raphael
Steiner (ETH Zürich) as part of London number theory seminar\n\n\nAbstra
ct\nIt is a classical problem in harmonic analysis to bound $L^p$-norms of
eigenfunctions of the Laplacian on (compact) Riemannian manifolds in term
s of the eigenvalue. A general sharp result in that direction was given by
Hörmander and Sogge. However\, in an arithmetic setting\, one ought to d
o better. Indeed\, it is a classical result of Iwaniec and Sarnak that exa
ctly that is true for Hecke-Maass forms on arithmetic hyperbolic surfaces.
They achieved their result by considering an amplified second moment of H
ecke eigenforms. Their technique has since been adapted to numerous other
settings. In my talk\, I shall explain how to use Shimizu's theta function
to express a fourth moment of Hecke eigenforms in geometric terms (second
moment of matrix counts). In joint work with Ilya Khayutin and Paul Nelso
n\, we give sharp bounds for said matrix counts and thus a sharp bound on
the fourth moment in the weight and level aspect. As a consequence\, we im
prove upon the best known bounds for the sup-norm in these aspects. In par
ticular\, we prove a stronger than Weyl-type sub-convexity result.\n
LOCATION:https://researchseminars.org/talk/LNTS/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lynnelle Ye (Stanford University)
DTSTART;VALUE=DATE-TIME:20210127T160000Z
DTEND;VALUE=DATE-TIME:20210127T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/26
DESCRIPTION:Title: Pr
operness for eigenvarieties\nby Lynnelle Ye (Stanford University) as p
art of London number theory seminar\n\n\nAbstract\nCan a family of finite-
slope modular Hecke eigenforms lying over a punctured disc in weight space
always be extended over the puncture? This was first asked by Coleman and
Mazur in 1998 and settled by Diao and Liu in 2016 using deep\, powerful G
alois-theoretic machinery. We will discuss a new proof which is geometric
and explicit and uses no Galois theory\, and which generalizes in some cas
es to Hilbert modular forms. We adapt an earlier method of Buzzard and Cal
egari based on elementary properties of overconvergent modular forms\, for
which we have to extend the construction of Andreatta-Iovita-Pilloni over
convergent forms farther into the supersingular locus.\n
LOCATION:https://researchseminars.org/talk/LNTS/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lassina Dembélé (University of Luxembourg)
DTSTART;VALUE=DATE-TIME:20210203T160000Z
DTEND;VALUE=DATE-TIME:20210203T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/27
DESCRIPTION:Title: Fi
nite flat $p$-group schemes over $\\mathbf{Z}$\nby Lassina Dembélé (
University of Luxembourg) as part of London number theory seminar\n\n\nAbs
tract\nConjecture (Abrashkin-Fontaine): For $p$ prime\, the only simple fi
nite flat group schemes of $p$-power order defined over $\\mathbf{Z}$ are
$\\mathbf{Z}/p\\mathbf{Z}$ and $\\mu_p$.\n\nAbrashkin and Fontaine separat
ely proved that this conjecture is true for $p \\le 17$. In this talk\, we
extend their result to the primes $p \\le 37$ under GRH. (This is joint w
ork with René Schoof.)\n
LOCATION:https://researchseminars.org/talk/LNTS/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juan Esteban Rodriguez Camargo (ENS de Lyon)
DTSTART;VALUE=DATE-TIME:20210210T160000Z
DTEND;VALUE=DATE-TIME:20210210T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/28
DESCRIPTION:Title: Du
al Eichler-Shimura maps for the modular curve\nby Juan Esteban Rodrigu
ez Camargo (ENS de Lyon) as part of London number theory seminar\n\n\nAbst
ract\nAndreatta-Iovita-Stevens have constructed interpolations of the sm
all slope part of the Eichler-Shimura decomposition for the modular curve.
Roughly speaking\, they defined in a geometric way a map from the overcon
vergent modular symbols of weight k\, to the overconvergent modular forms
of weight k+2. Then\, using classicality theorems of Coleman and Ash-Ste
vens\, they achieved a Hodge-Tate decomposition of the small slope part of
overconvergent modular symbols. On the other hand\, in a recent paper of
Boxer-Pilloni\, the authors proved that higher Coleman and Hida theories
exist for the modular curve. The aim of this talk is to construct geometri
cally a map from the higher cohomology of overconvergent modular forms of
weight -k to the modular symbols as above. We shall recover the Hodge-Tat
e decomposition of the small slope part of modular symbols\, with the addi
tion that all the maps involved are defined using the geometry of the modu
lar curve. If time permits\, we will discuss the compatibility of the prev
ious work with duality.\n
LOCATION:https://researchseminars.org/talk/LNTS/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chao Li (Columbia University)
DTSTART;VALUE=DATE-TIME:20210217T160000Z
DTEND;VALUE=DATE-TIME:20210217T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/29
DESCRIPTION:Title: Be
ilinson-Bloch conjecture for unitary Shimura varieties\nby Chao Li (Co
lumbia University) as part of London number theory seminar\n\n\nAbstract\n
For certain automorphic representations $\\pi$ on unitary groups\, we show
that if $L(s\, \\pi)$ vanishes to order one at the center $s=1/2$\, then
the associated $\\pi$-localized Chow group of a unitary Shimura variety is
nontrivial. This proves part of the Beilinson-Bloch conjecture for unitar
y Shimura varieties\, which generalizes the BSD conjecture. Assuming Kudla
's modularity conjecture\, we further prove the arithmetic inner product f
ormula for $L'(1/2\, \\pi)$\, which generalizes the Gross-Zagier formula.
We will motivate these conjectures and discuss some aspects of the proof.
We will also mention recent extensions applicable to certain symmetric pow
er L-functions of elliptic curves. This is joint work with Yifeng Liu.\n
LOCATION:https://researchseminars.org/talk/LNTS/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Charlotte Chan (MIT)
DTSTART;VALUE=DATE-TIME:20210224T160000Z
DTEND;VALUE=DATE-TIME:20210224T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/30
DESCRIPTION:Title: Ge
ometric L-packets of toral supercuspidal representations\nby Charlotte
Chan (MIT) as part of London number theory seminar\n\n\nAbstract\nIn 2001
\, Yu gave an algebraic construction of supercuspidal representations of p
-adic groups (now known to be exhaustive when the residual characteristic
is sufficiently large---Kim\, Fintzen). There has since been a lot of prog
ress towards explicitly constructing the local Langlands correspondence: K
azhdan-Varshavsky and DeBacker-Reeder (depth zero)\, Reeder and DeBacker-S
pice (toral)\, and Kaletha (regular supercuspidals). In this talk\, we pre
sent recent and ongoing work investigating a geometric counterpart to this
story. This is based on joint work with Alexander Ivanov and joint work w
ith Masao Oi.\n
LOCATION:https://researchseminars.org/talk/LNTS/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anders Södergren (Chalmers University of Technology)
DTSTART;VALUE=DATE-TIME:20210303T160000Z
DTEND;VALUE=DATE-TIME:20210303T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/31
DESCRIPTION:Title: Cu
bic fields\, low-lying zeros and the L-functions Ratios Conjecture\nby
Anders Södergren (Chalmers University of Technology) as part of London n
umber theory seminar\n\n\nAbstract\nIn this talk I will discuss recent wor
k on the low-lying zeros in the family of $L$-functions attached to non-Ga
lois cubic Dedekind zeta functions. In particular\, I will describe the cl
ose relation between these low-lying zeros and precise counting results fo
r cubic fields with local conditions. The main application of this investi
gation is a conditional omega result for cubic field counting functions. I
will also discuss the $L$-functions Ratios Conjecture associated to this
family of Dedekind zeta functions and the fact that the conjecture in its
standard form does not predict all the terms in the family's one-level den
sity of low-lying zeros. This is joint work with Peter Cho\, Daniel Fioril
li and Yoonbok Lee.\n
LOCATION:https://researchseminars.org/talk/LNTS/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Min Lee
DTSTART;VALUE=DATE-TIME:20210310T160000Z
DTEND;VALUE=DATE-TIME:20210310T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/32
DESCRIPTION:Title: Li
nnik problem for Maass-Hecke cusp forms and effective multiplicity one the
orem\nby Min Lee as part of London number theory seminar\n\n\nAbstract
\nThe strong multiplicity one theorem (for GL(2)\, proved by Jacquet and L
anglands) implies that if two Maass-Hecke cuspforms share the same Laplaci
an eigenvalue and the same Hecke eigenvalues for almost all primes then th
e two forms must be equal up to a constant multiple. In this talk we cons
ider the following question\, an analogue of Linnik’s question for Diric
hlet characters: if the two forms are not equal up to a constant multiple\
, how large can the first prime p be\, such that the corresponding Hecke e
igenvalues differ? Alternatively we can also ask: how large is the dimensi
on of the joint eigenspace of the given finite set of Hecke operators and
the Laplace operator? We approach these two questions with two different m
ethods. This is a joint work with Junehyuk Jung.\n
LOCATION:https://researchseminars.org/talk/LNTS/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emmanuel Lecouturier (Yau Mathematical Sciences Center and Tsinghu
a University (Beijing))
DTSTART;VALUE=DATE-TIME:20210317T160000Z
DTEND;VALUE=DATE-TIME:20210317T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/33
DESCRIPTION:Title: On
an analogue of a conjecture of Sharifi for imaginary quadratic fields
\nby Emmanuel Lecouturier (Yau Mathematical Sciences Center and Tsinghua U
niversity (Beijing)) as part of London number theory seminar\n\n\nAbstract
\nWe explore a relation between the cohomology of certain Bianchi 3-folds\
, modulo some Eisenstein ideal\, to the arithmetic of imaginary quadratic
fields.\nFor instance\, in the case of Euclidean imaginary quadratic field
s\, we get a relation between modular symbols and cup-products of elliptic
