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BEGIN:VEVENT
SUMMARY:Vesselin Dimitrov (University of Toronto)
DTSTART;VALUE=DATE-TIME:20200317T203000Z
DTEND;VALUE=DATE-TIME:20200317T213000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/1
DESCRIPTION:Title: An
arithmetic holonomicity criterion and irrationality of the 2-adic period
$\\zeta_2(5)$\nby Vesselin Dimitrov (University of Toronto) as part of
MIT number theory seminar\n\n\nAbstract\nI will present a new arithmetic
criterion for a formal power\nseries to satisfy a linear ODE on an affine
curve over a global field.\nThis result characterizes the holonomic functi
ons by a sharp positivity\ncondition on a suitably defined arithmetic degr
ee for an adelic set where\na given formal power series is analytic. As an
application\, based on\nCalegari's method with overconvergent p-adic modu
lar forms\, we derive an\nirrationality proof of the Leopoldt-Kubota 2-adi
c zeta value $\\zeta_2(5)$.\nThis is a joint work in progress with Frank C
alegari and Yunqing Tang.\n
LOCATION:https://researchseminars.org/talk/MITNT/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicole Looper (Brown University)
DTSTART;VALUE=DATE-TIME:20200331T203000Z
DTEND;VALUE=DATE-TIME:20200331T213000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/2
DESCRIPTION:Title: Eq
uidistribution techniques in arithmetic dynamics\nby Nicole Looper (Br
own University) as part of MIT number theory seminar\n\n\nAbstract\nThis t
alk is about the arithmetic of points of small canonical height\nrelative
to dynamical systems over number fields\, particularly those\naspects amen
able to the use of equidistribution techniques. Past milestones\nin the su
bject include the proof of the Manin-Mumford Conjecture given by\nSzpiro-U
llmo-Zhang\, and Baker-DeMarco's work on the finiteness of common\npreperi
odic points of rational functions. Recently\, quantitative\nequidistributi
on techniques have emerged both as a way of improving upon\nsome of these
old results\, and as an avenue to studying previously\ninaccessible proble
ms\, such as the Uniform Boundedness Conjecture of Morton\nand Silverman.
I will describe the key ideas behind these developments\, and\nraise relat
ed questions for future research.\n
LOCATION:https://researchseminars.org/talk/MITNT/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Uriya First (University of Haifa)
DTSTART;VALUE=DATE-TIME:20200428T203000Z
DTEND;VALUE=DATE-TIME:20200428T213000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/3
DESCRIPTION:Title: Ge
neration of algebras and versality of torsors\nby Uriya First (Univers
ity of Haifa) as part of MIT number theory seminar\n\n\nAbstract\nThe prim
itive element theorem states that every finite separable field\nextension
L/K is generated by a single element. An almost equally known\nfolklore fa
ct states that every central simple algebra over a field can be\ngenerated
by 2-elements.\n\nI will discuss two recent works with Zinovy Reichstein
(one is forthcoming)\nwhere we establish global analogues of these results
. In more detail\, over\na ring R (or a scheme X)\, separable field extens
ions and central simple\nalgebras globalize to finite etale algebras and A
zumaya algebras\,\nrespectively. We show that if R is of finite type over
an infinite field K\nand has Krull dimension d\, then every finite etale R
-algebra is generated\nby d+1 elements and every Azumaya R-algebra of degr
ee n is generated by\n2+floor(d/[n-1]) elements. The case d=0 recovers the
well-known facts\nstated above. Recent works of B. Williams\, A.K. Shukla
and M. Ojanguren\nshow that these bounds are tight in the etale case and
suggest that they\nshould also be tight in the Azumaya case.\n\nThe proof
makes use of principal homogeneous G-bundles T-->X (G is an\naffine algebr
aic group over K) which can specialize to any principal\nhomogeneous G-bun
dle over an affine K-variety of dimension at most d. In\nparticular\, such
G-bundles exist for all G and d.\n
LOCATION:https://researchseminars.org/talk/MITNT/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jan Vonk (Institute for Advanced Study)
DTSTART;VALUE=DATE-TIME:20200505T203000Z
DTEND;VALUE=DATE-TIME:20200505T213000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/4
DESCRIPTION:Title: Si
ngular moduli for real quadratic fields\nby Jan Vonk (Institute for Ad
vanced Study) as part of MIT number theory seminar\n\n\nAbstract\nIn the e
arly 20th century\, Hecke studied the diagonal restrictions of Eisenstein
series over real quadratic fields. An infamous sign error caused him to mi
ss an important feature\, which later lead to highly influential developme
nts in the theory of complex multiplication (CM) initiated by Gross and Za
gier in their famous work on Heegner points on elliptic curves. In this ta
lk\, we will explore what happens when we replace the imaginary quadratic
fields in CM theory with real quadratic fields\, and propose a framework f
or a tentative 'RM theory'\, based on the notion of rigid meromorphic cocy
cles\, introduced in joint work with Henri Darmon. I will discuss several
of their arithmetic properties\, and their apparent relevance in the study
of explicit class field theory of real quadratic fields\, the constructio
n of rational points on elliptic curves\, and the theory of Borcherds lift
s. This concerns various joint works\, with Henri Darmon\, Alice Pozzi\, a
nd Yingkun Li.\n
LOCATION:https://researchseminars.org/talk/MITNT/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jessica Fintzen (Cambridge/Duke/IAS)
DTSTART;VALUE=DATE-TIME:20200908T143000Z
DTEND;VALUE=DATE-TIME:20200908T153000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/6
DESCRIPTION:Title: Re
presentations of p-adic groups and applications\nby Jessica Fintzen (C
ambridge/Duke/IAS) as part of MIT number theory seminar\n\n\nAbstract\nThe
Langlands program is a far-reaching collection of conjectures that relate
