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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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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:20240719T153849Z
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
BEGIN:VEVENT
SUMMARY:Andrew Graham (Université Paris-Saclay)
DTSTART;VALUE=DATE-TIME:20220215T213000Z
DTEND;VALUE=DATE-TIME:20220215T223000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/40
DESCRIPTION:Title: F
unctoriality of higher Coleman theory and p-adic L-functions in the unitar
y setting\nby Andrew Graham (Université Paris-Saclay) as part of MIT
number theory seminar\n\nLecture held in Room 2-449 in the Simons Building
(building 2).\n\nAbstract\nI will describe the construction of a p-adic a
nalytic function interpolating unitary Friedberg--Jacquet periods\, which
are conjecturally related to central critical values of L-functions for cu
spidal automorphic representations of unitary groups. The construction inv
olves establishing functoriality of Boxer and Pilloni's higher Coleman the
ory\, and p-adically interpolating branching laws for a certain pair of un
itary groups. The motivation for such a p-adic analytic function arises fr
om the Bloch--Kato conjecture for twists of the associated Galois represen
tation by anticyclotomic characters.\n
LOCATION:https://researchseminars.org/talk/MITNT/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Holly Krieger (University of Cambridge)
DTSTART;VALUE=DATE-TIME:20220222T213000Z
DTEND;VALUE=DATE-TIME:20220222T223000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/41
DESCRIPTION:Title: T
ranscendental values of power series and dynamical degrees\nby Holly K
rieger (University of Cambridge) as part of MIT number theory seminar\n\nL
ecture held in Room 2-449 in the Simons Building (building 2).\n\nAbstract
\nI will explain the construction (joint with Bell\, Diller\, and Jonsson)
of a birational map of projective 3-space with transcendental dynamical d
egree. This number is a measure of algebraic complexity of the iterates o
f a rational map\, and was previously conjectured to be algebraic for all
birational maps. Our proof includes a more general statement on transcend
ental values of certain power series\, using techniques similar to those o
f Adamczewski-Bugeaud.\n
LOCATION:https://researchseminars.org/talk/MITNT/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Keerthi Madapusi (Boston College)
DTSTART;VALUE=DATE-TIME:20220308T213000Z
DTEND;VALUE=DATE-TIME:20220308T223000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/43
DESCRIPTION:Title: D
erived special cycles on Shimura varieties\nby Keerthi Madapusi (Bosto
n College) as part of MIT number theory seminar\n\nLecture held in Room 2-
449 in the Simons Building (building 2).\n\nAbstract\nWe employ methods fr
om derived algebraic geometry to give a uniform moduli-theoretic construct
ion of special cycle classes on many Shimura varieties of Hodge type. Our
results apply in particular to classes on GSpin Shimura varieties associat
ed with arbitrary positive semi-definite symmetric matrices\, as well as t
o certain unitary and quaternionic Shimura varieties. We show that these c
lasses agree with the ones constructed in work with B. Howard using $K$-th
eoretic methods.\n
LOCATION:https://researchseminars.org/talk/MITNT/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph Silverman (Brown University)
DTSTART;VALUE=DATE-TIME:20220412T203000Z
DTEND;VALUE=DATE-TIME:20220412T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/47
DESCRIPTION:Title: O
rbits on tri-involutive K3 surfaces\nby Joseph Silverman (Brown Univer
sity) as part of MIT number theory seminar\n\nLecture held in Room 2-449 i
n the Simons Building (building 2).\n\nAbstract\nLet $W$ be a surface in $
\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$ given by the va
nishing of a $(2\,2\,2)$ form. The three projections $W \\to \\mathbb{P}^1
\\times \\mathbb{P}^1$ are double covers that induce three non-commuting
involutions on $W$. Let $G$ be the group of automorphisms of $W$ generat
ed by these involutions. We investigate the $G$-orbit structure of the poi
nts of $W$. In particular\, we study $G$-orbital components over finite fi
elds and finite $G$-orbits in characteristic 0. This is joint work with El
ena Fuchs\, Matthew Litman\, and Austin Tran.\n
LOCATION:https://researchseminars.org/talk/MITNT/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Petrov (Harvard)
DTSTART;VALUE=DATE-TIME:20220419T203000Z
DTEND;VALUE=DATE-TIME:20220419T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/48
DESCRIPTION:Title: G
alois action on the pro-algebraic fundamental group\nby Alexander Petr
ov (Harvard) as part of MIT number theory seminar\n\nLecture held in Room
2-449 in the Simons Building (building 2).\n\nAbstract\nGiven a smooth var
iety X over a number field\, the action of the Galois group on the geometr
ic etale fundamental group of X makes the ring of functions on the pro-alg
ebraic completion of this fundamental group into a (usually infinite-dimen
sional) Galois representation. This Galois representation turns out to sat
isfy the following two properties:\n\n1)Every finite-dimensional subrepres
entation of it satisfies the assumptions of the Fontaine-Mazur conjecture:
it is de Rham an almost everywhere unramifed.\n\n2)If X is the projective
line with three punctures\, the semi-simplification of every Galois repre
sentation of geometric origin is a subquotient of the ring of regular func
tions on the pro-algebraic completion of the etale fundamental group of X.
\n\nI will also discuss a conjectural characterization of local systems of
geometric origin on complex algebraic varieties\, arising from property 1
) above.\n
LOCATION:https://researchseminars.org/talk/MITNT/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Melanie Matchett Wood (Harvard)
DTSTART;VALUE=DATE-TIME:20220426T203000Z
DTEND;VALUE=DATE-TIME:20220426T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/49
DESCRIPTION:Title: D
istributions of unramified extensions of global fields\nby Melanie Mat
chett Wood (Harvard) as part of MIT number theory seminar\n\nLecture held
in Room 2-449 in the Simons Building (building 2).\n\nAbstract\nEvery numb
er field K has a maximal unramified extension K^un\, with\nGalois group Ga
l(K^un/K) (whose abelianization is the class group of\nK). As K varies\,
we ask about the distribution of the groups\nGal(K^un/K). We prove some
results about the structure of Gal(K^un/K) \nthat motivate us to give a c
onjecture about this distribution\, which we\nalso conjecture in the funct
ion field analog. We give theorems in\nthe function field case (as the s
ize of the finite field goes to\ninfinity) that support these new conjectu
res. In particular\, our\ndistributions abelianize to the Cohen-Lenstra-
Martinet distributions\nfor class groups\, and so our function field theor
ems prove\n(suitably modified) versions of the Cohen-Lenstra-Martinet heur
istics\nover function fields as the size of the finite field goes to\ninfi
nity. This talk is on joint work with Yuan Liu and David Zureick-Brown.\
n
LOCATION:https://researchseminars.org/talk/MITNT/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Pollack (Boston University)
DTSTART;VALUE=DATE-TIME:20220503T203000Z
DTEND;VALUE=DATE-TIME:20220503T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/50
DESCRIPTION:Title: P
redicting slopes of modular forms and reductions of crystalline representa
tions\nby Robert Pollack (Boston University) as part of MIT number the
ory seminar\n\nLecture held in Room 2-449 in the Simons Building (building
2).\n\nAbstract\nThe ghost conjecture predicts slopes of modular forms wh
ose residual representation is locally reducible. In this talk\, we'll ex
amine locally irreducible representations and discuss recent progress on f
ormulating a conjecture in this case. It's a lot trickier and the story r
emains incomplete\, but we will discuss how an irregular ghost conjecture
is intimately related to reductions of crystalline representations.\n
LOCATION:https://researchseminars.org/talk/MITNT/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noam David Elkies (Harvard)
DTSTART;VALUE=DATE-TIME:20220510T203000Z
DTEND;VALUE=DATE-TIME:20220510T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/51
DESCRIPTION:Title: A
belian surfaces with a (1\,2) polarization and full level-2 structure\
nby Noam David Elkies (Harvard) as part of MIT number theory seminar\n\nLe
cture held in Room 2-449 in the Simons Building (building 2).\n\nAbstract\
nAbstract: The moduli threefold of principally polarized abelian surfaces\
nwith full level-2 structure is well understood thanks to its close\nconne
ction with the moduli space $M_{0\,6}$ of six points on ${\\bf P}^1$.\nThe
moduli threefolds of (1\,d)-polarized surfaces with d>1 are more elusive.