units.\nThis is similar to conjectures of Sharifi for classical modular c
urves\, relating modular symbols to cup-product of cyclotomic units. \nThi
s is work in progress with Jun Wang.\n
LOCATION:https://researchseminars.org/talk/LNTS/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akshay Venkatesh (Institute for Advanced Study)
DTSTART;VALUE=DATE-TIME:20210324T160000Z
DTEND;VALUE=DATE-TIME:20210324T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/34
DESCRIPTION:Title: Ce
ntral L-values up to squares\nby Akshay Venkatesh (Institute for Advan
ced Study) as part of London number theory seminar\n\n\nAbstract\nThis is
a report on joint work -- in progress -- with A. Abdurrahman. \nGiven an
everywhere unramified symplectic Galois representation\nover a function fi
eld\, we propose a conjectural formula for its central L-value\nup to squa
res in the coefficient field\, in terms of a certain cohomological invaria
nt\nof the representation. \n \nI'll describe three types of evidence f
or this conjecture\, coming\nfrom numerical examples\, topology\, and auto
morphic forms. \nThen I will discuss (much more speculatively) what the ra
mified/number field\nanalogue of the formula might be\, and its potential
relationship to a theory\nof "higher epsilon factors."\n
LOCATION:https://researchseminars.org/talk/LNTS/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eva Viehmann
DTSTART;VALUE=DATE-TIME:20210428T150000Z
DTEND;VALUE=DATE-TIME:20210428T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/36
DESCRIPTION:Title: Ne
wton strata on the B$_{dR}$-Grassmannian\nby Eva Viehmann as part of L
ondon number theory seminar\n\n\nAbstract\nRecently\, Fargues and Scholze
laid the foundations for\na geometric Langlands program on the Fargues-Fon
taine curve. One of the\ncentral objects of interest is the stack Bun$_G$
of $G$-bundles on the\ncurve. I will explain how to determine the underlyi
ng topological space\n|Bun$_{G}$| and its relation to the Newton stratific
ation on the\nB$_{dR}$-Grassmannian.\n
LOCATION:https://researchseminars.org/talk/LNTS/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dimitris Koukoulopoulos
DTSTART;VALUE=DATE-TIME:20210505T150000Z
DTEND;VALUE=DATE-TIME:20210505T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/37
DESCRIPTION:Title: An
atomy of integers\, polynomials and permutations\nby Dimitris Koukoulo
poulos as part of London number theory seminar\n\n\nAbstract\nThere is a f
amous analogy between the statistics of the prime factors of a random inte
ger\, of the irreducible factors of a random polynomial over a finite fiel
d\, and of the cycles of a random permutation. This analogy allows us to t
ransfer techniques and intuition from one setup to the other\, and it has
been in the center of a lot of recent activity in probabilistic number the
ory and group theory. I will survey some of this progress\, focusing in pa
rticular on results about the irreducibility of randomly chosen polynomial
s with 0\,1 coefficients (joint with Lior Bary-Soroker and Gady Kozma)\, a
s well as on results about the concentration of divisors of random integer
s and the size of the Hooley Delta function (joint with Ben Green and Kevi
n Ford).\n
LOCATION:https://researchseminars.org/talk/LNTS/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jef Laga
DTSTART;VALUE=DATE-TIME:20210512T150000Z
DTEND;VALUE=DATE-TIME:20210512T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/38
DESCRIPTION:Title: Ra
tional points and Selmer groups of genus 3 curves\nby Jef Laga as part
of London number theory seminar\n\n\nAbstract\nManjul Bhargava and Arul S
hankar have determined the average size of the n-Selmer group of the famil
y of all elliptic curves over Q ordered by height\, for n at most 5. They
used this to show that the average rank of elliptic curves is less than on
e. \n\nIn this talk we will consider a family of nonhyperelliptic genus 3
curves\, and bound the average size of the 2-Selmer group of their Jacobia
ns. This implies that a majority of curves in this family have relatively
few rational points. We also consider a family of abelian surfaces which a
re not principally polarized and obtain similar results.\n
LOCATION:https://researchseminars.org/talk/LNTS/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anna Medvedovsky
DTSTART;VALUE=DATE-TIME:20210526T150000Z
DTEND;VALUE=DATE-TIME:20210526T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/39
DESCRIPTION:Title: Co
unting modular forms with fixed mod-p Galois representation and Atkin-Lehn
er-at-p eigenvalue\nby Anna Medvedovsky as part of London number theor
y seminar\n\n\nAbstract\nWork in progress joint with Samuele Anni and Alex
andru Ghitza. For N prime to p\, we count the number of classical modular
forms of level Np and weight k with fixed residual Galois representation a
nd Atkin-Lehner-at-p sign\, generalizing both recent results of Martin gen
eralizing work of Wakatsuki (no residual representation constraint) and th
e rhobar-dimension-counting formulas of Bergdall-Pollack and Jochnowitz. T
o resolve tension between working mod p and the need to invert p\, we use
the trace formula to establish up-to-semisimplifcation isomorphisms betwee
n certain mod-p Hecke\nmodules (namely\, refinements of the weight-filtrat
ion graded pieces W_k) by exhibiting ever-deeper congruences between trace
s of prime-power Hecke operators acting on characteristic-zero Hecke\nmodu
les. This last technique is new\, purely algebraic\, and may be of indepen
dent interest\; it relies on a combinatorial theorem whose proof benefited
from a beautiful boost from Gessel.\n
LOCATION:https://researchseminars.org/talk/LNTS/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lillian Pierce
DTSTART;VALUE=DATE-TIME:20210602T150000Z
DTEND;VALUE=DATE-TIME:20210602T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/40
DESCRIPTION:Title: Co
unting problems\, from the perspective of moments\nby Lillian Pierce a
s part of London number theory seminar\n\n\nAbstract\nMany questions in nu
mber theory can be phrased as counting problems. How many number fields ar
e there? How many elliptic curves are there? How many integral solutions t
o this system of Diophantine equations are there? If the answer is “infi
nitely many\,” we want to understand the order of growth for the number
of objects we are counting in the “family." But in many settings we are
also interested in finer-grained questions\, like: how many number fields
are there\, with fixed degree and fixed discriminant? We know the answer i
s “finitely many\,” but it would have important consequences if we cou
ld show the answer is always “very few indeed.” In this talk\, we will
describe a way that these finer-grained questions can be related to the b
igger infinite-family questions. Then we will use this perspective to surv
ey interconnections between several big open conjectures in number theory\
, related in particular to class groups and number fields.\n
LOCATION:https://researchseminars.org/talk/LNTS/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pip Goodman
DTSTART;VALUE=DATE-TIME:20210609T150000Z
DTEND;VALUE=DATE-TIME:20210609T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/41
DESCRIPTION:Title: Su
perelliptic curves with large Galois images\nby Pip Goodman as part of
London number theory seminar\n\n\nAbstract\nLet $K$ be a number field. Th
e inverse Galois problem for $K$ asks if for every finite group $G$ there
exists a Galois extension $L/K$ whose Galois group is isomorphic to $G$. M
any people have used torsion points on abelian varieties to realise symple
ctic similitude groups (${\\rm GSp}_n(F_\\ell)$) over $Q$.\n\nIn this talk
\, we examine mod $\\ell$ Galois representations attached to superelliptic
curves and use them to realise general linear and unitary similitude grou
ps over cyclotomic fields. A variety of mathematics is involved\, includin
g group theory\, CM theory\, root discriminant bounds\, and models of curv
es.\n
LOCATION:https://researchseminars.org/talk/LNTS/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pär Kurlberg
DTSTART;VALUE=DATE-TIME:20210616T150000Z
DTEND;VALUE=DATE-TIME:20210616T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/42
DESCRIPTION:Title: Le
vel repulsion for arithmetic toral point scatterers\nby Pär Kurlberg
as part of London number theory seminar\n\n\nAbstract\nThe Seba billiard w
as introduced to study the transition between\n integrability and chaos
in quantum systems. The model seem to exhibit\n intermediate level sta
tistics with strong repulsion between nearby\n eigenvalues (consistent
with random matrix theory predictions for\n spectra of chaotic systems)
\, whereas large gaps seem to have "Poisson\n tails" (as for spectra of
integrable systems.)\n\n We investigate the closely related "toral poi
nt scatterer"-model\, i.e.\,\n the Laplacian perturbed by a delta-poten
tial\, on 3D tori of the form\n R^3/Z^3. This gives a rank one perturb
ation of the original Laplacian\,\n and it is natural to split the spec
trum/eigenspaces into two parts: the\n "old" (unperturbed) one spanned
by eigenfunctions vanishing at the\n scatterer location\, and the "new"
part (spanned by Green's functions).\n We show that there is strong re
pulsion between the new set of\n eigenvalues.\n
LOCATION:https://researchseminars.org/talk/LNTS/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sug Woo Shin
DTSTART;VALUE=DATE-TIME:20210623T100000Z
DTEND;VALUE=DATE-TIME:20210623T110000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/43
DESCRIPTION:Title: Fr
om Langlands–Rapoport to the trace formula\nby Sug Woo Shin as part
of London number theory seminar\n\n\nAbstract\nIn this talk\, I will repor
t on joint work with Mark Kisin and Yihang Zhu to establish a stabilized t
race formula computing the cohomology of abelian-type Shimura varieties at
a prime of good reduction. As a key intermediate step\, we prove a versio
n of the Langlands-Rapoport conjecture that is more precise than shown in
Kisin’s recent paper.\n