different areas of mathematics including number theory and representation
theory. A fundamental problem on the representation theory side of the La
nglands program is the construction of all (irreducible\, smooth\, complex
) representations of p-adic groups. \n\nI will provide an overview of our
understanding of the representations of p-adic groups\, with an emphasis o
n recent progress. \n\nI will also outline how new results about the repre
sentation theory of p-adic groups can be used to obtain congruences betwee
n arbitrary automorphic forms and automorphic forms which are supercuspida
l at p\, which is joint work with Sug Woo Shin. This simplifies earlier co
nstructions of attaching Galois representations to automorphic representat
ions\, i.e. the global Langlands correspondence\, for general linear group
s. Moreover\, our results apply to general p-adic groups and have therefor
e the potential to become widely applicable beyond the case of the general
linear group.\n\nNote the this talk will take place at 10:30 rather than
16:30 (Eastern time).\n
LOCATION:https://researchseminars.org/talk/MITNT/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Lawrence (University of Chicago)
DTSTART;VALUE=DATE-TIME:20200915T203000Z
DTEND;VALUE=DATE-TIME:20200915T213000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/7
DESCRIPTION:Title: Th
e Shafarevich conjecture for hypersurfaces in abelian varieties\nby Br
ian Lawrence (University of Chicago) as part of MIT number theory seminar\
n\n\nAbstract\nLet K be a number field\, S a finite set of primes of O_K\,
and g a positive integer. Shafarevich conjectured\, and Faltings proved\
, that there are only finitely many curves of genus g\, defined over K and
having good reduction outside S. Analogous results have been proven for
other families\, replacing "curves of genus g" with "K3 surfaces"\, "del P
ezzo surfaces" etc.\; these results are also called Shafarevich conjecture
s. There are good reasons to expect the Shafarevich conjecture to hold fo
r many families of varieties: the moduli space should have only finitely m
any integral points.\n\nWill Sawin and I prove this for hypersurfaces in a
belian varieties of dimension not equal to 3.\n
LOCATION:https://researchseminars.org/talk/MITNT/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shou-Wu Zhang (Princeton University)
DTSTART;VALUE=DATE-TIME:20200922T203000Z
DTEND;VALUE=DATE-TIME:20200922T213000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/8
DESCRIPTION:Title: De
composition theorems for arithmetic cycles\nby Shou-Wu Zhang (Princeto
n University) as part of MIT number theory seminar\n\n\nAbstract\nWe will
describe some decomposition theorems for cycles over polarized variet
ies in both local and global settings under some conjectures of Lefsch
etz type. In local settings\, our decomposition theorems are essentially
non-archimedean analogues of ``harmonic forms" on Kahler manifolds. As
an application\, we will define a notion of ``admissible pairings" of
algebraic cycles which is a simultaneous generalization of Beilinson--Bl
och height pairing\, and the local intersection pairings \ndeveloped by
Arakelov\, Faltings\, and Gillet--Soule on Kahler manifolds. In glob
al setting\,\nour decomposition theorems provide canonical splittings of
some canonical filtrations\, including canonical liftings of homological
cycles to algebraic cycles.\n
LOCATION:https://researchseminars.org/talk/MITNT/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brendan Creutz (University of Canterbury)
DTSTART;VALUE=DATE-TIME:20200929T203000Z
DTEND;VALUE=DATE-TIME:20200929T213000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/9
DESCRIPTION:Title: Qu
adratic points on del Pezzo surfaces of degree 4\nby Brendan Creutz (U
niversity of Canterbury) as part of MIT number theory seminar\n\n\nAbstrac
t\nI will report on joint work (in progress) with Bianca Viray concerning
the following question. If $X/k$ is a smooth complete intersection of $2$
quadrics in $\\mathbb{P}^n$ over a field $k$\, does $X$ have a rational po
int over some quadratic extension of $k$?\n
LOCATION:https://researchseminars.org/talk/MITNT/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Salim Tayou (Harvard)
DTSTART;VALUE=DATE-TIME:20201006T203000Z
DTEND;VALUE=DATE-TIME:20201006T213000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/10
DESCRIPTION:Title: E
xceptional jumps of Picard rank of K3 surfaces over number fields\nby
Salim Tayou (Harvard) as part of MIT number theory seminar\n\n\nAbstract\n
Given a K3 surface X over a number field K\, we prove that the set of prim
es of K where the geometric Picard rank jumps is infinite\, assuming that
X has everywhere potentially good reduction. This result is formulated in
the general framework of GSpin Shimura varieties and I will explain other
applications to abelian surfaces. I will also discuss applications to the
existence of rational curves on K3 surfaces. The results in this talk are
joint work with Ananth Shankar\, Arul Shankar and Yunqing Tang.\n
LOCATION:https://researchseminars.org/talk/MITNT/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesc Castella (UC Santa Barbara)
DTSTART;VALUE=DATE-TIME:20201020T203000Z
DTEND;VALUE=DATE-TIME:20201020T213000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/11
DESCRIPTION:Title: I
wasawa theory of elliptic curves at Eisenstein primes and applications
\nby Francesc Castella (UC Santa Barbara) as part of MIT number theory sem
inar\n\n\nAbstract\nIn the study of Iwasawa theory of elliptic curves $E/\
\mathbb{Q}$\, it is often assumed that $p$ is a non-Eisenstein prime\, mea
ning that $E[p]$ is irreducible as a $G_{\\mathbb{Q}}$-module. Because of
this\, most of the recent results on the $p$-converse to the theorem of Gr
oss–Zagier and Kolyvagin (following Skinner and Wei Zhang) and on the $p
$-part of the Birch–Swinnerton-Dyer formula in analytic rank 1 (followin
g Jetchev–Skinner–Wan) were only known for non-Eisenstein primes $p$.
