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BEGIN:VEVENT
SUMMARY:Jacob Tsimerman (University of Toronto)
DTSTART;VALUE=DATE-TIME:20200408T190000Z
DTEND;VALUE=DATE-TIME:20200408T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/1
DESCRIPTION:Title: Bounding torsion in class groups and families of local systems\nby J
acob Tsimerman (University of Toronto) as part of Harvard number theory se
minar\n\n\nAbstract\n(joint w/ Arul Shankar) We discuss a new method to bo
und 5-torsion in class groups of quadratic fields using the refined BSD co
njecture for elliptic curves. The most natural “trivial” bound on the
n-torsion is to bound it by the size of the entire class group\, for which
one has a global class number formula. We explain how to make sense of th
e n-torsion of a class group intrinsically as a selmer group of a Galois m
odule. We may then similarly bound its size by the Tate-Shafarevich group
of an appropriate elliptic curve\, which we can bound using the BSD conjec
ture. This fits into a general paradigm where one bounds selmer groups of
finite Galois modules by embedding into global objects\, and using class n
umber formulas. If time permits\, we explain how the function field pictur
e yields unconditional results and suggests further generalizations.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Kriz (MIT)
DTSTART;VALUE=DATE-TIME:20200415T190000Z
DTEND;VALUE=DATE-TIME:20200415T201500Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/2
DESCRIPTION:Title: Converse theorems for supersingular CM elliptic curves\nby Daniel Kr
iz (MIT) as part of Harvard number theory seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/HarvardNT/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Weinstein (BU)
DTSTART;VALUE=DATE-TIME:20200422T190000Z
DTEND;VALUE=DATE-TIME:20200422T201500Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/3
DESCRIPTION:Title: Modularity for self-products of elliptic curves over function fields
\nby Jared Weinstein (BU) as part of Harvard number theory seminar\n\nAbst
ract: TBA\n
LOCATION:https://researchseminars.org/talk/HarvardNT/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Newton (Kings College London)
DTSTART;VALUE=DATE-TIME:20200506T190000Z
DTEND;VALUE=DATE-TIME:20200506T201500Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/4
DESCRIPTION:Title: Symmetric power functoriality for modular forms\nby James Newton (Ki
ngs College London) as part of Harvard number theory seminar\n\nAbstract:
TBA\n
LOCATION:https://researchseminars.org/talk/HarvardNT/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arthur-Cesar Le Bras (CNRS/Paris-13)
DTSTART;VALUE=DATE-TIME:20200513T190000Z
DTEND;VALUE=DATE-TIME:20200513T201500Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/5
DESCRIPTION:Title: Prismatic Dieudonne theory\nby Arthur-Cesar Le Bras (CNRS/Paris-13)
as part of Harvard number theory seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/HarvardNT/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Preston Wake (Michigan State)
DTSTART;VALUE=DATE-TIME:20200520T190000Z
DTEND;VALUE=DATE-TIME:20200520T201500Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/6
DESCRIPTION:Title: Tame derivatives and the Eisenstein ideal\nby Preston Wake (Michigan
State) as part of Harvard number theory seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/HarvardNT/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Landesman (Stanford University)
DTSTART;VALUE=DATE-TIME:20201104T200000Z
DTEND;VALUE=DATE-TIME:20201104T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/7
DESCRIPTION:Title: A geometric approach to the Cohen-Lenstra heuristics\nby Aaron Lande
sman (Stanford University) as part of Harvard number theory seminar\n\n\nA
bstract\nFor any positive integer $n$\,\nwe explain why the total number o
f order $n$ elements\nin class groups of quadratic fields of discriminant\
nhaving absolute value at most $X$ is $O_n(X^{5/4})$.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karol Koziol (University of Michigan)
DTSTART;VALUE=DATE-TIME:20201028T190000Z
DTEND;VALUE=DATE-TIME:20201028T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/8
DESCRIPTION:Title: Supersingular representations of $p$-adic reductive groups\nby Karol
Koziol (University of Michigan) as part of Harvard number theory seminar\
n\n\nAbstract\nThe local Langlands conjectures predict that (packets of) i
rreducible complex representations of $p$-adic reductive groups (such as $
\\mathrm{GL}_n(\\mathbb{Q}_p)$\, $\\mathrm{GSp}_{2n}(\\mathbb{Q}_p)$\, etc
.) should be parametrized by certain representations of the Weil-Deligne g
roup. A special role in this hypothetical correspondence is held by the
supercuspidal representations\, which generically are expected to correspo
nd to irreducible objects on the Galois side\, and which serve as building
blocks for all irreducible representations. Motivated by recent advance
s in the mod-$p$ local Langlands program (i.e.\, with mod-$p$ coefficients
instead of complex coefficients)\, I will give an overview of what is kno
wn about supersingular representations of $p$-adic reductive groups\, whic
h are the "mod-$p$ coefficients" analogs of supercuspidal representations.