LOCATION:https://researchseminars.org/talk/LNTS/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matt Young (Texas A&M University)
DTSTART;VALUE=DATE-TIME:20210421T150000Z
DTEND;VALUE=DATE-TIME:20210421T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/44
DESCRIPTION:Title: An
improved spectral large sieve inequality for $\\text{SL}_3(\\mathbb Z)$.<
/a>\nby Matt Young (Texas A&M University) as part of London number theory
seminar\n\n\nAbstract\nI will discuss recent progress on the spectral larg
e sieve problem for $\\text{SL}_3(\\mathbb Z)$. The method of proof uses
duality and its structure reveals unexpected connections to Heath-Brown's
large sieve for cubic characters.\n
LOCATION:https://researchseminars.org/talk/LNTS/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Natalie Evans
DTSTART;VALUE=DATE-TIME:20210630T150000Z
DTEND;VALUE=DATE-TIME:20210630T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/45
DESCRIPTION:Title: Co
rrelations of almost primes\nby Natalie Evans as part of London number
theory seminar\n\n\nAbstract\nThe Hardy-Littlewood generalised twin prime
conjecture states an asymptotic formula for the number of primes $p\\le X
$ such that $p+h$ is prime for any non-zero even integer $h$. While this c
onjecture remains wide open\, Matomaki\, Radziwill and Tao proved that it
holds on average over $h$\, improving on a previous result of Mikawa. In t
his talk we will discuss an almost prime analogue of the Hardy-Littlewood
conjecture for which we can go beyond what is known for primes. We will de
scribe some recent work in which we prove an asymptotic formula for the nu
mber of almost primes $n=p_1p_2 \\le X$ such that $n+h$ has exactly two pr
ime factors which holds for a very short average over $h$.\n
LOCATION:https://researchseminars.org/talk/LNTS/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vaidehee Thatte (King's College London)
DTSTART;VALUE=DATE-TIME:20211208T150000Z
DTEND;VALUE=DATE-TIME:20211208T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/48
DESCRIPTION:Title: Un
derstanding the Defect via Ramification Theory\nby Vaidehee Thatte (Ki
ng's College London) as part of London number theory seminar\n\nLecture he
ld in Huxley 144\, Imperial.\n\nAbstract\nClassical ramification theory de
als with complete discrete valuation fields $k((X))$ with perfect residue
fields $k$. Invariants such as the Swan conductor capture important inform
ation about extensions of these fields. Many fascinating complications ari
se when we allow non-discrete valuations and imperfect residue fields $k$.
Particularly in positive residue characteristic\, we encounter the myster
ious phenomenon of the defect (or ramification deficiency). The occurrence
of a non-trivial defect is one of the main obstacles to long-standing pro
blems\, such as obtaining resolution of singularities in positive characte
ristic.\n\nDegree p extensions of valuation fields are building blocks of
the general case. In this talk\, we will present a generalization of ramif
ication invariants for such extensions and discuss how this leads to a bet
ter understanding of the defect. If time permits\, we will briefly discuss
their connection with some recent work (joint with K. Kato) on upper rami
fication groups.\n
LOCATION:https://researchseminars.org/talk/LNTS/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matteo Tamiozzo (Imperial College London)
DTSTART;VALUE=DATE-TIME:20211013T140000Z
DTEND;VALUE=DATE-TIME:20211013T150000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/50
DESCRIPTION:Title: Pe
rfectoid Jacquet-Langlands and the cohomology of Hilbert modular varieties
\nby Matteo Tamiozzo (Imperial College London) as part of London numbe
r theory seminar\n\nLecture held in Huxley 139\, Imperial.\n\nAbstract\nDe
uring and Serre showed that the supersingular locus in a special fibre of
a modular curve can be identified with a Shimura set attached to a definit
e quaternion algebra. I will discuss a perfectoid version of this result o
ver totally real fields\, comparing the cohomology of fibres of the Hodge-
Tate period maps attached to different quaternionic Shimura varieties. I w
ill then explain how this can be used to prove vanishing theorems for the
cohomology with torsion coefficients of Hilbert modular varieties. This is
joint work with Ana Caraiani.\n
LOCATION:https://researchseminars.org/talk/LNTS/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rachel Newton (King's College London)
DTSTART;VALUE=DATE-TIME:20211020T140000Z
DTEND;VALUE=DATE-TIME:20211020T150000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/51
DESCRIPTION:Title: Nu
mber fields with prescribed norms\nby Rachel Newton (King's College Lo
ndon) as part of London number theory seminar\n\nLecture held in Huxley 14
4\, Imperial.\n\nAbstract\nLet $G$ be a finite abelian group\, let $k$ be
a number field\, and let $\\alpha\\in k^\\times$. We count Galois extensio
ns $K/k$ with Galois group $G$ such that $\\alpha$ is a norm from $K/k$.\n
In particular\, we show that such extensions always exist. This is joint w
ork with Christopher\nFrei and Daniel Loughran.\n
LOCATION:https://researchseminars.org/talk/LNTS/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chris Williams (University of Warwick)
DTSTART;VALUE=DATE-TIME:20211110T150000Z
DTEND;VALUE=DATE-TIME:20211110T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/52
DESCRIPTION:Title: $p
$-adic $L$-functions for GL(3)\nby Chris Williams (University of Warwi
ck) as part of London number theory seminar\n\nLecture held in Huxley 144\
, Imperial.\n\nAbstract\nLet $\\pi$ be a $p$-ordinary cohomological cuspid
al automorphic representation of $GL_n(\\mathbb{A}_\\mathbb{Q})$. A conjec
ture of Coates--Perrin-Riou predicts that the (twisted) critical values of
its $L$-function $L(\\pi \\times \\chi\,s)$\, for Dirichlet characters $\
\chi$ of $p$-power conductor\, satisfy systematic congruence properties mo
dulo powers of $p$\, captured in the existence of a $p$-adic $L$-function.
For $n = 1\,2$ this conjecture has been known for decades\, but for $n \\
geq 3$ it is known only in special cases\, e.g. symmetric squares of modul
ar forms\; and in all known cases\, $\\pi$ is a functorial transfer from a
proper subgroup of $GL_n$. I will explain what a $p$-adic $L$-function is
\, state the conjecture more precisely\, and then report on ongoing joint
work with David Loeffler\, in which we prove this conjecture for $n=3$ (wi
thout any transfer or self-duality assumptions).\n
LOCATION:https://researchseminars.org/talk/LNTS/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Torzewski (King's College London)
DTSTART;VALUE=DATE-TIME:20211124T150000Z
DTEND;VALUE=DATE-TIME:20211124T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/53
DESCRIPTION:Title: La
wrence-Venkatesh in families\nby Alex Torzewski (King's College London
) as part of London number theory seminar\n\nLecture held in Huxley 144\,
Imperial.\n\nAbstract\nWe outline how the method of Lawrence-Venkatesh can
be used in families to obtain upper bounds on the number of rational poin
ts on curves of genus > 1 depending only on the reduction modulo a well ch
osen prime and the primes of bad reduction. This was first shown by Faltin
gs as a consequence of the Mordell and Shafarevich Conjectures.\n
LOCATION:https://researchseminars.org/talk/LNTS/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jack Sempliner (Imperial College London)
DTSTART;VALUE=DATE-TIME:20211201T150000Z
DTEND;VALUE=DATE-TIME:20211201T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/54
DESCRIPTION:Title: On
the almost-product structure on the moduli of bounded global $G$-shtuka\nby Jack Sempliner (Imperial College London) as part of London number t
heory seminar\n\nLecture held in Huxley 144\, Imperial.\n\nAbstract\nLet $
X$ be an algebraic curve over $\\mathbb{F}_q$ and $G$ be a reductive algeb
raic group over $\\mathbb{F}_q(X)$. Under mild technical hypotheses we con
struct families of stacks over the moduli $\\text{Sht}_{G\, X\, I}^{\\mu_*
}$ of bounded global $G$-shtuka (a small generalization of the stacks stud
ied by Lafforgue and Varshavsky) which provide natural analogues of Igusa
varieties in the function field setting. Our main result is an isomorphism
between certain Igusa varieties associated to moduli of shtuka for reduct
ive groups $G\, G'$ which are related by an inner twist. Along the way we
prove an almost-product formula computing the compactly supported cohomolo
gy of the special fibers of $\\text{Sht}_{G\, X\, I}^{\\mu_*}$ with trivia
l coefficients in terms of the cohomology of our Igusa stacks and a functi
on-field analogue of Rapoport-Zink spaces constructed in previous work of
Hartl and Arasteh Rad.\n
LOCATION:https://researchseminars.org/talk/LNTS/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rosa Winter
DTSTART;VALUE=DATE-TIME:20211117T150000Z
DTEND;VALUE=DATE-TIME:20211117T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/55
DESCRIPTION:Title: De
nsity of rational points on del Pezzo surfaces of degree 1\nby Rosa Wi
nter as part of London number theory seminar\n\nLecture held in Huxley 144
\, Imperial.\n\nAbstract\nDel Pezzo surfaces are surfaces classified by th
eir degree $d$\, which is an integer between 1 and 9 (for $d\\geq3$\, thes
e are the smooth surfaces of degree $d$ in $\\mathbb{P}^d$). For del Pezzo
surfaces of degree at least 2 over a field $k$\, we know that the set of
$k$-rational points is Zariski dense provided that the surface has one $k$
-rational point to start with (that lies outside a specific subset of the
surface for degree 2). However\, for del Pezzo surfaces of degree 1 over a
field $k$\, even though we know that they always contain at least one $k$
-rational point\, we do not know if the set of $k$-rational points is Zari
ski dense in general. I will talk about a result that is joint work with J
ulie Desjardins\, in which we give sufficient conditions for the set of $k
$-rational points on a specific family of del Pezzo surfaces of degree 1 t
o be Zariski dense\, where $k$ is any infinite field of characteristic 0.