\nWe report on our recent work on the d=2 case with full level-2 structure
.\nHere the moduli threefold is still rational\, and comes with\nan action
of a group G isomorphic with ${\\rm Aut}(S_4^2)$ instead of $S_6$.\nWe us
e elliptic fibrations of the Kummer surface to give\nseveral models of thi
s moduli threefold together with the G-action.\n
LOCATION:https://researchseminars.org/talk/MITNT/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kimball Martin (University of Oklahoma)
DTSTART;VALUE=DATE-TIME:20220315T203000Z
DTEND;VALUE=DATE-TIME:20220315T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/52
DESCRIPTION:Title: Q
uaternionic and algebraic modular forms: structure and applications\nb
y Kimball Martin (University of Oklahoma) as part of MIT number theory sem
inar\n\nLecture held in Room 2-449 in the Simons Building (building 2).\n\
nAbstract\nModular forms on definite quaternion algebras are amenable to e
xact calculation by algebraic methods\, and are related to classical modul
ar forms via the Jacquet-Langlands correspondence. I will describe some s
tructural results on quaternionic modular forms and applications to comput
ing modular forms\, Eisenstein congruences and central $L$-values. Along
the way\, I will discuss issues and progress toward analogues for algebrai
c modular forms on higher rank groups.\n
LOCATION:https://researchseminars.org/talk/MITNT/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kiran Kedlaya (University of California San Diego)
DTSTART;VALUE=DATE-TIME:20220915T190000Z
DTEND;VALUE=DATE-TIME:20220915T200000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/53
DESCRIPTION:Title: T
he relative class number one problem for function fields\nby Kiran Ked
laya (University of California San Diego) as part of MIT number theory sem
inar\n\nLecture held in Room 4-145.\n\nAbstract\nBuilding on my lecture fr
om ANTS-XV\, we classify extensions of function fields (of curves over fin
ite fields) with relative class number 1. Many of the ingredients come fro
m the study of the maximum number of points on a curve over a finite field
\, such as the function field analogue of Weil's explicit formulas (a/k/a
the "linear programming method"). Additional tools include the classificat
ion of abelian varieties of order 1 and the geometry of moduli spaces of c
urves of genus up to 7.\n
LOCATION:https://researchseminars.org/talk/MITNT/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Disegni (Ben-Gurion University of the Negev)
DTSTART;VALUE=DATE-TIME:20220920T214500Z
DTEND;VALUE=DATE-TIME:20220920T224500Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/54
DESCRIPTION:Title: A
lgebraic cycles and p-adic L-functions for conjugate-symplectic motives\nby Daniel Disegni (Ben-Gurion University of the Negev) as part of MIT n
umber theory seminar\n\nLecture held in Room 2-143 in the Simons Building
(building 2).\n\nAbstract\nI will introduce ‘canonical’ algebraic cyc
les for motives $M$ enjoying a certain symmetry - for instance\, some sy
mmetric powers of elliptic curves. The construction is based on works of K
udla and Liu on some (conjecturally modular) theta series valued in Chow
groups of Shimura varieties. The cycles have Heegner-point-like features
that allow\, under some assumptions\, to support an analogue of the BSD co
njecture for M at an ordinary prime $p$. Namely: if the $p$-adic $L$-funct
ion of $M$ vanishes at the center to order exactly 1\, then the ${\\bf Q}_
p$-Selmer group of $M$ has rank 1\, and it is generated by classes of alge
braic cycles. Partly joint work with Yifeng Liu.\n
LOCATION:https://researchseminars.org/talk/MITNT/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frank Calegari (University of Chicago)
DTSTART;VALUE=DATE-TIME:20220927T203000Z
DTEND;VALUE=DATE-TIME:20220927T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/55
DESCRIPTION:Title: T
he arithmetic of power series\nby Frank Calegari (University of Chicag
o) as part of MIT number theory seminar\n\nLecture held in Room 2-143 in t
he Simons Building (building 2).\n\nAbstract\nAbstract: A function of a co
mplex variable $P(z)$ which is holomorphic around $z=0$ has a power series
expansion $P(z)=\\sum a_n z^n$. Suppose that the $a_n$ are all integers:
what restrictions does that place on the function $P(z)$? We explore the r
elationship between this problem to questions in complex analysis\, number
theory\, and to Klein’s famous observation that not all finite index su
bgroups of $\\mathrm{SL}_2(\\mathbf{Z})$ are determined by congruence cond
itions. This talk is based on joint work with Vesselin Dimitrov and Yunqin
g Tang.\n
LOCATION:https://researchseminars.org/talk/MITNT/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yujie Xu (MIT)
DTSTART;VALUE=DATE-TIME:20221004T203000Z
DTEND;VALUE=DATE-TIME:20221004T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/56
DESCRIPTION:Title: N
ormalization in the integral models of Shimura varieties of abelian type\nby Yujie Xu (MIT) as part of MIT number theory seminar\n\nLecture held
in Room 2-143 in the Simons Building (building 2).\n\nAbstract\nShimura v
arieties are moduli spaces of abelian varieties with extra structures. Man
y interesting questions about abelian varieties have been answered by stud
ying the geometry of Shimura varieties. \n\nIn order to study the mod $p$
points of Shimura varieties\, over the decades\, various mathematicians (e
.g. Rapoport\, Kottwitz\, etc.) have constructed nice integral models of S
himura varieties. \nIn this talk\, I will discuss some motivic aspects of
integral models of Hodge type (or more generally abelian type) constructed
by Kisin and Kisin-Pappas. I will talk about my recent work on removing t
he normalization step in the construction of such integral models\, which
gives closed embeddings of Hodge type integral models into Siegel integral
models. I will also mention an application to toroidal compactifications
of such integral models. Such results (and their proof techniques) have fo
und interesting applications to the Kudla program (and various other progr
ams!).\n\nIf time permits\, I will also mention a new result on connected
components of affine Deligne–Lusztig varieties\, which gives us a new CM
lifting result for integral models of Shimura varieties at parahoric leve
ls and serves as an ingredient for my main theorem at parahoric levels.\n
LOCATION:https://researchseminars.org/talk/MITNT/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashvin Swaminathan (Harvard)
DTSTART;VALUE=DATE-TIME:20221011T203000Z
DTEND;VALUE=DATE-TIME:20221011T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/57
DESCRIPTION:Title: G
eometry-of-numbers in the cusp\, and class groups of orders in number fiel
ds\nby Ashvin Swaminathan (Harvard) as part of MIT number theory semin
ar\n\nLecture held in Room 2-143 in the Simons Building (building 2).\n\nA
bstract\nIn this talk\, we discuss the distributions of class groups of or