This is joint work with Florian Herzig and Marie-France Vigneras.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arul Shankar (University of Toronto)
DTSTART;VALUE=DATE-TIME:20201202T200000Z
DTEND;VALUE=DATE-TIME:20201202T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/9
DESCRIPTION:Title: The 2-torsion subgroups of the class groups in families of cubic fields<
/a>\nby Arul Shankar (University of Toronto) as part of Harvard number the
ory seminar\n\n\nAbstract\nThe Cohen--Lenstra--Martinet conjectures have b
een verified in\nonly two cases. Davenport--Heilbronn compute the average
size of the\n3-torsion subgroups in the class group of quadratic fields an
d Bhargava\ncomputes the average size of the 2-torsion subgroups in the cl
ass groups of\ncubic fields. The values computed in the above two results
are remarkably\nstable. In particular\, work of Bhargava--Varma shows that
they do not\nchange if one instead averages over the family of quadratic
or cubic fields\nsatisfying any finite set of splitting conditions.\n\nHow
ever for certain "thin" families of cubic fields\, namely\, families of\nm
onogenic and n-monogenic cubic fields\, the story is very different. In\nt
his talk\, we will determine the average size of the 2-torsion subgroups o
f\nthe class groups of fields in these thin families. Surprisingly\, these
\nvalues differ from the Cohen--Lenstra--Martinet predictions! We will als
o\nprovide an explanation for this difference in terms of the Tamagawa num
bers\nof naturally arising reductive groups. This is joint work with Manju
l\nBhargava and Jon Hanke.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean-Marc Couveignes (University of Bordeaux)
DTSTART;VALUE=DATE-TIME:20201209T200000Z
DTEND;VALUE=DATE-TIME:20201209T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/10
DESCRIPTION:Title: Hermite interpolation and counting number fields\nby Jean-Marc Couv
eignes (University of Bordeaux) as part of Harvard number theory seminar\n
\n\nAbstract\nThere are several ways to specify a number\nfield. One can p
rovide the minimal polynomial\nof a primitive element\, the multiplication
\ntable of a $\\bf Q$-basis\, the traces of a large enough\nfamily of elem
ents\, etc.\nFrom any way of specifying a number field\none can hope to
deduce a bound on the number\n$N_n(H)$ of number\nfields of given degree
$n$ and discriminant bounded by $H$.\nExperimental data\nsuggest that th
e number\nof isomorphism classes of number fields of degree $n$\nand discr
iminant bounded by $H$ is equivalent to $c(n)H$\nwhen $n\\geqslant 2$ is f
ixed and $H$ tends to infinity.\nSuch an estimate has been proved for $n=3
$\nby Davenport and Heilbronn and for $n=4$\, $5$ by\n Bhargava. For an
arbitrary $n$ Schmidt proved\na bound of the form $c(n)H^{(n+2)/4}$\nusin
g Minkowski's theorem.\nEllenberg et Venkatesh have proved that the expone
nt of\n$H$ in $N_n(H)$ is less than sub-exponential in $\\log (n)$.\nI wil
l explain how Hermite interpolation (a theorem\nof Alexander and Hirschowi
tz) and geometry of numbers\ncombine to produce short models for number fi
elds\nand sharper bounds for $N_n(H)$.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Cass (Harvard University)
DTSTART;VALUE=DATE-TIME:20200909T190000Z
DTEND;VALUE=DATE-TIME:20200909T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/11
DESCRIPTION:Title: A mod p geometric Satake isomorphism\nby Robert Cass (Harvard Unive
rsity) as part of Harvard number theory seminar\n\n\nAbstract\nWe apply me
thods from geometric representation theory toward the mod p\nLanglands pro
gram.\nMore specifically\, we explain a mod p version of the geometric Sat
ake\nisomorphism\, which gives a sheaf-theoretic description of the spheri
cal mod\np Hecke algebra. In our setup the mod p Satake category is not co