In this talk\, I’ll explain some of the ingredients in a joint work with
Giada Grossi\, Jaehoon Lee\, and Christopher Skinner in which we study th
e (anticyclotomic) Iwasawa theory of elliptic curves over $\\mathbb{Q}$ at
Eisenstein primes. As a consequence of our study\, we obtain an extension
of the aforementioned results to the Eisenstein case. In particular\, for
$p=3$ this leads to an improvement on the best known results towards Gold
feld’s conjecture in the case of elliptic curves over $\\mathbb{Q}$ with
a rational $3$-isogeny.\n
LOCATION:https://researchseminars.org/talk/MITNT/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Winnie Li (Pennsylvania State University)
DTSTART;VALUE=DATE-TIME:20201027T203000Z
DTEND;VALUE=DATE-TIME:20201027T213000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/12
DESCRIPTION:Title: P
air arithmetical equivalence for quadratic fields\nby Winnie Li (Penns
ylvania State University) as part of MIT number theory seminar\n\n\nAbstra
ct\nGiven two distinct number fields $K$ and $M$\, and two finite order He
cke characters $\\chi$ of $K$ and $\\eta$ of $M$ respectively\, we say tha
t the pairs $(\\chi\, K)$ and $(\\eta\, M)$ are arithmetically equivalent
if the associated L-functions coincide: $L(s\, \\chi\, K) = L(s\, \\eta\,
M)$. When the characters are trivial\, this reduces to the question of fie
lds with the same Dedekind zeta function\, investigated by Gassmann in 192
6\, who found such fields of degree 180\, and by Perlis in 1977 and others
\, who showed that there are no nonisomorphic fields of degree less than 7
.\n\nIn this talk we discuss arithmetically equivalent pairs where the fie
lds are quadratic. They give rise to dihedral automorphic forms induced fr
om characters of different quadratic fields. We characterize when a given
pair is arithmetically equivalent to another pair\, explicitly construct s
uch pairs for infinitely many quadratic extensions with odd class number\,
and classify such characters of order 2.\n\nThis is a joint work with Zee
v Rudnick.\n
LOCATION:https://researchseminars.org/talk/MITNT/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ana Caraiani (Imperial College London)
DTSTART;VALUE=DATE-TIME:20201103T153000Z
DTEND;VALUE=DATE-TIME:20201103T163000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/13
DESCRIPTION:Title: V
anishing theorems for Shimura varieties\nby Ana Caraiani (Imperial Col
lege London) as part of MIT number theory seminar\n\n\nAbstract\nThe Langl
ands program is a vast network of conjectures that connect number theory t
o other areas of mathematics\, such as representation theory and harmonic
analysis. The global Langlands correspondence can often be realised throug
h the cohomology of Shimura varieties\, which are certain moduli spaces eq
uipped with many symmetries. In this talk\, I will survey some recent vani
shing results for the cohomology of Shimura varieties with mod $p$ coeffic
ients and mention several applications to the Langlands program and beyond
. I will discuss some results that have an $\\ell$-adic flavour\, where
$\\ell$ is a prime different from $p$\, that are primarily joint work wit
h Peter Scholze. I will then mention some results that have a $p$-adic
flavour\, that are primarily joint work with Dan Gulotta and Christian Joh
ansson. I will highlight the different kinds of techniques that are needed
in these different settings using the toy model of the modular curve.\n\n
There are two papers that contain work related to this talk: arXiv:1909.01898 and arXiv:1910.0914.\n
LOCATION:https://researchseminars.org/talk/MITNT/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cong Xue (CNRS and IMJ-PRG)
DTSTART;VALUE=DATE-TIME:20201110T153000Z
DTEND;VALUE=DATE-TIME:20201110T163000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/14
DESCRIPTION:Title: S
moothness of the cohomology sheaves of stacks of shtukas\nby Cong Xue
(CNRS and IMJ-PRG) as part of MIT number theory seminar\n\n\nAbstract\nLet
$X$ be a smooth projective geometrically connected curve over a finite fi
eld $\\mathbb{F}_q$. Let $G$ be a connected reductive group over the funct
ion field of $X$. For every finite set $I$ and every representation of $(\
\check{G})^I$\, where $\\check{G}$ is the Langlands dual group of $G$\, we
have a stack of shtukas over $X^I$. For every degree\, we have a compact
support $\\ell$-adic cohomology sheaf over $X^I$.\n\nIn this talk\, I will
recall some properties of these sheaves. I will talk about a work in prog
ress which proves that these sheaves are ind-smooth over $X^I$.\n
LOCATION:https://researchseminars.org/talk/MITNT/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiuya Wang (Duke University)
DTSTART;VALUE=DATE-TIME:20201117T213000Z
DTEND;VALUE=DATE-TIME:20201117T223000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/15
DESCRIPTION:Title: A
verage of $3$-torsion in class groups of $2$-extensions\nby Jiuya Wang