ders in number fields. We explain how studying such distributions is relat
ed to counting integral orbits having bounded invariants that lie inside t
he cusps of fundamental domains for coregular representations. We introduc
e two new methods to solve this counting problem\, and as an example\, we
demonstrate how one of these methods can be used to determine the average
size of the 2-torsion in the class groups of totally real or complex cubic
orders\, when such orders are enumerated by discriminant. Much of this wo
rk is joint with Arul Shankar\, Artane Siad\, and Ila Varma.\n
LOCATION:https://researchseminars.org/talk/MITNT/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Rüd (MIT)
DTSTART;VALUE=DATE-TIME:20221018T203000Z
DTEND;VALUE=DATE-TIME:20221018T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/58
DESCRIPTION:Title: T
amagawa numbers for maximal symplectic tori and mass formulae for abelian
varieties\nby Thomas Rüd (MIT) as part of MIT number theory seminar\n
\nLecture held in Room 2-143 in the Simons Building (building 2).\n\nAbstr
act\nTamagawa numbers are defined as a specific volumes of algebraic group
\, encapsulating the size of their adelic points modulo rational points. U
nsurprisingly\, such numbers are intrinsically linked to mass formulae of
various kinds. More recently\, Tamagawa numbers of centralizers of element
s within some algebraic groups also appear in the context of the stable tr
ace formula.\n\nGekeler's result on a mass formula for elliptic curves def
ined over $\\mathbb{F}_p$ was extended by Achter-Gordon and then Achter-Al
tug-Garcia-Gordon to mass formulae for certain principally polarized abeli
an varieties over finite fields\, using orbital integrals appearing in Lan
glands-Kottwitz formula. \nGekeler's work uses the analytic class number f
ormula\, which can be restated as $\\tau_\\mathbb{Q}(\\mathrm{R}_{K/\\math
bb{Q}}\\mathbb{G}_m)=1$ ($\\mathrm{R}$ denotes the Weil restriction of sca
lars)\, but in the case of higher-dimensional abelian varieties the tori i
nvolved are more complicated and their Tamagawa numbers were not known.\n\
n\nThe work presented aims at showing techniques to compute such numbers a
s well as many general results on Tamagawa numbers of a vast class of tori
\, which includes maximal tori of $\\mathrm{GSp}_{2n}$ over any global fie
ld. In particular we will give extensive results for tori splitting over C
M-fields and the possible range of such Tamagawa numbers.\n
LOCATION:https://researchseminars.org/talk/MITNT/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Congling Qiu (Yale)
DTSTART;VALUE=DATE-TIME:20221025T203000Z
DTEND;VALUE=DATE-TIME:20221025T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/59
DESCRIPTION:Title: A
rithmetic mixed Siegel-Weil formulas and modular form of arithmetic diviso
rs\nby Congling Qiu (Yale) as part of MIT number theory seminar\n\nLec
ture held in Room 2-143 in the Simons Building (building 2).\n\nAbstract\n
The classical Siegel–Weil formula relates theta series to Eisenstein
series and its arithmetic version is central in Kudla's program. I will di
scuss arithmetic mixed Siegel-Weil formulas. I will focus on the one in th
e work of Gross and Zagier\, and the one in my recent work. As an applicat
ion\, I obtained modular generating series of arithmetic extensions of Ku
dla's special divisors for unitary Shimura varieties over CM fields with a
rbitrary split level. This provides a partial solution to a problem of Kud
la.\n
LOCATION:https://researchseminars.org/talk/MITNT/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Allechar Serrano López (Harvard University)
DTSTART;VALUE=DATE-TIME:20221101T203000Z
DTEND;VALUE=DATE-TIME:20221101T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/60
DESCRIPTION:Title: C
ounting fields generated by points on plane curves\nby Allechar Serran
o López (Harvard University) as part of MIT number theory seminar\n\nLect
ure held in Room 2-143 in the Simons Building (building 2).\n\nAbstract\nF
or a smooth projective curve $C/\\mathbb{Q}$\, how many field extensions o
f $\\mathbb{Q}$ -- of given degree and bounded discriminant --- arise from
adjoining a point of $C(\\overline{\\mathbb{Q}})$? Can we further count t
he number of such extensions with a specified Galois group? Asymptotic low
er bounds for these quantities have been found for elliptic curves by Lemk
e Oliver and Thorne\, for hyperelliptic curves by Keyes\, and for superell
iptic curves by Beneish and Keyes. We discuss similar asymptotic lower bou
nds that hold for all smooth plane curves $C$. This is joint work with Mic
hael\, Allen\, Renee Bell\, Robert Lemke Oliver\, and Tian An Wong.\n
LOCATION:https://researchseminars.org/talk/MITNT/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Cowan (Harvard)
DTSTART;VALUE=DATE-TIME:20221108T213000Z
DTEND;VALUE=DATE-TIME:20221108T223000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/61
DESCRIPTION:Title: A
twisted additive divisor problem\nby Alex Cowan (Harvard) as part of
MIT number theory seminar\n\nLecture held in Room 2-143 in the Simons Buil
ding (building 2).\n\nAbstract\nWe give asymptotics for a shifted convolut
ion of sum-of-divisors functions twisted by Dirichlet characters and with
nonzero powers. We'll use the technique of "automorphic regularization" to
find a spectral decomposition of a combination of Eisenstein series which
is not obviously square-integrable.\n
LOCATION:https://researchseminars.org/talk/MITNT/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Barinder Banwait (Boston University)
DTSTART;VALUE=DATE-TIME:20221115T213000Z
DTEND;VALUE=DATE-TIME:20221115T223000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/62
DESCRIPTION:Title: T
owards strong uniformity for isogenies of prime degree\nby Barinder Ba
nwait (Boston University) as part of MIT number theory seminar\n\nLecture
held in Room 2-143 in the Simons Building (building 2).\n\nAbstract\nLet $
E$ be an elliptic curve over a number field $k$ of degree $d$ that admits
a $k$-rational isogeny of prime degree $p$. We study the question of findi
ng uniform bounds on $p$ that depend only $d$\, and\, under a certain cond
ition on the signature of the isogeny\, explicitly construct non-zero inte
gers that $p$ must divide. As a corollary\, we find a bound on prime order
torsion points defined over unramified extensions of the base field. This
is work in progress joint with Maarten Derickx.\n
LOCATION:https://researchseminars.org/talk/MITNT/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Smith (Stanford)
DTSTART;VALUE=DATE-TIME:20221122T213000Z
DTEND;VALUE=DATE-TIME:20221122T223000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/63
DESCRIPTION:Title: S
imple abelian varieties over finite fields with extreme point counts\n
by Alexander Smith (Stanford) as part of MIT number theory seminar\n\nLect
ure held in Room 2-143 in the Simons Building (building 2).\n\nAbstract\nG
iven a compactly supported probability measure on the reals\, we will give