These conditions are necessary if $k$ is finitely generated over $\\mathbb
{Q}$. I will compare this to previous results.\n
LOCATION:https://researchseminars.org/talk/LNTS/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarah Zerbes (University College London)
DTSTART;VALUE=DATE-TIME:20211103T150000Z
DTEND;VALUE=DATE-TIME:20211103T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/56
DESCRIPTION:Title: Eu
ler systems and the BSD conjecture for abelian surfaces\nby Sarah Zerb
es (University College London) as part of London number theory seminar\n\n
Lecture held in Huxley 144\, Imperial.\n\nAbstract\nEuler systems are one
of the most powerful tools for proving cases of the Bloch--Kato conjecture
\, and other related problems such as the Birch and Swinnerton-Dyer conjec
ture. I will recall a series of recent works (variously joint with Loeffle
r\, Pilloni\, Skinner) giving rise to an Euler system in the cohomology of
Shimura varieties for GSp(4)\, and an explicit reciprocity law relating t
his to values of L-functions. I will then explain work in progress with Lo
effler\, in which we use this Euler system to prove new cases of the BSD c
onjecture for modular abelian surfaces over Q\, and modular elliptic curve
s over imaginary quadratic fields.\n
LOCATION:https://researchseminars.org/talk/LNTS/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Jossen
DTSTART;VALUE=DATE-TIME:20220112T160000Z
DTEND;VALUE=DATE-TIME:20220112T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/57
DESCRIPTION:Title: A
non-hypergeometric E-function\nby Peter Jossen as part of London numbe
r theory seminar\n\n\nAbstract\nWith the goal of generalising the theorems
of Hermite\, Lindemann\, and Weierstrass about transcendence of values of
the exponential function\, Siegel\nintroduced the notion of E-function in
his landmark 1929 paper "Über einige Anwendungen diophantischer Approxim
ationen". Hypergeometric functions\nprovide a rich class of E-functions\,
and Siegel asked whether in fact every E-function is a polynomial expressi
on in hypergeometric E-functions. In a\njoint work with Javier Fresán\, w
e answer Siegel's question in the negative.\n
LOCATION:https://researchseminars.org/talk/LNTS/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Yafaev
DTSTART;VALUE=DATE-TIME:20220119T160000Z
DTEND;VALUE=DATE-TIME:20220119T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/58
DESCRIPTION:Title: He
ights of special points and the Andre-Oort conjecture\nby Andrei Yafae
v as part of London number theory seminar\n\n\nAbstract\nThe Andre-Oort co
njecture states that components of the Zariski closure of a set of\nspecia
l points in a Shimura variety\, are special subvarieties.\nThis conjecture
has been a subject of active research in \nrecent years.\nThe last remain
ing step was to obtain lower bounds for Galois\ndegrees of special points.
\n\nIn a joint work with Gal Biniyamini and Harry Schmidt\, we have formul
ated a conjecture \non heights of special points and deduced from it the r
equired bounds.\nVery recently\, J.Pila\, A.Shankar and J.Tsimerman \nanno
unced a proof of our height conjecture\, thus completing the proof\nof the
Andre-Oort conjecture in full generality.\n
LOCATION:https://researchseminars.org/talk/LNTS/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Caleb Springer
DTSTART;VALUE=DATE-TIME:20220126T160000Z
DTEND;VALUE=DATE-TIME:20220126T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/59
DESCRIPTION:Title: Ev
ery finite abelian group arises as the group of rational points of an ordi
nary abelian variety over $\\mathbb{F}_2$\, $\\mathbb{F}_3$\, and $\\math
bb{F}_5$\nby Caleb Springer as part of London number theory seminar\n\
n\nAbstract\nWe will show that every finite abelian group arises as the gr
oup of rational points of an ordinary abelian variety over a finite field
with 2\, 3 or 5 elements. Similar results hold over finite fields of larg
er cardinality. On our way to proving these results\, we will view the gr
oup of rational points of an abelian variety as a module over its endomorp
hism ring. By describing this module structure in important cases\, we obt
ain (a fortiori) an understanding of the underlying groups. Combining this
description of structure with recent results on the cardinalities of grou
ps of rational points of abelian varieties over finite fields\, we will de
duce the main theorem. This work is joint with Stefano Marseglia.\n
LOCATION:https://researchseminars.org/talk/LNTS/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Walker
DTSTART;VALUE=DATE-TIME:20220202T160000Z
DTEND;VALUE=DATE-TIME:20220202T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/60
DESCRIPTION:Title: Th
e Average Number of Divisors of the Output of a Quadratic Polynomial\n
by Alex Walker as part of London number theory seminar\n\n\nAbstract\nLet
$d(n)$ count the number of divisors of $n$. In 1963\, Hooley studied parti
al sums $n < X$ of $d(n^2+h)$ and showed that the result was asymptotic to
$c X \\log X + c' X + O(X^{8/9})$ as $X$ tends to infinity (assuming $h$
not a negative square). In other words\, the irreducible polynomial $Q(x)
= x^2 + h$ has outputs with\, on average\, $\\sim \\log x$ many divisors.