(Duke University) as part of MIT number theory seminar\n\n\nAbstract\nIn
1971\, Davenport and Heilbronn proved the celebrated theorem determining t
he average of $3$-torsion in class groups of quadratic extensions. In this
talk\, we will study the average of $3$-torsion in class groups of $2$-ex
tensions\, which are towers of relative quadratic extensions. As an exampl
e\, we determine the average of $3$-torsion in class groups of $D_4$ quart
ic extensions. This is a joint work with Robert J. Lemke Oliver and Melani
e Matchett Wood.\n
LOCATION:https://researchseminars.org/talk/MITNT/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yiannis Sakellaridis (Johns Hopkins University)
DTSTART;VALUE=DATE-TIME:20210216T213000Z
DTEND;VALUE=DATE-TIME:20210216T223000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/16
DESCRIPTION:Title: P
eriods\, L-functions\, and duality of Hamiltonian spaces\nby Yiannis S
akellaridis (Johns Hopkins University) as part of MIT number theory semina
r\n\n\nAbstract\nThe relationship between periods of automorphic forms and
L-functions has been studied since the times of Riemann\, but remains mys
terious. In this talk\, I will explain how periods and L-functions arise a
s quantizations of certain Hamiltonian spaces\, and will propose a conject
ural duality between certain Hamiltonian spaces for a group $G$\, and its
Langlands dual group $\\check G$\, in the context of the geometric Langlan
ds program\, recovering known and conjectural instances of the aforementio
ned relationship. This is joint work with David Ben-Zvi and Akshay Venkate
sh.\n
LOCATION:https://researchseminars.org/talk/MITNT/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lola Thompson (Utrecht University)
DTSTART;VALUE=DATE-TIME:20201208T213000Z
DTEND;VALUE=DATE-TIME:20201208T223000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/17
DESCRIPTION:Title: S
umming $\\mu(n)$: an even faster elementary algorithm\nby Lola Thompso
n (Utrecht University) as part of MIT number theory seminar\n\n\nAbstract\
nWe present a new-and-improved elementary algorithm for computing $M(x) =
\\sum_{n \\leq x} \\mu(n)\,$ where $\\mu(n)$ is the Moebius function. Our
algorithm takes time $O\\left(x^{\\frac{3}{5}} \\log \\log x \\right)$ and
space $O\\left(x^{\\frac{3}{10}} \\log x \\right)$\, which improves on e
xisting combinatorial algorithms. While there is an analytic algorithm due
to Lagarias-Odlyzko with computations based\non integrals of $\\zeta(s)$
that only takes time $O(x^{1/2 + \\epsilon})$\, our algorithm has the adva
ntage of being easier to implement. The new approach roughly amounts to an
alyzing the difference between a model that we obtain via Diophantine appr
oximation and reality\, and showing that it has a simple description in te
rms of congruence classes and segments. This simple description allows us
to compute the difference quickly by means of a table lookup. This talk is
based on joint work with Harald Andres Helfgott.\n
LOCATION:https://researchseminars.org/talk/MITNT/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lillian Pierce (Duke University)
DTSTART;VALUE=DATE-TIME:20201215T213000Z
DTEND;VALUE=DATE-TIME:20201215T223000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/18
DESCRIPTION:Title: O
n superorthogonality\nby Lillian Pierce (Duke University) as part of M
IT number theory seminar\n\n\nAbstract\nThe Burgess bound is a well-known
upper bound for short multiplicative character sums\, which implies for ex
ample a subconvexity bound for Dirichlet L-functions. Since the 1950's\, p
eople have tried to improve the Burgess method. In order to try to improve
a method\, it makes sense to understand the bigger “proofscape” in wh
ich a method fits. The Burgess method didn’t seem to fit well into a big
ger proofscape. In this talk we will show that in fact it can be regarded
as an application of “superorthogonality.” This perspective links topi
cs from harmonic analysis and number theory\, such as Khintchine’s inequ
ality\, Walsh-Paley series\, square function estimates and decoupling\, mu
lti-correlation sums of trace functions\, and the Burgess method. We will
survey these connections in an accessible way\, with a focus on the number
theoretic side.\n
LOCATION:https://researchseminars.org/talk/MITNT/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lue Pan (University of Chicago)
DTSTART;VALUE=DATE-TIME:20201124T213000Z
DTEND;VALUE=DATE-TIME:20201124T223000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/19
DESCRIPTION:Title: O
n the locally analytic vectors of the completed cohomology of modular curv
es\nby Lue Pan (University of Chicago) as part of MIT number theory se
minar\n\n\nAbstract\nA classical result identifies holomorphic modular for
ms with\nhighest weight vectors of certain representations of $SL_2(\\math
bb{R})$. We\nstudy locally analytic vectors of the (p-adically) completed
cohomology of\nmodular curves and prove a p-adic analogue of this result.