a necessary and sufficient condition for there to be a sequence of totall
y real algebraic integers whose distribution of conjugates approaches the
measure. We use this result to prove that there are infinitely many totall
y positive algebraic integers X satisfying tr(X)/deg(X) < 1.899\; previous
ly\, there were only known to be infinitely many such integers satisfying
tr(X)/deg(X) < 2. We also will explain how our method can be used in the s
earch for simple abelian varieties with extreme point counts.\n
LOCATION:https://researchseminars.org/talk/MITNT/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Álvaro Lozano-Robledo (University of Connecticut)
DTSTART;VALUE=DATE-TIME:20221129T213000Z
DTEND;VALUE=DATE-TIME:20221129T223000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/64
DESCRIPTION:Title: T
he adelic image of a Galois representation attached to a CM elliptic curve
\nby Álvaro Lozano-Robledo (University of Connecticut) as part of MIT
number theory seminar\n\nLecture held in Room 2-143 in the Simons Buildin
g (building 2).\n\nAbstract\nIn this talk we will discuss recent work on t
he classification of $\\ell$-adic images of Galois representations attache
d to elliptic curves with complex multiplication\, and applications. In pa
rticular\, we will show how to construct the adelic image of representatio
ns attached to CM elliptic curves (joint work with Benjamin York)\, and we
will discuss results on the minimal degree of definition of torsion struc
tures (joint work with Enrique Gonzalez Jimenez).\n
LOCATION:https://researchseminars.org/talk/MITNT/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roy Zhao (University of California at Berkeley)
DTSTART;VALUE=DATE-TIME:20221206T213000Z
DTEND;VALUE=DATE-TIME:20221206T223000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/65
DESCRIPTION:Title: H
eights on quaternionic Shimura varieties\nby Roy Zhao (University of C
alifornia at Berkeley) as part of MIT number theory seminar\n\nLecture hel
d in Room 2-143 in the Simons Building (building 2).\n\nAbstract\nWe give
an explicit formula for the height of a special point on a quaternionic Sh
imura variety in terms of Faltings heights of CM abelian varieties. This i
s a generalization of the work of Yuan and Zhang on proving the averaged C
olmez conjecture. We also show an application of this formula to the Andre
-Oort conjecture\, which was recently proven by Pila\, Shankar\, and Tsime
rman.\n
LOCATION:https://researchseminars.org/talk/MITNT/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicholas M. Katz and Pham Huu Tiep (Princeton\, Rutgers)
DTSTART;VALUE=DATE-TIME:20230228T213000Z
DTEND;VALUE=DATE-TIME:20230228T230000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/66
DESCRIPTION:Title: L
ocal systems\, exponential sums\, and simple groups\nby Nicholas M. Ka
tz and Pham Huu Tiep (Princeton\, Rutgers) as part of MIT number theory se
minar\n\nLecture held in Room 2-449 in the Simons Building (building 2).\n
\nAbstract\nWe study the possible structure of monodromy groups of Airy\,
Kloosterman\, and hypergeometric $\\ell$-adic sheaves in characteristic $p
$. We also discuss explicit constructions of local systems and their relat
ed exponential sums on $\\mathbb{G}_m$ or $\\mathbb{A}^1$ that realize var
ious (close to be) simple groups.\n\nThis will be a 2-part talk\, with Kat
z giving part 1 starting at 4:35pm Eastern (via zoom\, projected on the sc
reen in 2-449)\, and Tiep giving part 2 starting at around 5:20pm in perso
n in 2-449.\n
LOCATION:https://researchseminars.org/talk/MITNT/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Li-Huerta (Harvard University)
DTSTART;VALUE=DATE-TIME:20230418T203000Z
DTEND;VALUE=DATE-TIME:20230418T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/67
DESCRIPTION:Title: O
n the plectic conjecture\nby Daniel Li-Huerta (Harvard University) as
part of MIT number theory seminar\n\nLecture held in Room 2-449 in the Sim
ons Building (building 2).\n\nAbstract\nNekovář–Scholl observed that t
he étale cohomology groups of Hilbert modular varieties enjoy the action
of a much larger profinite group than the absolute Galois group of $\\math
bb{Q}$: the *plectic Galois group*. They conjectured that this action
extends to the level of complexes\, which would give a construction of ca
nonical classes in higher wedge powers of Selmer groups. I'll explain how
this works\, as well as discuss analogues over local fields and global fun
ction fields.\n
LOCATION:https://researchseminars.org/talk/MITNT/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linus Hamann (Princeton University)
DTSTART;VALUE=DATE-TIME:20230404T203000Z
DTEND;VALUE=DATE-TIME:20230404T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/68
DESCRIPTION:Title: G
eometric Eisenstein series and the Fargues-Fontaine curve\nby Linus Ha
mann (Princeton University) as part of MIT number theory seminar\n\nLectur
e held in Room 2-449 in the Simons Building (building 2).\n\nAbstract\nGiv
en a connected reductive group $G$ and a Levi subgroup $M$\,\nBraverman-Ga
itsgory and Laumon constructed geometric Eisenstein\nfunctors which take H
ecke eigensheaves on the moduli stack $\\operatorname{Bun}_{M}$ of\n$M$-bu
ndles on a smooth projective curve to eigensheaves on the moduli stack\n$\
\operatorname{Bun}_{G}$ of\n$G$-bundles. Recently\, Fargues and Scholze co
nstructed a general\ncandidate for the local Langlands correspondence by d
oing geometric\nLanglands on the Fargues-Fontaine curve. In this talk\, we
explain recent work\non carrying the theory of geometric Eisenstein serie
s over to the\nFargues-Scholze setting. In particular\, we explain how\, g
iven the\neigensheaf $S_{\\chi}$ on $\\operatorname{Bun}_{T}$ attached to
a smooth character $\\chi$ of\nthe maximal torus $T$\, one can construct a
n eigensheaf on $\\operatorname{Bun}_{G}$ under\na certain genericity hypo
thesis on $\\chi$\, by applying a geometric\nEisenstein functor to $S_{\\c
hi}$. Assuming the Fargues-Scholze\ncorrespondence satisfies certain expec
ted properties\, we fully\ndescribe the stalks of this eigensheaf in terms
of normalized\nparabolic inductions of the generic $\\chi$. This eigenshe
af has\nseveral useful applications to the study of the cohomology of\nloc
al and global Shimura varieties\, and time permitting we will\nexplain suc
h applications.\n
LOCATION:https://researchseminars.org/talk/MITNT/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lie Qian (Stanford University)
DTSTART;VALUE=DATE-TIME:20230411T203000Z
DTEND;VALUE=DATE-TIME:20230411T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/69
DESCRIPTION:Title: L
ocal Compatibility for Trianguline Representations\nby Lie Qian (Stanf
ord University) as part of MIT number theory seminar\n\nLecture held in Ro
om 2-449 in the Simons Building (building 2).\n\nAbstract\nTrianguline rep
resentations are a big class of $p$-adic representations that contains all
nice enough (cristalline) ones but allow a continuous variation of weight