Hooley's error bound was improved by Bykoskii in 1987 to $O(X^{2/3})$ usin
g the spectral theory of automorphic forms. This talk describes a new proo
f of Bykovskii's result in a new framework\, now using Dirichlet series an
d automorphic forms of half-integral weight. This new framework has limita
tions but is also quite flexible. To demonstrate this\, we develop in tand
em counts for the average number of divisors of $Q(x\,y) = x^2+y^2+h$ for
$x^2+y^2+h < X$.\n
LOCATION:https://researchseminars.org/talk/LNTS/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Streeter
DTSTART;VALUE=DATE-TIME:20220209T160000Z
DTEND;VALUE=DATE-TIME:20220209T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/61
DESCRIPTION:Title: We
ak approximation for del Pezzo surfaces of low degree\nby Sam Streeter
as part of London number theory seminar\n\n\nAbstract\nConjecturally\, th
e rational points of a del Pezzo surface over a number field are well-dist
ributed among the local points over all but finitely completions of the gr
ound field—that is\, the surface satisfies weak weak approximation. Howe
ver\, describing the rational points becomes harder as the degree of the d
el Pezzo surface decreases. As such\, many questions remain unanswered for
del Pezzo surfaces of low degree. In this talk\, I will report on recent
joint work with Julian Demeio\, in which we prove that del Pezzo surfaces
of degrees 1 and 2 satisfy weak weak approximation\, provided that we assu
me some additional geometric structure in the form of conic fibrations.\n
LOCATION:https://researchseminars.org/talk/LNTS/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nirvana Coppola (Vrije Universiteit Amsterdam)
DTSTART;VALUE=DATE-TIME:20220223T160000Z
DTEND;VALUE=DATE-TIME:20220223T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/62
DESCRIPTION:Title: Co
leman integrals over number fields: a computational approach\nby Nirva
na Coppola (Vrije Universiteit Amsterdam) as part of London number theory
seminar\n\n\nAbstract\nOne of the deepest mathematical results is Faltings
's Theorem on the finiteness of rational points on an algebraic curve of g
enus $g \\geq 2$. A much more difficult question\, still not completely an
swered\, is whether given a curve of genus $g \\geq 2$\, we can find all i
ts rational points\, or\, more in general\, all points defined over a cert
ain number field. An entire (currently very active!) area of research is d
evoted to find an answer to such questions\, using the "method of Chabauty
".\n\nIn this seminar\, I will talk about one of the first tools employed
in Chabauty method\, namely Coleman integrals\, which Coleman used to comp
ute an explicit bound on the number of rational points on a curve. After e
xplaining how this is defined\, I will give a generalisation of this defin
ition for curves defined over number fields\, and explain how to explicitl
y compute these integrals. This is based on an ongoing project\, which sta
rted during the Arizona Winter School 2020\, joint with E. Kaya\, T. Kelle
r\, N. Müller\, S. Muselli.\n
LOCATION:https://researchseminars.org/talk/LNTS/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Miriam Norris (King's College London)
DTSTART;VALUE=DATE-TIME:20220302T160000Z
DTEND;VALUE=DATE-TIME:20220302T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/63
DESCRIPTION:Title: La
ttice graphs for representations of $GL_3(\\F_p)$\nby Miriam Norris (K
ing's College London) as part of London number theory seminar\n\n\nAbstrac
t\nIn a recent paper Le\, Le Hung\, Levin and Morra proved a generalisatio
n of Breuil's Lattice conjecture in dimension three. This involved showing
that lattices inside representations of $GL_3(\\F_p)$ coming from both a
global and a local construction coincide. Motivated by this we consider th
e following graph. For an irreducible representation $\\tau$ of a group $G
$ over a finite extension $K$ of $\\Q_p$ we define a graph on the $\\mathc
al{O}_K$-lattices inside $\\tau$ whose edges encapsulate the relationship
between lattices in terms of irreducible modular representations of $G$ (o
r Serre weights in the context of the paper by Le et al.). \n\nIn this tal
k\, I will demonstrate how one can apply the theory of graduated orders an
d their lattices\, established by Zassenhaus and Plesken\, to understand t
he lattice graphs of residually multiplicity free representation over suit
ably large fields in terms of a matrix called an exponent matrix. Furtherm
ore I will explain how I have been able to show that one can determine the
exponent matrices for suitably generic representation go $GL_3(\\F_p)$ al
lowing us to construct their lattice graphs.\n
LOCATION:https://researchseminars.org/talk/LNTS/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:María Inés de Frutos Fernández (Imperial College London)
DTSTART;VALUE=DATE-TIME:20220309T160000Z
DTEND;VALUE=DATE-TIME:20220309T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/64
DESCRIPTION:Title: Fo
rmalizing the ring of adèles and some applications in Lean\nby María
Inés de Frutos Fernández (Imperial College London) as part of London nu
mber theory seminar\n\n\nAbstract\nI will present a formalization of the r
ing of adèles and group of idèles of a global field in the Lean 3 theore
m prover. Lean is an interactive theorem prover with an ever-growing mathe
matics library. I will give a quick introduction to Lean and explain how t
hese definitions were formalized\, with a focus on the kind of decisions o
ne has to make during the formalization process.\n\nBesides the definition
of the adèles\, we will discuss the formalization of applications includ
ing the statement of the main theorem of global class field theory and a p
roof that the ideal class group of a number field is isomorphic to an expl
icit quotient of its idèle class group.\n
LOCATION:https://researchseminars.org/talk/LNTS/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Rockwood (University of Warwick)
DTSTART;VALUE=DATE-TIME:20220316T160000Z
DTEND;VALUE=DATE-TIME:20220316T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/65
DESCRIPTION:Title: Sp
herical varieties and non-ordinary families of cohomology classes\nby
Robert Rockwood (University of Warwick) as part of London number theory se
minar\n\n\nAbstract\nThe theory of norm compatible cohomology classes is o
f fundamental importance in Iwasawa theory\, encompassing both the theory
of Euler systems and p-adic L-functions. Loeffler has developed a systemat
ic approach to constructing norm-compatible classes using the theory of sp
herical varieties. We show that classes constructed in this way vary natur
ally in Coleman families and give some concrete applications.\n
LOCATION:https://researchseminars.org/talk/LNTS/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Magee
DTSTART;VALUE=DATE-TIME:20220323T160000Z
DTEND;VALUE=DATE-TIME:20220323T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/66
DESCRIPTION:Title: Th
e maximal spectral gap of a hyperbolic surface\nby Michael Magee as pa
rt of London number theory seminar\n\n\nAbstract\nA hyperbolic surface is
a surface with metric of constant curvature -1. The spectral gap between\n
the first two eigenvalues of the Laplacian on a closed hyperbolic surface
contains a good deal of\ninformation about the surface\, including its con
nectivity\, dynamical properties of its geodesic flow\,\nand error terms i
n geodesic counting problems. For arithmetic hyperbolic surfaces the spect
ral gap\nis also the subject of one of the biggest open problems in automo
rphic forms: Selberg’s eigenvalue\nconjecture.\nIt was an open problem f
rom the 1970s whether there exist a sequence of closed hyperbolic sur-\nfa
ces with genera tending to infinity and spectral gap tending to 1/4. (The
value 1/4 here is the\nasymptotically optimal one.) Recently we proved tha
t this is indeed possible. I’ll discuss the very\ninteresting background
of this problem in detail as well as some ideas of the proof. This is joi
nt work\nwith Will Hide.\n
LOCATION:https://researchseminars.org/talk/LNTS/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Best (VU Amsterdam)
DTSTART;VALUE=DATE-TIME:20220427T150000Z
DTEND;VALUE=DATE-TIME:20220427T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/67
DESCRIPTION:Title: Th
e S-unit equation and non-abelian Chabauty in depth 2\nby Alex Best (V
U Amsterdam) as part of London number theory seminar\n\nLecture held in Bu
sh House S2.03\, King's College London.\n\nAbstract\nThe S-unit equation i
s a classical and well-studied Diophantine equation\, with numerous connec
tions to other Diophantine problems.\nRecent work of Kim and refinements d
ue to Betts-Dogra have suggested new cohomological strategies to find rati
onal and integral points on curves\, based on but massively extending the
classical method of Chabauty. At present\, these methods are only conjectu
rally guaranteed to succeed in general\, but they promise several applicat
ions in arithmetic geometry if they could be proved to always work.\nIn or
der to better understand the conjectures of Kim that suggest that this met
hod should work\, we consider the case of the thrice punctured projective
line\, in "depth 2"\, the "smallest" non-trivial extension of the classica
l method. In doing so we get very explicit results for some S-unit equatio
ns\, demonstrating the usability of the aforementioned cohomological metho
ds in this setting. To do this we determine explicitly equations for (maps
between) the (refined) Selmer schemes defined by Kim\, and Betts-Dogra\,
which turn out to have some particularly simple forms.\nThis is joint work
with Alexander Betts\, Theresa Kumpitsch\, Martin Lüdtke\, Angus McAndre
w\, Lie Qian\, Elie Studnia\, and Yujie Xu .\n
LOCATION:https://researchseminars.org/talk/LNTS/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aled Walker (King's College London)
DTSTART;VALUE=DATE-TIME:20220504T150000Z
DTEND;VALUE=DATE-TIME:20220504T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/68
DESCRIPTION:Title: Co
rrelations of sieve weights and distributions of zeros\nby Aled Walker
(King's College London) as part of London number theory seminar\n\nLectur
e held in King's Building K0.18\, King's College London.\n\nAbstract\nIn t
his talk we will discuss Montgomery's pair correlation conjecture for the
zeros of the Riemann zeta function. This is a deep spectral conjecture\, c
losely related to several arithmetic conjectures on the distribution of pr
imes. For example\, even assuming a strong form of the twin prime conjectu
re\, one would only resolve Montgomery's conjecture in a limited range. Ye
t\, building on work of Goldston and Gonek from the late 1990s\, we will p
resent a recent conditional lower bound on the Fourier transform of Montgo
mery's pair correlation function\, valid under milder hypotheses. The new
technical ingredient is a correlation estimate for the Selberg sieve weigh
ts\, for which the level of support of the weights lies beyond the classic
al square-root barrier.\n
LOCATION:https://researchseminars.org/talk/LNTS/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Zenz (McGill)
DTSTART;VALUE=DATE-TIME:20220511T150000Z
DTEND;VALUE=DATE-TIME:20220511T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/69
DESCRIPTION:Title: Ho
lomorphic Hecke Cusp Forms and Quantum Chaos\nby Peter Zenz (McGill) a
s part of London number theory seminar\n\nLecture held in King's Building\
, K0.18.\n\nAbstract\nArithmetic Quantum Chaos (AQC) is an active area of
research at the intersection of number theory and physics. One major goal
in AQC is to study the mass distribution and behaviour of Hecke Maass cusp
forms on hyperbolic surfaces as the Laplace eigenvalue tends to infinity.