As\napplications\, we are able to prove a classicality result for\novercon
vergent eigenforms and give a new proof of Fontaine-Mazur\nconjecture in t
he irregular case under some mild hypothesis. One technical\ntool is relat
ive Sen theory which allows us to relate infinitesimal group\naction with
Hodge(-Tate) structure.\n
LOCATION:https://researchseminars.org/talk/MITNT/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Nelson (ETH Zurich)
DTSTART;VALUE=DATE-TIME:20210223T153000Z
DTEND;VALUE=DATE-TIME:20210223T163000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/20
DESCRIPTION:Title: T
he orbit method\, microlocal analysis and applications to L-functions\
nby Paul Nelson (ETH Zurich) as part of MIT number theory seminar\n\n\nAbs
tract\nI will describe how the orbit method can be developed in a quantita
tive form\, along the lines of microlocal analysis\, and applied to local
problems in representation theory and global problems concerning automorph
ic forms. The local applications include asymptotic expansions of relativ
e characters. The global applications include moment estimates and subcon
vex bounds for L-functions. These results are the subject of two papers\,
the first joint with Akshay Venkatesh:\n\nhttps://arxiv.org/abs/1805.0775
0\n\nhttps://arxiv.org/abs/2012.02187\n
LOCATION:https://researchseminars.org/talk/MITNT/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Huajie Li (Aix-Marseille Université)
DTSTART;VALUE=DATE-TIME:20210302T153000Z
DTEND;VALUE=DATE-TIME:20210302T163000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/21
DESCRIPTION:Title: A
n infinitesimal variant of Guo-Jacquet trace formulae and its comparison\nby Huajie Li (Aix-Marseille Université) as part of MIT number theory
seminar\n\n\nAbstract\nThe Guo-Jacquet conjecture is a promising generaliz
ation to higher dimensions of Waldspurger’s well-known theorem relating
toric periods to central values of automorphic L-functions for $GL(2)$. Fe
igon-Martin-Whitehouse have proved some cases of this conjecture using sim
ple relative trace formulae\, Guo’s work on the fundamental lemma and C.
Zhang’s work on the transfer. However\, if we want to obtain more gener
al results\, we have to establish and compare more general relative trace
formulae\, where some analytic difficulties such as the divergence issue s
hould be addressed. \n\nIn this talk\, we plan to study analogues of these
problems at the infinitesimal level. After briefly introducing the backgr
ound\, we shall present an infinitesimal variant of Guo-Jacquet trace form
ulae. To compare regular semi-simple terms in these formulae\, we shall di
scuss the weighted fundamental lemma and certain identities between Fourie
r transforms of local weighted orbital integrals. During the proof\, we al
so need some results in local harmonic analysis such as local trace formul
ae for some $p$-adic infinitesimal symmetric spaces. This talk is based on
my thesis supervised by P.-H. Chaudouard.\n
LOCATION:https://researchseminars.org/talk/MITNT/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ian Gleason (UC Berkeley)
DTSTART;VALUE=DATE-TIME:20210309T213000Z
DTEND;VALUE=DATE-TIME:20210309T223000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/22
DESCRIPTION:Title: O
n the geometric connected components of moduli of p-adic shtukas.\nby
Ian Gleason (UC Berkeley) as part of MIT number theory seminar\n\n\nAbstra
ct\nThrough the recent theory of diamonds\, P. Scholze constructs local Sh
imura varieties and moduli of p-adic shtukas attached to any reductive gro
up. These are diamonds that generalize the generic fiber of a Rapoport–Z
ink space. These interesting spaces realize in their cohomology instances
of the local Langlands correspondence. In this talk\, we describe the set
of connected components of moduli spaces of p-adic shtukas (with one paw).
The new ingredient of this work is the use of specialization maps in the
context of diamonds.\n
LOCATION:https://researchseminars.org/talk/MITNT/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarah Zerbes (University College London)
DTSTART;VALUE=DATE-TIME:20210316T143000Z
DTEND;VALUE=DATE-TIME:20210316T153000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/23
DESCRIPTION:Title: E
uler systems and explicit reciprocity laws for GSp(4)\nby Sarah Zerbes
(University College London) as part of MIT number theory seminar\n\n\nAbs
tract\nEuler systems are a very powerful tool for attacking the Bloch—Ka
to conjecture\, which is one of the central open problems in number theory
. In this talk\, I will sketch the construction of an Euler system for the
spin Galois representation of a genus 2 Siegel modular form. I will then
explain how to prove an explicit reciprocity law\, relating the image of t
he Euler system under the Bloch—Kato logarithm map to values of the comp
lex L-function of the Siegel modular form. The applications of this result
to the Bloch—Kato conjecture and the Iwasawa Main Conjecture will be di
scussed by David Loeffler in the following week.\n
LOCATION:https://researchseminars.org/talk/MITNT/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Loeffler (University of Warwick)
DTSTART;VALUE=DATE-TIME:20210323T143000Z
DTEND;VALUE=DATE-TIME:20210323T153000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/24
DESCRIPTION:Title: T
he Bloch--Kato conjecture for critical values of GSp(4) L-functions\nb