s. Global consideration suggests that the $GL_2(\\mathbb{Q}_p)$ representa
tion arising from a trianguline representation should have nonzero eigensp
ace under Emerton's Jacquet functor. We prove this result using purely loc
al method as well as a generalization to $p$-adic representation of $G_F$
for $F$ unramified over $\\mathbb{Q}_p$.\n
LOCATION:https://researchseminars.org/talk/MITNT/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sameera Vemulapalli (Princeton University)
DTSTART;VALUE=DATE-TIME:20230321T203000Z
DTEND;VALUE=DATE-TIME:20230321T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/70
DESCRIPTION:Title: C
ounting low degree number fields with almost prescribed successive minima<
/a>\nby Sameera Vemulapalli (Princeton University) as part of MIT number t
heory seminar\n\nLecture held in Room 2-449 in the Simons Building (buildi
ng 2).\n\nAbstract\nThe successive minima of an order in a degree $n$ numb
er field are $n$ real numbers encoding information about the Euclidean str
ucture of the order. How many orders in degree n number fields are there w
ith almost prescribed successive minima\, fixed Galois group\, and bounded
discriminant? In this talk\, I will address this question for $n = 3\,4\,
5$. The answers\, appropriately interpreted\, turn out to be piecewise lin
ear functions on certain convex bodies. If time permits\, I will also disc
uss a geometric analogue of this problem: scrollar invariants of covers of
$\\mathbb{P}^1$.\n
LOCATION:https://researchseminars.org/talk/MITNT/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vadim Vologodsky (MIT)
DTSTART;VALUE=DATE-TIME:20230307T213000Z
DTEND;VALUE=DATE-TIME:20230307T223000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/71
DESCRIPTION:Title: D
ual abelian varieties over a local field have equal volumes\nby Vadim
Vologodsky (MIT) as part of MIT number theory seminar\n\nLecture held in R
oom 2-449 in the Simons Building (building 2).\n\nAbstract\nA top degree d
ifferential form $\\omega$ on a smooth algebraic variety $X$ over a local
field $K$ gives rise to a (real valued) measure on $X(K)$. The Serre duali
ty yields a natural isomorphism between the vector spaces of global top de
gree forms on an abelian variety and the dual abelian variety. I will prov
e that the corresponding volumes are equal.\n
LOCATION:https://researchseminars.org/talk/MITNT/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stephen D. Miller (Rutgers University)
DTSTART;VALUE=DATE-TIME:20230509T203000Z
DTEND;VALUE=DATE-TIME:20230509T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/72
DESCRIPTION:Title: T
he number theory and modular forms behind 8- and 24-dimensional sphere pac
king\nby Stephen D. Miller (Rutgers University) as part of MIT number
theory seminar\n\nLecture held in Room 2-449 in the Simons Building (build
ing 2).\n\nAbstract\nAlthough the solution to the sphere packing and "univ
ersal optimality" energy minimization problems in dimensions 8 and 24 have
a very analytic flavor\, number theory is pervasive behind the scenes. I
'll describe the rationality conjectures with Cohn which first pointed to
the appearance of modular forms\, as well as Viazovska's interpolation ans
atz which more directly linked with modular forms\, especially for energy
minimization. (Joint work with Henry Cohn\, Abhinav Kumar\, Danylo Radche
nko\, and Maryna Viazovska.)\n
LOCATION:https://researchseminars.org/talk/MITNT/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amnon Besser (Ben-Gurion University/Boston University)
DTSTART;VALUE=DATE-TIME:20230502T203000Z
DTEND;VALUE=DATE-TIME:20230502T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/73
DESCRIPTION:Title: L
ocal contributions to Quadratic Chabauty functions and derivatives of Volo
godsky functions with respect to $log(p)$\nby Amnon Besser (Ben-Gurion
University/Boston University) as part of MIT number theory seminar\n\nLec
ture held in Room 2-449 in the Simons Building (building 2).\n\nAbstract\n
Quadratic Chabauty is a method for finding rational points on curves using
$p$-adic methods. The quadratic Chabauty function is a function on these
rational points\, usually derived from some $p$-adic height\, which is a s
um of local terms at finite primes. The main term is the term at $p$ which
is a Coleman function\, but in order to make the method work one needs to
be able to compute the finite list of possible values of the other contri
butions at primes of bad reduction.\n\nVologodsky functions are the genera
lisation of Coleman functions to varieties with bad reduction. In this tal
k\, which is based on ongoing work with Steffen Muller and Padma Srinivasa
n\, I would like to promote the general (and vague) idea that the derivati
ve of a Vologodsky integral with respect to the branch of log parameter $l
og(p)$ is arithmetically interesting.\n\nAs an example I will show how the
local contribution above a prime $q$ to a $p$ adic height can be computed
by deriving the $q$-adic contribution to a $q$-adic height and use this t
o obtain a computable formula for this contribution using the work of Katz
and Litt. In particular\, I will recover a formula of Betts and Dogra for
the local contribution to the Quadratic Chabauty function at a prime wher
e the completion is a Mumford curve.\n
LOCATION:https://researchseminars.org/talk/MITNT/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefano Marseglia (Utrecht University)
DTSTART;VALUE=DATE-TIME:20230516T203000Z
DTEND;VALUE=DATE-TIME:20230516T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/74
DESCRIPTION:Title: C
ohen-Macaulay type of endomorphism rings of abelian varieties over finite
fields\nby Stefano Marseglia (Utrecht University) as part of MIT numbe
r theory seminar\n\nLecture held in Room 2-449 in the Simons Building (bui
lding 2).\n\nAbstract\nIn this talk\, we will speak about the (Cohen-Macau
lay) type of the endomorphism ring of abelian varieties over a finite fiel
d with commutative endomorphism algebra. We will exhibit a condition on th
e type of $\\mathrm{End}(A)$ implying that $A$ cannot be isomorphic to its
dual. In particular\, such an $A$ cannot be principally polarised or a Ja
cobian. This is partly joint work with Caleb Springer.\n
LOCATION:https://researchseminars.org/talk/MITNT/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robin Zhang (MIT)
DTSTART;VALUE=DATE-TIME:20230912T203000Z
DTEND;VALUE=DATE-TIME:20230912T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/75
DESCRIPTION:Title: M
odular Gelfand pairs\, multiplicity-free triples\, and maybe some gamma fa
ctors\nby Robin Zhang (MIT) as part of MIT number theory seminar\n\nLe
cture held in Room 2-449 in the Simons Building (building 2).\n\nAbstract\
nThe classical theory of Gelfand pairs and its generalizations over the co
mplex numbers has many applications to number theory and automorphic forms
\, such as the uniqueness of Whittaker models and the non-vanishing of the
central value of a triple product $L$-function. With an eye towards simil