In this talk we will focus on analogous questions for holomorphic Hecke c
usp forms. First\, we will review some of the important solved and unsolve
d questions in the area\, like the Quantum Unique Ergodicity problem or th
e Gaussian Moment Conjecture. We then elaborate on a sharp bound for the f
ourth moment of holomorphic cusp forms and ongoing work on evaluating the
averaged sixth moment of holomorphic cusp forms. These are special instanc
es of the Gaussian Moment Conjecture.\n
LOCATION:https://researchseminars.org/talk/LNTS/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emilia Alvarez (Bristol)
DTSTART;VALUE=DATE-TIME:20220518T150000Z
DTEND;VALUE=DATE-TIME:20220518T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/70
DESCRIPTION:Title: Mo
ment computations in the classical compact ensembles\nby Emilia Alvare
z (Bristol) as part of London number theory seminar\n\n\nAbstract\nAfter a
brief introduction on the random matrix applications to number theory\, I
will present a collection of moment computations over the unitary\, sympl
ectic and special orthogonal random matrix ensembles that I've done throug
hout my thesis. I will highlight work on the asymptotics of moments of the
logarithmic derivative of characteristic polynomials evaluated near the p
oint 1. Throughout\, the focus will be on the methods used\, the motivatio
n from number theory and directions for future work.\n
LOCATION:https://researchseminars.org/talk/LNTS/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Justin Trias-Batle (Imperial College London)
DTSTART;VALUE=DATE-TIME:20220525T150000Z
DTEND;VALUE=DATE-TIME:20220525T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/71
DESCRIPTION:Title: To
wards a theta correspondence in families for type II dual pairs\nby Ju
stin Trias-Batle (Imperial College London) as part of London number theory
seminar\n\nLecture held in K0.18 King's building KCL.\n\nAbstract\nThis i
s current work with Gil Moss. The classical local theta correspondence for
p-adic reductive dual pairs defines a bijection between prescribed subset
s of irreducible smooth complex representations coming from two groups (H\
,H')\, forming a dual pair in a symplectic group. Alberto Mínguez extende
d this result for type II dual pairs\, i.e. when (H\,H') is made of genera
l linear groups\, to representations with coefficients in an algebraically
closed field of characteristic l as long as the characteristic l does not
divide the pro-orders of H and H'. For coefficients rings like Z[1/p]\, w
e explain how to build a theory in families for type II dual pairs that is
compatible with reduction to residue fields of the base coefficient ring\
, where central to this approach is the integral Bernstein centre. We tran
slate some weaker properties of the classical correspondence\, such as com
patibility with supercuspidal support\, as a morphism between the integral
Bernstein centres of H and H' and interpret it for the Weil representatio
n. In general\, we only know that this morphism is finite though we may ex
pect it to be surjective. This would result in a closed immersion between
the associated affine schemes as well as a correspondence between characte
rs of the Bernstein centre.\n
LOCATION:https://researchseminars.org/talk/LNTS/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raman Parimala (Emory)
DTSTART;VALUE=DATE-TIME:20220601T150000Z
DTEND;VALUE=DATE-TIME:20220601T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/72
DESCRIPTION:Title: Th
e Brauer group of hyperelliptic curves over number fields\nby Raman Pa
rimala (Emory) as part of London number theory seminar\n\nLecture held in
King's building K0.18\, King's College London.\n\nAbstract\nWe discuss per
iod-index bounds for the unramified Brauer group of function fields of hyp
erelliptic curves over number fields. We describe a link to the question
of Hasse principle for smooth intersection of quadrics.\n
LOCATION:https://researchseminars.org/talk/LNTS/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sacha Mangerel (Durham)
DTSTART;VALUE=DATE-TIME:20220615T150000Z
DTEND;VALUE=DATE-TIME:20220615T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/73
DESCRIPTION:Title: Ga
ussian distribution of squarefree and B-free numbers in short intervals\nby Sacha Mangerel (Durham) as part of London number theory seminar\n\nL
ecture held in Room K0.18 in the King's Building.\n\nAbstract\n(Joint with
O. Gorodetsky and B. Rodgers) It is of classical interest in analytic num
ber theory to understand the fine-scale distribution of arithmetic sequenc
es such as the primes. For a given length scale h\, the number of elements
of a ``nice'' sequence in a uniformly randomly selected interval $(x\,x+h
]\, 1 \\leq x \\leq X$\, might be expected to follow the statistics of a n
ormally distributed random variable (in suitable ranges of $1 \\leq h \\le
q X$). Following the work of Montgomery and Soundararajan\, this is known
to be true for the primes\, but only if we assume several deep and long-s
tanding conjectures among which the Riemann Hypothesis. \n\nAs a model for
the primes\, in this talk I will address such statistical questions for t
he sequence of squarefree numbers\, i.e.\, numbers not divisible by the sq
uare of any prime\, among other related ``sifted'' sequences called B-free
numbers. I hope to further motivate and explain our main result that show
s\, unconditionally\, that short interval counts of squarefree numbers do
satisfy Gaussian statistics\, answering several questions of R.R. Hall.\n
LOCATION:https://researchseminars.org/talk/LNTS/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lassina Dembele (King's College London)
DTSTART;VALUE=DATE-TIME:20220622T150000Z
DTEND;VALUE=DATE-TIME:20220622T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/74
DESCRIPTION:Title: Ex
plicit inertial local Langlands correspondence for ${\\rm GL_2}$ and arith
metic applications\nby Lassina Dembele (King's College London) as part
of London number theory seminar\n\n\nAbstract\nIn this talk\, we describe
an algorithm for computing automorphic and inertial types for ${\\rm GL_2
}$\, and gives several applications. (This is joint work with Nuno Freitas
and John Voight.)\n
LOCATION:https://researchseminars.org/talk/LNTS/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yukako Kezuka (Jussieu)
DTSTART;VALUE=DATE-TIME:20220629T150000Z
DTEND;VALUE=DATE-TIME:20220629T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/75
DESCRIPTION:Title: Ar
ithmetic of elliptic curves with complex multiplication at small primes\nby Yukako Kezuka (Jussieu) as part of London number theory seminar\n\nL
ecture held in King's Building K0.18.\n\nAbstract\nThe equation E: x^3+y^3
=N defines a classical family of elliptic curves as N varies over cube-fre
e positive integers. They admit complex multiplication\, which allows us t
o tackle the conjecture of Birch and Swinnerton-Dyer for E effectively. In
deed\, using Iwasawa theory\, Rubin was able to show the p-part of the con
jecture for E for all primes p\, except for the primes 2 and 3. The theory
becomes much more complex at these small primes\, but at the same time we
can observe some interesting phenomena. I will explain a method to study
the p-adic valuation of the algebraic part of the central L-value of E\, a
nd I will establish the 3-part of the conjecture for E in special cases. I
will then explain a relation between the 2-part of a certain ideal class
group and the Tate-Shafarevich group of E. Part of this talk is based on j
oint work with Yongxiong Li.\n
LOCATION:https://researchseminars.org/talk/LNTS/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Si Ying Lee (MPIM Bonn)
DTSTART;VALUE=DATE-TIME:20221012T150000Z
DTEND;VALUE=DATE-TIME:20221012T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/76
DESCRIPTION:Title: Ei
chler-Shimura relations of Hodge type Shimura varieties\nby Si Ying Le
e (MPIM Bonn) as part of London number theory seminar\n\nLecture held in L
ecture held in Huxley 144\, Imperial.\n\nAbstract\nThe well-known classica
l Eichler-Shimura relation for modular curves asserts that the Hecke opera
tor $T_p$ is equal\, as an algebraic correspondence over the special fiber
\, to the sum of Frobenius and Verschiebung. Blasius and Rogawski proposed
a generalization of this result for Shimura varieties with good reduction
at $p$\, and conjectured that the Frobenius satisfies a certain Hecke pol
ynomial. I will talk about a recent proof of this for some Shimura varieti
es of Hodge type.\n
LOCATION:https://researchseminars.org/talk/LNTS/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peiyi Cui (University of East Anglia)
DTSTART;VALUE=DATE-TIME:20221102T160000Z
DTEND;VALUE=DATE-TIME:20221102T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/77
DESCRIPTION:Title: A
decomposition of the category of l-modular representations of SL_n(F).
\nby Peiyi Cui (University of East Anglia) as part of London number theory
seminar\n\nLecture held in Huxley 139\, Imperial.\n\nAbstract\nLet F be a
p-adic field\, and k an algebraically closed field of characteristic l di
fferent from p. In this talk\, we will first give a category decomposition
of Rep_k(SL_n(F))\, the category of smooth k-representations of SL_n(F)\,
with respect to the GL_n(F)-equivalent supercuspidal classes of SL_n(F)\,
which is not always a block decomposition in general. We then give a bloc
k decomposition of the supercuspidal subcategory\, by introducing a partit
ion on each GL_n(F)-equivalent supercuspidal class through type theory\, a
nd we interpret this partition by the sense of l-blocks of finite groups.