y David Loeffler (University of Warwick) as part of MIT number theory semi
nar\n\n\nAbstract\nIn Sarah's talk last week\, she explained the construct
ion of a family of Galois cohomology \nclasses (an Euler system) attached
to Siegel modular forms\, and related the localisations of these classes a
t p to non-critical values of p-adic L-functions. In this talk\, I will ex
plain how to 'analytically continue' this relation to obtain an explicit r
eciprocity law relating Galois cohomology classes to critical values of L-
functions\; and I will discuss applications of this result to the Bloch--K
ato conjecture.\n
LOCATION:https://researchseminars.org/talk/MITNT/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Teppei Takamatsu (University of Tokyo)
DTSTART;VALUE=DATE-TIME:20210330T203000Z
DTEND;VALUE=DATE-TIME:20210330T213000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/25
DESCRIPTION:Title: M
inimal model program for semi-stable threefolds in mixed characteristic\nby Teppei Takamatsu (University of Tokyo) as part of MIT number theory
seminar\n\n\nAbstract\nThe minimal model program\, which is a theory to co
nstruct a birational model of a variety which is as simple as possible\, i
s a very strong method in algebraic geometry.\nThe minimal model program i
s also studied for more general schemes not necessarily defined over a f
ield\, and play an important role in studies of reductions of varieties.\n
Kawamata showed that the minimal model program holds for strictly semi-sta
ble schemes over an excellent Dedekind scheme of relative dimension two
whose each residue characteristic is neither 2 nor 3.\nIn this talk\, I w
ill introduce a generalization of the result of Kawamata without any assum
ption on the residue characteristic.\nThis talk is based on a joint work w
ith Shou Yoshikawa.\n
LOCATION:https://researchseminars.org/talk/MITNT/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kaisa Matomäki (University of Turku)
DTSTART;VALUE=DATE-TIME:20210406T143000Z
DTEND;VALUE=DATE-TIME:20210406T153000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/26
DESCRIPTION:Title: A
lmost primes in almost all very short intervals\nby Kaisa Matomäki (U
niversity of Turku) as part of MIT number theory seminar\n\n\nAbstract\nBy
probabilistic models one expects that\, as soon as $h \\to \\infty$ with
$X \\to \\infty$\, short intervals of the type $(x- h \\log X\, x]$ contai
n primes for almost all $x \\in (X/2\, X]$. However\, this is far from bei
ng established. In the talk I discuss related questions and in particular
describe how to prove the above claim when one is satisfied with finding $
P_2$-numbers (numbers that have at most two prime factors) instead of prim
es.\n
LOCATION:https://researchseminars.org/talk/MITNT/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pol van Hoften (King's College London)
DTSTART;VALUE=DATE-TIME:20210413T143000Z
DTEND;VALUE=DATE-TIME:20210413T153000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/27
DESCRIPTION:Title: M
od $p$ points on Shimura varieties of parahoric level\nby Pol van Hoft
en (King's College London) as part of MIT number theory seminar\n\n\nAbstr
act\nThe conjecture of Langlands-Rapoport gives a conjectural description
of the mod $p$ points of Shimura varieties\, with applications towards com
puting the (semi-simple) zeta function of these Shimura varieties. The con
jecture was proven by Kisin for abelian type Shimura varieties at primes o
f (hyperspecial) good reduction\, after having constructed smooth integral
models. For primes of (parahoric) bad reduction\, Kisin and Pappas have c
onstructed a good integral model and the conjecture was generalised to thi
s setting by Rapoport. In this talk I will discuss recent results towards
the conjecture for these integral models\, under minor hypothesis\, buildi
ng on earlier work of Zhou. Along the way we will see irreducibility resul
ts for various stratifications on special fibers of Shimura varieties\, in
cluding irreducibility of central leaves and Ekedahl-Oort strata.\n
LOCATION:https://researchseminars.org/talk/MITNT/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ila Varma (University of Toronto)
DTSTART;VALUE=DATE-TIME:20210427T203000Z
DTEND;VALUE=DATE-TIME:20210427T213000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/28
DESCRIPTION:Title: M
alle's conjecture for octic $D_4$-fields.\nby Ila Varma (University of
Toronto) as part of MIT number theory seminar\n\n\nAbstract\nWe consider
the family of normal octic fields with Galois group $D_4$\, ordered by the
ir discriminant. In forthcoming joint work with Arul Shankar\, we verify t
he strong form of Malle's conjecture for this family of number fields\, ob
taining the order 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 results.\n
LOCATION:https://researchseminars.org/talk/MITNT/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eva Bayer-Fluckiger (EPFL)
DTSTART;VALUE=DATE-TIME:20210504T143000Z
DTEND;VALUE=DATE-TIME:20210504T153000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/29
DESCRIPTION:Title: I
sometries of lattices and Hasse principle\nby Eva Bayer-Fluckiger (EPF
L) as part of MIT number theory seminar\n\n\nAbstract\nWe give necessary a
nd sufficient conditions for an integral polynomial to be the characterist
ic polynomial of an isometry of some even\, unimodular lattice of given si
gnature.\n\nRelated papers: arX
iv:2001.07094\, arXiv:2107.