ar applications in the modular setting\, this talk presents an extension o
f the classical theory for representations of finite and compact groups to
such representations over algebraically closed fields with arbitrary char
acteristic. Time permitting\, I will also mention an analogue (joint with
J. Bakeberg\, M. Gerbelli-Gauthier\, H. Goodson\, A. Iyengar\, and G. Moss
) of the local converse theorem for Jacquet–Piatetski-Shapiro–Shalika
gamma factors of mod $\\ell \\neq p$ representations of finite general lin
ear groups.\n
LOCATION:https://researchseminars.org/talk/MITNT/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Omer Offen (Brandeis University)
DTSTART;VALUE=DATE-TIME:20230926T203000Z
DTEND;VALUE=DATE-TIME:20230926T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/76
DESCRIPTION:Title: A
new application of the residue method\nby Omer Offen (Brandeis Univer
sity) as part of MIT number theory seminar\n\nLecture held in Room 2-449 i
n the Simons Building (building 2).\n\nAbstract\nThe relative Langlands pr
ogram studies the relations between functorial transfers of automorphic re
presentations from one group $G'$ to another group $G$\, period integrals
over a subgroup $H$ of $G$ and special $L$-values.\nWhen $(G\,H)$ is a van
ishing pair\, that is\, every cusp form of G has a vanishing $H$-period\,
it is of interest to study discrete automorphic representations that admit
a non-vanishing $H$-period.\nFor this task\, Jacquet and Rallis developed
the residue method. \n It has since been used extensively\, mostly for re
presentations with cuspidal data lying in a maximal Levi subgroup of G. In
this talk we focus on the case where $H=Sp(a)\\times Sp(b)$ lies in $G=Sp
(a+b)$. We will introduce a new construction of some residual representati
ons of G that admit H-periods. This is joint work in progress with Sol Fri
edberg and David Ginzburg.\n
LOCATION:https://researchseminars.org/talk/MITNT/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dubi Kelmer (Boston College)
DTSTART;VALUE=DATE-TIME:20231024T203000Z
DTEND;VALUE=DATE-TIME:20231024T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/77
DESCRIPTION:Title: N
orm bounds on Eisenstein series\nby Dubi Kelmer (Boston College) as pa
rt of MIT number theory seminar\n\nLecture held in Room 2-449 in the Simon
s Building (building 2).\n\nAbstract\nIn this talk I will describe some ne
w results on the magnitude Eisenstein series corresponding to arithmetic l
attices in hyperbolic space.\nAll new results are based on joint work with
Shucheng Yu as well as with Alex Kontorovich and Cristopher Lutsko.\n
LOCATION:https://researchseminars.org/talk/MITNT/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jerson Caro (Boston University)
DTSTART;VALUE=DATE-TIME:20231003T203000Z
DTEND;VALUE=DATE-TIME:20231003T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/78
DESCRIPTION:Title: A
Chabauty-Coleman bound for surfaces\nby Jerson Caro (Boston Universit
y) as part of MIT number theory seminar\n\nLecture held in Room 2-449 in t
he Simons Building (building 2).\n\nAbstract\nA celebrated result of Colem
an gives a completely explicit version of Chabauty's finiteness theorem fo
r rational points in curves over a number field\, by a study of zeros of p
-adic analytic functions. \nAfter several developments around this result\
, the problem of proving an analogous explicit bound for higher dimensiona
l subvarieties of abelian varieties remains elusive. In this talk\, I'll s
ketch the proof of such a bound for surfaces contained in abelian varietie
s. This is a joint work with Hector Pasten.\n\nIn addition\, I'll present
an application of this method to give an upper bound for the number of une
xpected quadratic points of hyperelliptic curves of genus 3 defined over $
\\mathbb{Q}$. This is a joint work in progress with Jennifer Balakrishnan.
\n
LOCATION:https://researchseminars.org/talk/MITNT/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kestutis Cesnavicius (Université Paris-Saclay)
DTSTART;VALUE=DATE-TIME:20231031T203000Z
DTEND;VALUE=DATE-TIME:20231031T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/79
DESCRIPTION:Title: T
he affine Grassmannian as a presheaf quotient\nby Kestutis Cesnavicius
(Université Paris-Saclay) as part of MIT number theory seminar\n\nLectur
e held in Room 2-449 in the Simons Building (building 2).\n\nAbstract\nThe
affine Grassmannian of a reductive group $G$ is usually defined as the é
tale sheafification of the quotient of the loop group $LG$ by the positive
loop subgroup. I will discuss various triviality results for $G$-torsors
which imply that this sheafification is often not necessary.\n
LOCATION:https://researchseminars.org/talk/MITNT/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Spencer Leslie (Boston College)
DTSTART;VALUE=DATE-TIME:20231107T213000Z
DTEND;VALUE=DATE-TIME:20231107T223000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/80
DESCRIPTION:Title: R
elative Langlands and Endoscopy\nby Spencer Leslie (Boston College) as
part of MIT number theory seminar\n\nLecture held in Room 2-449 in the Si
mons Building (building 2).\n\nAbstract\nIn this talk\, I will discuss the
motivation for a theory of endoscopy in the context of the relative Langl
ands program. I then outline the construction of endoscopic varieties\, wh
ich are spherical varieties of an associated endoscopic group. The constru
ction works for most hyperspherical varieties induced from symmetric varie
ties\, and relies on new rationality results for such varieties. We highli
ght the role played by the dual Hamiltonian variety associated to a symmet
ric variety a la Ben-Zvi\, Sakellaridis\, and Venkatesh.\n
LOCATION:https://researchseminars.org/talk/MITNT/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rafael von Känel (IAS\, Tsinghua University)
DTSTART;VALUE=DATE-TIME:20231121T210000Z
DTEND;VALUE=DATE-TIME:20231121T220000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/81
DESCRIPTION:Title: I
ntegral points on the Clebsch-Klein surfaces\nby Rafael von Känel (IA
S\, Tsinghua University) as part of MIT number theory seminar\n\nLecture h
eld in Room 2-449 in the Simons Building (building 2).\n\nAbstract\nIn thi
s talk we present explicit bounds for the Weil height and the number of in
tegral points on classical surfaces first studied by Clebsch (1871) and Kl
ein (1873). Building on Hirzebruch’s work in which he related these surf
aces to a Hilbert modular surface\, we deduced our bounds from a general r
esult for integral points on coarse Hilbert moduli schemes. After explaini
ng this deduction\, we discuss the strategy of proof of the general result
which combines the method of Faltings (Arakelov\,\nParsin\, Szpiro) with
modularity\, Masser-Wuestholz isogeny estimates\, and results based on eff
ective analytic estimates and/or Arakelov theory. Joint work with Arno Kre
t.\n
LOCATION:https://researchseminars.org/talk/MITNT/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hélène Esnault (FU Berlin/Harvard/Copenhagen)