We give an example where a block of Rep_k(SL_2(F)) is defined with respect
to several SL_2(F)-equivalent supercuspidal classes\, which is different
from the case where l is zero. We end this talk by giving a prediction on
the block decomposition of Rep_k(A) for a general p-adic group A.\n
LOCATION:https://researchseminars.org/talk/LNTS/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yu Min (Imperial College London)
DTSTART;VALUE=DATE-TIME:20221116T160000Z
DTEND;VALUE=DATE-TIME:20221116T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/78
DESCRIPTION:Title: Ho
dge--Tate crystals and Sen theory\nby Yu Min (Imperial College London)
as part of London number theory seminar\n\nLecture held in Huxley 139\, I
mperial.\n\nAbstract\nLet $K$ be a finite extension of $\\mathbb Q_p$. Bha
tt and Scholze have proved that the category of prismatic $F$-crystals on
the absolute prismatic site of $\\mathcal O_K$ is equivalent to the catego
ry of crystalline $\\mathbb Z_p$-representations of the absolute Galois gr
oup of $K$. In this talk\, we will instead consider the (rational) Hodge--
Tate crystals on the absolute prismatic site of $\\mathcal O_K$ or more ge
nerally of a smooth $p$-adic formal scheme. We will show how Hodge--Tate c
rystals are related to the Sen theory. If time permits\, we will also disc
uss its application in the arithmetic $p$-adic Simpson correspondence. Thi
s is joint work with Yupeng Wang.\n
LOCATION:https://researchseminars.org/talk/LNTS/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeff Manning (Imperial College London)
DTSTART;VALUE=DATE-TIME:20221123T160000Z
DTEND;VALUE=DATE-TIME:20221123T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/79
DESCRIPTION:Title: Th
e Wiles-Lenstra-Diamond numerical criterion over imaginary quadratic field
s\nby Jeff Manning (Imperial College London) as part of London number
theory seminar\n\nLecture held in Huxley 139\, Imperial.\n\nAbstract\nWile
s' modularity lifting theorem was the central argument in his proof of mod
ularity of (semistable) elliptic curves over Q\, and hence of Fermat's Las
t Theorem. His proof relied on two key components: his "patching" argument
(developed in collaboration with Taylor) and his numerical isomorphism cr
iterion.\n\nIn the time since Wiles' proof\, the patching argument has bee
n generalized extensively to prove a wide variety of modularity lifting re
sults. In particular Calegari and Geraghty have found a way to generalize
it to prove potential modularity of elliptic curves over imaginary quadrat
ic fields (contingent on some standard conjectures). The numerical criteri
on on the other hand has proved far more difficult to generalize\, althoug
h in situations where it can be used it can prove stronger results than wh
at can be proven purely via patching.\n\nIn this talk I will present joint
work with Srikanth Iyengar and Chandrashekhar Khare which proves a genera
lization of the numerical criterion to the context considered by Calegari
and Geraghty (and contingent on the same conjectures). This allows us to p
rove integral "R=T" theorems at non-minimal levels over imaginary quadrati
c fields\, which are inaccessible by Calegari and Geraghty's method. The r
esults provide new evidence in favor of a torsion analog of the classical
Langlands correspondence.\n
LOCATION:https://researchseminars.org/talk/LNTS/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Toby Gee (IC)
DTSTART;VALUE=DATE-TIME:20221130T160000Z
DTEND;VALUE=DATE-TIME:20221130T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/80
DESCRIPTION:Title: Co
ngruences between modular forms and the categorical p-adic Langlands progr
am\nby Toby Gee (IC) as part of London number theory seminar\n\nLectur
e held in Huxley 139\, Imperial.\n\nAbstract\nI will attempt to give a gen
tle introduction to the categorical p-adic Langlands program and its conne
ctions to questions about congruences between modular forms.\n
LOCATION:https://researchseminars.org/talk/LNTS/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hanneke Wiersema (University of Cambridge)
DTSTART;VALUE=DATE-TIME:20221207T160000Z
DTEND;VALUE=DATE-TIME:20221207T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/81
DESCRIPTION:Title: Mo
dularity in the partial weight one case\nby Hanneke Wiersema (Universi
ty of Cambridge) as part of London number theory seminar\n\nLecture held i
n Huxley 139\, Imperial.\n\nAbstract\nThe strong form of Serre's conjectur
e states that a two-dimensional mod $p$ representation of the absolute Gal
ois group of $\\mathbb{Q}$ arises from a modular form of a specific weight
\, level and character. Serre restricted to modular forms of weight at lea
st 2\, but Edixhoven later refined this conjecture to include weight one m
odular forms. In this talk we explore analogues of Edixhoven's refinement
for Galois representations of totally real fields\, extending recent work
of Diamond–-Sasaki. In particular\, we show how modularity of partial we
ight one Hilbert modular forms can be related to modularity of Hilbert mod
ular forms with regular weights\, and vice versa. Time permitting\, we wil
l also discuss a $p$-adic Hodge theoretic version of this.\n
LOCATION:https://researchseminars.org/talk/LNTS/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rong Zhou (University of Cambridge)
DTSTART;VALUE=DATE-TIME:20221214T160000Z
DTEND;VALUE=DATE-TIME:20221214T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/82
DESCRIPTION:Title: In
dependence of $\\ell$ for $G$-valued Weil--Deligne representations associa
ted to abelian varieties\nby Rong Zhou (University of Cambridge) as pa
rt of London number theory seminar\n\nLecture held in Huxley 139\, Imperia
l.\n\nAbstract\nLet $A$ be an abelian variety over a number field $E$ of d
imension $g$ and $\\rho_\\ell:\\mathrm{Gal}(\\overline{E}/E)\\rightarrow \
\mathrm{GL}_{2g}(\\mathbb{Q}_\\ell)$ the Galois representation on the $\\e
ll$-adic Tate module of $A$. For a place $v$ of $E$ not dividing $\\ell$\,
upon fixing an isomorphism $\\overline{\\mathbb{Q}}_\\ell\\cong \\mathbb{
C}$\, Grothendieck’s $\\ell$-adic monodromy theorem associates to $\\rho
_\\ell$ a $\\mathrm{GL}_{2g}(\\mathbb{C})$-valued Weil-Deligne representat
ion $\\rho_{\\ell\,v}^{WD}$. Then it is known that the conjugacy class of
$\\rho_{\\ell\,v}^{WD}$ is defined over $\\mathbb{Q}$ and independent of $
\\ell.$ When $v$ is a place a good reduction\, this is just the result tha
t the characteristic polynomial of Frobenius is defined over $\\mathbb{Z}$
and independent of $\\ell$.\n\nWe consider a refinement of this result. A
Theorem of Deligne implies that upon replacing $E$ by a finite extension\
, the representations $\\rho_{\\ell\,v}^{WD}$ can be refined to a $G(\\mat
hbb{C})$-valued Weil-Deligne representation $\\rho^{WD\,G}_{\\ell\,v}$\, w
here $G$ is the Mumford--Tate group of $A$. We prove that for $p>2$ and $v
|p$ a place of $E$ where $A$ has semistable reduction\, the conjugacy clas
s of $\\rho^{WD\,G}_{\\ell\,v}$ is defined over $\\mathbb{Q}$ and independ
ent of $\\ell$. This is joint work with Mark Kisin.\n
LOCATION:https://researchseminars.org/talk/LNTS/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jack Thorne (University of Cambridge)
DTSTART;VALUE=DATE-TIME:20221026T150000Z
DTEND;VALUE=DATE-TIME:20221026T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/83
DESCRIPTION:Title: Sy
mmetric power functoriality for Hilbert modular forms\nby Jack Thorne
(University of Cambridge) as part of London number theory seminar\n\nLectu
re held in Huxley 139\, Imperial.\n\nAbstract\nSymmetric power functoriali
ty is one of the basic cases of Langlands' functoriality conjectures and i
s the route to the proof of the Sato-Tate conjecture (concerning the distr
ibution of the modulo $p$ point counts of an elliptic curve over $\\mathbb
{Q}$\, as the prime $p$ varies).\n\nI will discuss the proof of the existe
nce of the symmetric power liftings of Hilbert modular forms of regular we
ight. This is joint work with James Newton.\n
LOCATION:https://researchseminars.org/talk/LNTS/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chung-Hang (Kevin) Kwan (University College London (UCL))
DTSTART;VALUE=DATE-TIME:20221019T150000Z
DTEND;VALUE=DATE-TIME:20221019T160000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/84
DESCRIPTION:Title: Mo
ments and Periods for GL(3)\nby Chung-Hang (Kevin) Kwan (University Co
llege London (UCL)) as part of London number theory seminar\n\nLecture hel
d in Huxley 144\, Imperial.\n\nAbstract\nIn the past century\, moments of
L-functions have been important in number theory and are well-motivated by
a variety of arithmetic applications. In this talk\, we will begin with t
wo elementary counting problems of Diophantine nature as motivation\, foll
owed by a survey of techniques in the past and the present. The main goal
is to demonstrate how period integrals can be used to study moments of aut
omorphic L-functions and uncover the interesting underlying structures (so
me of them can be modeled by random matrix theory).\n
LOCATION:https://researchseminars.org/talk/LNTS/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:George Boxer (Imperial College London)
DTSTART;VALUE=DATE-TIME:20221109T160000Z
DTEND;VALUE=DATE-TIME:20221109T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/85
DESCRIPTION:Title: Hi
gher Hida theory for Siegel modular varieties\nby George Boxer (Imperi