07583.\n
LOCATION:https://researchseminars.org/talk/MITNT/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eugen Hellmann (Mathematisches Institut Münster)
DTSTART;VALUE=DATE-TIME:20210511T143000Z
DTEND;VALUE=DATE-TIME:20210511T153000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/30
DESCRIPTION:Title: O
n classicality of overconvergent $p$-adic automorphic forms\nby Eugen
Hellmann (Mathematisches Institut Münster) as part of MIT number theory s
eminar\n\n\nAbstract\nI will report on some positive and a negative result
concerning the question whether a given overconvergent $p$-adic eigenform
of finite slope is classical or not. \nThe positive result is the general
ization of a classicality statement (obtained in earlier joint work with B
reuil and Schraen) to the case of semi-stable Galois representations. This
classicality result is rather a statement about the Galois representation
attached to a $p$-adic automorphic form than a statement about the $p$-ad
ic automorphic form itself. The negative result concerns the classicality
problem for the $p$-adic automorphic form itself. If time permits we will
discuss some conjectural picture explaining this negative result.\n
LOCATION:https://researchseminars.org/talk/MITNT/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shiva Chidambaram (MIT)
DTSTART;VALUE=DATE-TIME:20210921T203000Z
DTEND;VALUE=DATE-TIME:20210921T213000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/31
DESCRIPTION:Title: A
belian Varieties with given $p$-torsion representations\nby Shiva Chid
ambaram (MIT) as part of MIT number theory seminar\n\nLecture held in Room
2-143 in the Simons building (building 2).\n\nAbstract\nThe Siegel modula
r variety $\\mathcal{A}_2(3)$\, which parametrizes abelian surfaces with f
ull level $3$ structure\, was shown to be rational over $\\Q$ by Bruin and
Nasserden. What can we say about its twist $\\mathcal{A}_2(\\rho)$\, para
metrizing abelian surfaces $A$ with $\\rho_{A\,3} \\simeq \\rho$\, for a g
iven mod $3$ Galois representation $\\rho : G_{\\Q} \\rightarrow \\GSp(4\,
\\F_3)$? While it is not rational in general\, it is unirational over $\\
Q$ by a map of degree at most $6$\, if $\\rho$ satisfies the necessary con
dition of having cyclotomic similitude. In joint work with Frank Calegari
and David Roberts\, we obtain an explicit description of the universal obj
ect over a degree $6$ cover of $\\mathcal{A}_2(\\rho)$\, using invariant t
heoretic ideas. One application of this result is towards an explicit tran
sfer of modularity\, yielding infinitely many examples of modular abelian
surfaces with no extra endomorphisms. Similar ideas work in a few other ca
ses\, showing in particular that whenever $(g\,p) = (1\,2)$\, $(1\,3)$\, $
(1\,5)$\, $(2\,2)$\, $(2\,3)$ and $(3\,2)$\, the cyclotomic similitude con
dition is also sufficient for a mod $p$ Galois representation to arise fro
m the $p$-torsion of a $g$-dimensional abelian variety. When $(g\,p)$ is n
ot one of these six tuples\, we will discuss a local obstruction for repre
sentations to arise as torsion.\n
LOCATION:https://researchseminars.org/talk/MITNT/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bianca Viray (University of Washington)
DTSTART;VALUE=DATE-TIME:20210928T203000Z
DTEND;VALUE=DATE-TIME:20210928T213000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/32
DESCRIPTION:Title: Q
uadratic points on intersections of quadrics\nby Bianca Viray (Univers
ity of Washington) as part of MIT number theory seminar\n\nLecture held in
Room 2-143 in the Simons building (building 2).\n\nAbstract\nA projective
degree $d$ variety always has a point defined over a degree $d$ field ext
ension. For many degree $d$ varieties\, this is the best possible stateme
nt\, that is\, there exist classes of degree $d$ varieties that never have
points over extensions of degree less than $d$ (nor even over extensions
whose degree is nonzero modulo $d$). However\, there are some classes of
degree $d$ varieties that obtain points over extensions of smaller degree\
, for example\, degree $9$ surfaces in $\\mathbb{P}^9$\, and $6$-dimension
al intersections of quadrics over local fields. In this talk\, we explore
this question for intersections of quadrics. In particular\, we prove th
at a smooth complete intersection of two quadrics of dimension at least $2
$ over a number field has index dividing $2$\, i.e.\, that it possesses a
rational $0$-cycle of degree $2$. This is joint work with Brendan Creutz.
\n
LOCATION:https://researchseminars.org/talk/MITNT/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean Kieffer (Harvard)
DTSTART;VALUE=DATE-TIME:20211005T203000Z
DTEND;VALUE=DATE-TIME:20211005T213000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/33
DESCRIPTION:Title: H
igher-dimensional modular equations and point counting on abelian surfaces
\nby Jean Kieffer (Harvard) as part of MIT number theory seminar\n\nLe
cture held in Room 2-143 in the Simons building (building 2).\n\nAbstract\
nGiven a prime number $\\ell$\, the elliptic modular polynomial of level\n
$\\ell$ is an explicit equation for the locus of elliptic curves\nrelated
by an $\\ell$-isogeny. These polynomials have a large number of\nalgorithm
ic applications: in particular\, they are an essential\ningredient in the
celebrated SEA algorithm for counting points on\nelliptic curves over fini
te fields of large characteristic.\n\nMore generally\, modular equations d
escribe the locus of isogenous\nabelian varieties in certain moduli spaces
called PEL Shimura\nvarieties. We will present upper bounds on the size o
f modular\nequations in terms of their level\, and outline a general strat
egy to\ncompute an isogeny $A\\to A'$ given a point $(A\,A')$ where the mo
dular\nequations are satisfied. This generalizes well-known properties of\
nelliptic modular polynomials to higher dimensions.\n\nThe isogeny algorit
hm is made fully explicit for Jacobians of genus 2\ncurves. In combination
with efficient evaluation methods for modular\nequations in genus 2 via c
omplex approximations\, this gives rise to\npoint counting algorithms for
(Jacobians of) genus 2 curves whose\nasymptotic costs\, under standard heu
ristics\, improve on previous\nresults.\n
LOCATION:https://researchseminars.org/talk/MITNT/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Kisin (Harvard)
DTSTART;VALUE=DATE-TIME:20211014T190000Z
DTEND;VALUE=DATE-TIME:20211014T200000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/34
DESCRIPTION:Title: E
ssential dimension via prismatic cohomology\nby Mark Kisin (Harvard) a
s part of MIT number theory seminar\n\nLecture held in Room 2-449 in the S
imons building.\n\nAbstract\nLet $f:Y \\rightarrow X$ be a finite covering
map of complex algebraic varieties. The essential dimension of $f$ is the
smallest integer $e$ such that\, birationally\, $f$ arises as the pullbac
k \nof a covering $Y' \\rightarrow X'$ of dimension $e\,$ via a map $X \\r
ightarrow X'.$ This invariant goes back to classical questions about reduc
ing the number of parameters in a solution to a general $n^{\\rm th}$ degr
ee polynomial\, and appeared in work of Kronecker and Klein on solutions o
f the quintic. \n\nI will report on joint work with Benson Farb and Jesse
Wolfson\, where we introduce a new technique\, using prismatic cohomology\
, to obtain lower bounds on the essential dimension of certain coverings.