DTSTART;VALUE=DATE-TIME:20231205T210000Z
DTEND;VALUE=DATE-TIME:20231205T220000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/83
DESCRIPTION:Title: S
urvey on some arithmetic properties of rigid local systems\nby Hélèn
e Esnault (FU Berlin/Harvard/Copenhagen) as part of MIT number theory semi
nar\n\nLecture held in Room 2-449 in the Simons Building (building 2).\n\n
Abstract\nA central conjecture of Simpson predicts that complex rigid loca
l systems on a smooth complex variety come from geometry. In the last coup
le of years\, we proved some arithmetic consequences of it: integrality (u
sing the arithmetic Langlands program)\, F-isocrystal properties\, crystal
linity of the underlying p-adic representation (using the Cartier operator
over the Witt vectors and the Higgs-de Rham flow) (for Shimura varieties
of real rank at least 2\, this is the corner piece of Pila-Shankar-Tsimerm
an's proof of the André-Oort conjecture)\, weak integrality of the charac
ter variety (using de Jong's conjecture proved with the geometric Langland
s program) (yielding a new obstruction for a finitely presented group to
be the topological fundamental group of a smooth complex variety).\n\nWe'l
l survey some aspects of this (please ask if there is something on which y
ou would like me to focus on). The talk is based mostly on joint work with
Michael Groechenig\, also\, even if less\, with Johan de Jong.\n
LOCATION:https://researchseminars.org/talk/MITNT/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marco Sangiovanni Vincentelli (Princeton University)
DTSTART;VALUE=DATE-TIME:20231212T210000Z
DTEND;VALUE=DATE-TIME:20231212T220000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/84
DESCRIPTION:Title: S
elmer groups\, p-adic L-functions and Euler Systems: A Unified Framework.<
/a>\nby Marco Sangiovanni Vincentelli (Princeton University) as part of MI
T number theory seminar\n\nLecture held in Room 2-449 in the Simons Buildi
ng (building 2).\n\nAbstract\nSelmer groups are key invariants attached to
p-adic Galois representations. The Bloch—Kato conjecture predicts a pre
cise relationship between the size of certain Selmer groups and the leadin
g term of the L-function of the Galois representation under consideration.
In particular\, when the L-function does not have a zero at s=0\, it pred
icts that the Selmer group is finite and its order is controlled by the va
lue of the L-function at s=0. Historically\, one of the most powerful tool
s to prove such relationships is by constructing an Euler System (ES). \nA
n Euler System is a collection of Galois cohomology classes over ramified
abelian extensions of the base field that verify some co-restriction compa
tibilities. The key feature of ESs is that they provide a way to bound Sel
mer groups\, thanks to the machinery developed by Rubin\, inspired by earl
ier work of Thaine\, Kolyvagin\, and Kato. In this talk\, I will present j
oint work with C. Skinner\, in which we develop a new method for construct
ing Euler Systems and apply it to build an ES for the Galois representatio
n attached to the symmetric square of an elliptic modular form. I will str
ess how this method gives a unifying approach to constructing ESs\, in tha
t it can be successfully applied to retrieve most classical ESs (the cyclo
tomic units ES\, the elliptic units ES\, Kato’s ES…).\n
LOCATION:https://researchseminars.org/talk/MITNT/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jordan Ellenberg (University of Wisconsin)
DTSTART;VALUE=DATE-TIME:20240206T210000Z
DTEND;VALUE=DATE-TIME:20240206T220000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/85
DESCRIPTION:Title: A
round Smyth’s conjecture\nby Jordan Ellenberg (University of Wiscons
in) as part of MIT number theory seminar\n\nLecture held in Room 2-449 in
the Simons Building (building 2).\n\nAbstract\nAre there algebraic numbers
$x\,y\,z$ which are Galois conjugate to each other over $\\mathbb{Q}$ and
which satisfy the equation $5x + 6y + 7z = 0$? In 1986\, Chris Smyth prop
osed an appealingly simple conjecture about linear relations between Galoi
s conjugates\, which would provide answers to the above questions and all
questions of the same form\, and which has remained unsolved. My experien
ce is that most people\, upon seeing Smyth’s conjecture\, immediately th
ink it must be false (I certainly did!)\, but I have come to think it’s
true\, and I’ll talk about a provisional solution\, joint with Will Hard
t\, in the case of three conjugates. I’ll explain why (as Smyth observe
d) this is really a conjecture about linear combinations of permutation ma
trices (related question\, solved by Speyer: which algebraic numbers can
be eigenvalues of the sum of two permutation matrices?)\, and why our appr
oach can be thought of as proving a “Hasse principle for probability dis
tributions” in a particular case\, plus a bit of additive number theory.
Much of this talk\, maybe all\, will be suitable for undergraduates.\n
LOCATION:https://researchseminars.org/talk/MITNT/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Minhyong Kim (International Centre for Mathematical Sciences)
DTSTART;VALUE=DATE-TIME:20240220T210000Z
DTEND;VALUE=DATE-TIME:20240220T220000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/86
DESCRIPTION:Title: A
rithmetic Quantum Field Theory?\nby Minhyong Kim (International Centre
for Mathematical Sciences) as part of MIT number theory seminar\n\nLectur
e held in Room 2-449 in the Simons Building (building 2).\n\nAbstract\nMat
hematical structures suggested by quantum field theory have revolutionised
important areas of algebraic geometry\, differential geometry\, as well a
s topology in the last three decades. This talk will introduce a few of th
e recent ideas for applying structures inspired by physics to arithmetic g
eometry.\n
LOCATION:https://researchseminars.org/talk/MITNT/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ulrich Derenthal (Leibniz Universität Hannover)
DTSTART;VALUE=DATE-TIME:20240227T210000Z
DTEND;VALUE=DATE-TIME:20240227T220000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/87
DESCRIPTION:Title: M
anin's conjecture for spherical Fano threefolds\nby Ulrich Derenthal (
Leibniz Universität Hannover) as part of MIT number theory seminar\n\nLec
ture held in Room 2-449 in the Simons Building (building 2).\n\nAbstract\n
When an algebraic variety over the rational numbers contains infinitely ma
ny rational points\, we may study their distribution. In particular\, for
Fano varieties\, the asymptotic behavior of the number of rational points
of bounded height is predicted by Manin's conjecture.\n\nIn this talk\, we
discuss a proof of Manin's conjecture for smooth spherical Fano threefold
s. In one case\, in order to obtain the expected asymptotic formula\, it i
s necessary to exclude a thin subset with exceptionally many rational poin
ts from the count. This is joint work with V. Blomer\, J. Brüdern and G.