al College London) as part of London number theory seminar\n\nLecture held
in Huxley 139\, Imperial.\n\nAbstract\nThe goal of higher Hida theory is
to study the ordinary part of coherent cohomology of Shimura varieties int
egrally. We introduce a higher coherent cohomological analog of Hida's sp
ace of ordinary p-adic modular forms\, which is defined as the "ordinary p
art" of the coherent cohomology with "partial compact support" of the ordi
nary Igusa variety. Then we give an analog of Hida's classicality theorem
in this setting. This is joint work with Vincent Pilloni.\n
LOCATION:https://researchseminars.org/talk/LNTS/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Holly Green (University College London)
DTSTART;VALUE=DATE-TIME:20230111T160000Z
DTEND;VALUE=DATE-TIME:20230111T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/86
DESCRIPTION:Title: An
arithmetic analogue of the parity conjecture\nby Holly Green (Univers
ity College London) as part of London number theory seminar\n\nLecture hel
d in Rm. 505\, UCL Department of Mathematics (UCL Union Building).\n\nAbst
ract\nI will present a new method to compute the parity of the rank of an
elliptic curve and will comment on how this construction generalises to Ja
cobians of curves. This method involves studying the local arithmetic atta
ched to covers of the curve. In addition\, I will discuss applications to
the Birch and Swinnerton-Dyer conjecture\, including a new proof of the pa
rity conjecture for elliptic curves. This is joint work with Vladimir Dokc
hitser\, Alexandros Konstantinou\, Céline Maistret and Adam Morgan.\n
LOCATION:https://researchseminars.org/talk/LNTS/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akshat Mudgal (University of Oxford)
DTSTART;VALUE=DATE-TIME:20230118T160000Z
DTEND;VALUE=DATE-TIME:20230118T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/87
DESCRIPTION:Title: Ad
ditive equations over lattice points on spheres\nby Akshat Mudgal (Uni
versity of Oxford) as part of London number theory seminar\n\nLecture held
in Rm. 706\, UCL Department of Mathematics (UCL Union Building).\n\nAbstr
act\nIn this talk\, we will consider additive properties of lattice points
on spheres. Thus\, defining $S_m$ to be the set of lattice points on the
sphere $x^2 + y^2 + z^2 + w^2 = m$\, we are interested in counting the num
ber of solutions to the equation\n$$a_1 + a_2 = a_3 + a_4\,$$\nwhere $a_1\
,\\dots\, a_4$ lie in some arbitrary subset $A$ of $S_m$. Such an inquiry
is closely related to various problems in harmonic analysis and analytic n
umber theory\, including Bourgain's discrete restriction conjecture for sp
heres. We will survey some recent results in this direction\, as well as d
escribe some of the various techniques\, arising from areas such as incide
nce geometry\, analytic number theory and arithmetic combinatorics\, that
have been employed to tackle this type of problem.\n
LOCATION:https://researchseminars.org/talk/LNTS/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jens Funke (Durham University)
DTSTART;VALUE=DATE-TIME:20230125T160000Z
DTEND;VALUE=DATE-TIME:20230125T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/88
DESCRIPTION:Title: In
definite theta series via incomplete theta integrals\nby Jens Funke (D
urham University) as part of London number theory seminar\n\nLecture held
in Rm. 706\, UCL Department of Mathematics (UCL Union Building).\n\nAbstra
ct\nPositive definite theta series have been a classical tool in the arith
metic of quadratic forms and also in the theory of modular forms. In compa
rison\, the indefinite case has been less studied. \nIn this talk we will
explain how indefinite theta series naturally arise in the context of symm
etric spaces of orthogonal type and discuss recent developments inspired b
y mathematical physics. This is joint work with Steve Kudla.\n
LOCATION:https://researchseminars.org/talk/LNTS/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Min Lee (University of Bristol)
DTSTART;VALUE=DATE-TIME:20230201T160000Z
DTEND;VALUE=DATE-TIME:20230201T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/89
DESCRIPTION:Title: An
extension of converse theorems to the Selberg class\nby Min Lee (Univ
ersity of Bristol) as part of London number theory seminar\n\nLecture held
in Rm. 706\, UCL Department of Mathematics (UCL Union Building).\n\nAbstr
act\nThe converse theorem for automorphic forms has a long history beginni
ng with the work of Hecke (1936) and a work of Weil (1967): relating the a
utomorphy relations satisfied by classical modular forms to analytic prope
rties of their L-functions and the L-functions twisted by Dirichlet charac
ters. The classical converse theorems were reformulated and generalised in
the setting of automorphic representations for GL(2) by Jacquet and Langl
ands (1970). Since then\, the converse theorem has been a cornerstone of t
he theory of automorphic representations. \n\nVenkatesh (2002)\, in his th
esis\, gave new proof of the classical converse theorem for modular forms
of level 1 in the context of Langlands’ “Beyond Endoscopy”. In this
talk\, we extend Venkatesh’s proof of the converse theorem to forms of a
rbitrary levels and characters with the gamma factors of the Selberg class
type. \n\n\nThis is joint work with Andrew R. Booker and Michael Farmer.\
n
LOCATION:https://researchseminars.org/talk/LNTS/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steven Groen (University of Warwick)
DTSTART;VALUE=DATE-TIME:20230301T160000Z
DTEND;VALUE=DATE-TIME:20230301T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/90
DESCRIPTION:by Steven Groen (University of Warwick) as part of London numb
er theory seminar\n\nLecture held in Rm. 706\, UCL Department of Mathemati
cs (UCL Union Building).\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/LNTS/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarah Arpin (Leiden University)
DTSTART;VALUE=DATE-TIME:20230315T160000Z
DTEND;VALUE=DATE-TIME:20230315T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/91
DESCRIPTION:by Sarah Arpin (Leiden University) as part of London number th
eory seminar\n\nLecture held in Rm. 706\, UCL Department of Mathematics (U
CL Union Building).\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/LNTS/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hung Bui (University of Manchester)
DTSTART;VALUE=DATE-TIME:20230322T160000Z
DTEND;VALUE=DATE-TIME:20230322T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/92
DESCRIPTION:by Hung Bui (University of Manchester) as part of London numbe
r theory seminar\n\nLecture held in Rm. 706\, UCL Department of Mathematic
s (UCL Union Building).\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/LNTS/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lassina Dembélé (King's College London)
DTSTART;VALUE=DATE-TIME:20230222T160000Z
DTEND;VALUE=DATE-TIME:20230222T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/93
DESCRIPTION:by Lassina Dembélé (King's College London) as part of London
number theory seminar\n\nLecture held in Rm. 706\, UCL Department of Math
ematics (UCL Union Building).\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/LNTS/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Tyrell (University of Oxford)
DTSTART;VALUE=DATE-TIME:20230208T160000Z
DTEND;VALUE=DATE-TIME:20230208T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/94
DESCRIPTION:Title: Fu
rther afield and further\, a field: remarks on undecidability\nby Bria
n Tyrell (University of Oxford) as part of London number theory seminar\n\
nLecture held in Rm. 706\, UCL Department of Mathematics (UCL Union Buildi
ng).\n\nAbstract\nGiven a field $K$\, one can ask "what first-order senten
ces are true in $K$"? E.g. for $K = \\mathbb{C}$\, "$\\exists x (x^2 = -1)
$" is true\, but for $K = \\mathbb{Q}$ this is false. One major area of st
udy at the intersection of logic and number theory is\, given a field $K$
of number-theoretic interest\, whether there is an algorithmic process whi
ch can decide the truth or falsity of a given first-order sentence in $K$.
For $K = \\mathbb{C}$\, there exists such an algorithmic process\; for $K
= \\mathbb{Q}$ there cannot (due to work of Gödel & Julia Robinson).\n\n
I will pose a related question: whether the logical consequences of a give
n sentence in a field may be decided algorithmically. Often the answer is
no\; so e.g. we cannot algorithmically detect general properties of fields
$K$ with a Galois extension $L$ such that $\\mathrm{Gal}(L/K) \\cong S_5$
\, or e.g. general properties of characteristic $p$ fields that admit poin
ts on a given rationally parameterisable curve over $\\mathbb{F}_p$. I wil
l focus on those fields whose behaviour is tightly controlled by their abs
olute Galois group\, and prove some precise limitations.\n\nI will aim for
this talk to be self-contained on the logic side of things!\n
LOCATION:https://researchseminars.org/talk/LNTS/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pak-Hin Lee (University of Warwick)
DTSTART;VALUE=DATE-TIME:20230308T160000Z
DTEND;VALUE=DATE-TIME:20230308T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T212940Z
UID:LNTS/95
DESCRIPTION:by Pak-Hin Lee (University of Warwick) as part of London numbe
r theory seminar\n\nLecture held in Rm. 706\, UCL Department of Mathematic
s (UCL Union Building).\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/LNTS/95/
END:VEVENT
END:VCALENDAR