For example\, we show that for an abelian variety $A$ of dimension $g$ the
multiplication by $p$ map $A \\rightarrow A$ has essential dimension $g$
for almost all primes $p.$\n\nNote the unusual time and place: Thursday at
3pm in 2-449.\n
LOCATION:https://researchseminars.org/talk/MITNT/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Myrto Mavraki (Harvard)
DTSTART;VALUE=DATE-TIME:20211026T203000Z
DTEND;VALUE=DATE-TIME:20211026T213000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/36
DESCRIPTION:Title: O
n the dynamical Bogomolov conjecture\nby Myrto Mavraki (Harvard) as pa
rt of MIT number theory seminar\n\nLecture held in Room 2-143 in the Simon
s building.\n\nAbstract\nMotivated by the Manin-Mumford conjecture\, estab
lished by Raynaud\, and following the analogy of torsion with preperiodic
points\, Zhang posed a dynamical Manin-Mumford conjecture. Using a canonic
al height introduced by Call and Silverman he further formulated a dynamic
al Bogomolov conjecture. A special case of these conjectures has recently
been established by Nguyen\, Ghioca and Ye. In particular\, they show that
two rational maps have at most finitely many common preperiodic points\,
unless they are 'related'. In this talk we discuss relative and uniform ve
rsions of such results. This is joint work with Harry Schmidt.\n
LOCATION:https://researchseminars.org/talk/MITNT/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Landesman (Harvard)
DTSTART;VALUE=DATE-TIME:20211102T203000Z
DTEND;VALUE=DATE-TIME:20211102T213000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/37
DESCRIPTION:Title: T
he geometric distribution of Selmer groups of Elliptic curves over functio
n fields\nby Aaron Landesman (Harvard) as part of MIT number theory se
minar\n\nLecture held in Room 2-143 in the Simons building.\n\nAbstract\nB
hargava\, Kane\, Lenstra\, Poonen\, and Rains proposed heuristics for the
distribution of arithmetic data relating to elliptic curves\, such as thei
r ranks\, Selmer groups\, and Tate-Shafarevich groups.\nAs a special case
of their heuristics\, they obtain the minimalist conjecture\, which predic
ts that $50\\%$ of elliptic curves have rank $0$ and $50\\%$ of elliptic c
urves have rank $1$. \nAfter surveying these conjectures\, we will explain
joint work with Tony Feng and Eric Rains\, \nverifying a variant of these
conjectures over function fields of the form $\\mathbb F_q(t)$\, after ta
king a certain large $q$ limit.\n
LOCATION:https://researchseminars.org/talk/MITNT/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Shusterman (Harvard)
DTSTART;VALUE=DATE-TIME:20211109T213000Z
DTEND;VALUE=DATE-TIME:20211109T223000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/38
DESCRIPTION:Title: F
initely Presented Groups in Arithmetic Geometry\nby Mark Shusterman (H
arvard) as part of MIT number theory seminar\n\nLecture held in Room 2-143
in the Simons building.\n\nAbstract\nWe discuss the problem of determinin
g the number of generators and relations of several profinite groups of ar
ithmetic and geometric origin. \nThese include etale fundamental groups of
smooth projective varieties\, absolute Galois groups of local fields\, an
d Galois groups of maximal unramified extensions of number fields. The res
ults are based on a cohomological presentability criterion of Lubotzky\, a
nd draw inspiration from well-known facts about three-dimensional manifold
s\, as in arithmetic topology. \n\nThe talk is based in part on collabor
ations with Esnault\, Jarden\, and Srinivas.\n
LOCATION:https://researchseminars.org/talk/MITNT/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Levent Alpöge (Harvard)
DTSTART;VALUE=DATE-TIME:20211116T213000Z
DTEND;VALUE=DATE-TIME:20211116T223000Z
DTSTAMP;VALUE=DATE-TIME:20211128T090609Z
UID:MITNT/39
DESCRIPTION:Title: A
"height-free" effective isogeny estimate for abelian varieties of $\\GL_2
$-type.\nby Levent Alpöge (Harvard) as part of MIT number theory semi
nar\n\nLecture held in Room 2-143 in the Simons building.\n\nAbstract\nLet
$g\\in \\mathbb{Z}^+$\, $K$ a number field\, $S$ a finite set of places o
f $K$\, and $A\,B/K$ $g$-dimensional abelian varieties with good reduction
outside $S$ which are $K$-isogenous and of $\\GL_2$-type over $\\overline
{\\mathbb{Q}}$. We show that there is a $K$-isogeny $A\\rightarrow B$ of d
egree effectively bounded in terms of $g$\, $K$\, and $S$ only.\n\nWe dedu
ce among other things an effective upper bound on the number of $S$-integr
al $K$-points on a Hilbert modular variety.\n
LOCATION:https://researchseminars.org/talk/MITNT/39/
END:VEVENT
END:VCALENDAR