Gagliardi.\n
LOCATION:https://researchseminars.org/talk/MITNT/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ryan Chen (MIT)
DTSTART;VALUE=DATE-TIME:20240305T210000Z
DTEND;VALUE=DATE-TIME:20240305T220000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/88
DESCRIPTION:Title: C
o-rank 1 Arithmetic Siegel–Weil\nby Ryan Chen (MIT) as part of MIT n
umber theory seminar\n\nLecture held in Room 2-449 in the Simons Building
(building 2).\n\nAbstract\nI will introduce my recent work on an arithmeti
c Siegel–Weil formula for Kudla–Rapoport $1$-cycles on integral models
of some unitary Shimura varieties. This formula implies that degrees of K
udla–Rapoport arithmetic special $1$-cycles are encoded in the first der
ivatives of unitary Eisenstein series Fourier coefficients. In the simples
t case\, this can be rephrased in terms of Faltings heights of Hecke trans
lates of CM elliptic curves\, and the classical weight $2$ Eisenstein seri
es.\n\nThe key input is a new local limiting method which relates (a) degr
ees of local special 0-cycles and (b) local contributions to heights of sp
ecial $1$-cycles.\n
LOCATION:https://researchseminars.org/talk/MITNT/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laura DeMarco (Harvard University)
DTSTART;VALUE=DATE-TIME:20240312T203000Z
DTEND;VALUE=DATE-TIME:20240312T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/89
DESCRIPTION:Title: F
rom Manin–Mumford to dynamical rigidity\nby Laura DeMarco (Harvard U
niversity) as part of MIT number theory seminar\n\nLecture held in Room 2-
449 in the Simons Building (building 2).\n\nAbstract\nIn the early 1980s\,
Raynaud proved a theorem (the Manin–Mumford Conjecture) about the geome
try of torsion points in abelian varieties\, using number-theoretic method
s. Around the same time\, and with completely different methods\, McMulle
n proved a theorem about dynamical stability for maps on $\\mathbb{P}^1$.
In new work\, joint with Myrto Mavraki\, we view these results as special
cases of a unifying conjecture. The conjectural statement is directly in
spired by a recent theorem of Gao and Habegger (called Relative Manin–Mu
mford) and results in complex dynamics of Dujardin\, Gauthier\, Vigny\, an
d others.\n
LOCATION:https://researchseminars.org/talk/MITNT/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Borys Kadets (Hebrew University)
DTSTART;VALUE=DATE-TIME:20240402T203000Z
DTEND;VALUE=DATE-TIME:20240402T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/90
DESCRIPTION:Title: C
urves with many degree $d$ points\nby Borys Kadets (Hebrew University)
as part of MIT number theory seminar\n\nLecture held in Room 2-449 in the
Simons Building (building 2).\n\nAbstract\nWhen does a nice curve $X$ ove
r a number field $k$ have infinitely many closed points of degree $d$?\nFa
ltings' theorem allows us to rephrase this problem in purely algebro-geome
tric terms\, though the resulting geometric question is far from being ful
ly solved. Previous work gave easy to state answers to the problem for deg
rees $2$ (Harris-Silverman) and $3$ (Abramovich-Harris)\, but also uncover
ed exotic constructions of such curves in all degrees $d \\geqslant 4$ (De
barre-Fahlaoui). I will describe recent progress on the problem\, which an
swers the question in the large genus case. Along the way we uncover syste
matic explanations for the Debarre-Fahlaoui counstructions and provide a c
omplete geometric answer for $d \\leqslant 5$. The talk is based on joint
work with Isabel Vogt.\n
LOCATION:https://researchseminars.org/talk/MITNT/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Padmavathi Srinivasan (Boston University)
DTSTART;VALUE=DATE-TIME:20240416T203000Z
DTEND;VALUE=DATE-TIME:20240416T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/92
DESCRIPTION:Title: A
canonical algebraic cycle associated to a curve in its Jacobian\nby P
admavathi Srinivasan (Boston University) as part of MIT number theory semi
nar\n\nLecture held in Room 2-449 in the Simons Building (building 2).\n\n
Abstract\nThe Ceresa cycle is a canonical homologically trivial algebraic
cycle associated to a curve in its Jacobian. In his 1983 thesis\, Ceresa s
howed that this cycle is algebraically nontrivial for the generic curve ov
er genus at least 3. Strategies for proving Fermat curves have infinite or
der Ceresa cycles due to B. Harris\, Bloch\, Bertolini-Darmon-Prasanna\, E
skandari-Murty use a variety of ideas ranging from computation of explicit
iterated period integrals\, special values of p-adic L functions and poin
ts of infinite order on the Jacobian of Fermat curves. We will survey many
recent results around the Ceresa cycle\, and present ongoing work with Jo
rdan Ellenberg\, Adam Logan and Akshay Venkatesh where we produce many new
explicit examples of curves over number fields with infinite order Ceresa
cycles.\n
LOCATION:https://researchseminars.org/talk/MITNT/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vincent Pilloni (CNRS / Institut de Mathématiques d'Orsay)
DTSTART;VALUE=DATE-TIME:20240430T203000Z
DTEND;VALUE=DATE-TIME:20240430T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/93
DESCRIPTION:Title: O
n the modularity of abelian surfaces\nby Vincent Pilloni (CNRS / Insti
tut de Mathématiques d'Orsay) as part of MIT number theory seminar\n\nLec
ture held in Room 2-449 in the Simons Building (building 2).\n\nAbstract\n
We prove that a positive proportion of abelian surfaces over the rationals
are modular. This is joint work with G. Boxer\, F. Calegari and T. Gee.\n
LOCATION:https://researchseminars.org/talk/MITNT/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tangli Ge (Princeton University)
DTSTART;VALUE=DATE-TIME:20240507T203000Z
DTEND;VALUE=DATE-TIME:20240507T213000Z
DTSTAMP;VALUE=DATE-TIME:20240719T153849Z
UID:MITNT/94
DESCRIPTION:Title: S
ome height conjectures on abelian schemes\nby Tangli Ge (Princeton Uni
versity) as part of MIT number theory seminar\n\nLecture held in Room 2-44
9 in the Simons Building (building 2).\n\nAbstract\nMotivated by the conje
ctures of S. Zhang and Zilber–Pink\, I would like to formulate three con
jectures about height functions on abelian schemes. These conjectures will
represent intersections of respectively unlikely\, just likely and very l
ikely kinds. The first one is a Bogomolov type conjecture. The second one
is about boundedness of height. The third one is related to uniformity of
the height bound. Some known results will be mentioned during the talk. I
will then discuss an interesting implication: a specialization theorem for
the Mordell–Weil group of an elliptic surface of Silverman.\n
LOCATION:https://researchseminars.org/talk/MITNT/94/
END:VEVENT
END:VCALENDAR