ntrolled\nby the dual group but rather a certain affine monoid scheme. Alo
ng the way\nwe will discuss some new results about the F-singularities of
affine\nSchubert varieties. Time permitting\, we will explain how to geome
trically\nconstruct central elements in the Iwahori mod p Hecke algebra by
adapting a\nmethod due to Gaitsgory.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zijian Yao (CNRS/Harvard)
DTSTART;VALUE=DATE-TIME:20201111T200000Z
DTEND;VALUE=DATE-TIME:20201111T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/12
DESCRIPTION:Title: Frobenius and the Hodge numbers of the generic fiber\nby Zijian Yao
(CNRS/Harvard) as part of Harvard number theory seminar\n\n\nAbstract\nFo
r a smooth proper (formal) scheme $\\mathfrak{X}$ defined over a valuation
\nring of mixed characteristic\, the crystalline cohomology H of its speci
al\nfiber has the structure of an F-crystal\, to which one can attach a Ne
wton\npolygon and a Hodge polygon that describe the ''slopes of the Froben
ius\naction on H''. The shape of these polygons are constrained by the geo
metry\nof $\\mathfrak{X}$ -- in particular by the Hodge numbers of both th
e special\nfiber and the generic fiber of $\\mathfrak{X}$. One instance of
such\nconstraints is given by a beautiful conjecture of Katz (now a theor
em of\nMazur\, Ogus\, Nygaard etc.)\, another constraint comes from the no
tion of\n"weakly admissible" Galois representations.\n\nIn this talk\, I w
ill discuss some results regarding the shape of the\nFrobenius action on t
he F-crystal H and the Hodge numbers of the generic\nfiber of $\\mathfrak{
X}$\, along with generalizations in several directions.\nIn particular\,
we give a new proof of the fact that the Newton polygon of\nthe special fi
ber of $\\mathfrak{X}$ lies on or above the Hodge polygon of\nits generic
fiber\, without appealing to Galois representations. A new\ningredient tha
t appears is (a generalized version of) the Nygaard\nfiltration of the pri
smatic/Ainf cohomology\, developed by Bhatt\, Morrow and\nScholze.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elena Mantovan (Caltech)
DTSTART;VALUE=DATE-TIME:20201021T190000Z
DTEND;VALUE=DATE-TIME:20201021T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/13
DESCRIPTION:Title: p-adic differential operators on automorphic forms\, and mod p Galois r
epresentations\nby Elena Mantovan (Caltech) as part of Harvard number
theory seminar\n\n\nAbstract\nIn this talk\, we will discuss a geometric c
onstruction of p-adic analogues of Maass--Shimura differential operators o
n automorphic forms on Shimura varieties of PEL type A or C (that is\, uni
tary or symplectic)\, at p an unramified prime. Maass--Shimura operators a
re smooth weight raising differential operators used in the study of speci
al values of L-functions\, and in the arithmetic setting for the construct
ion of p-adic L-functions. In this talk\, we will focus in particular on
the case of unitary groups of arbitrary signature\, when new phenomena ari
se for p non split. We will also discuss an application to the study of
modular mod p Galois representations. This talk is based on joint work wit
h Ellen Eischen (in the unitary case for p non split)\, and with Eischen\,
Flanders\, Ghitza\, and Mc Andrew (in the other cases).\n
LOCATION:https://researchseminars.org/talk/HarvardNT/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Si Ying Lee (Harvard University)
DTSTART;VALUE=DATE-TIME:20201118T200000Z
DTEND;VALUE=DATE-TIME:20201118T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/14
DESCRIPTION:Title: Eichler-Shimura relations for Hodge type Shimura varieties\nby Si Y
ing Lee (Harvard University) as part of Harvard number theory seminar\n\n\
nAbstract\nThe well-known classical Eichler-Shimura relation for modular c
urves asserts that the Hecke operator $T_p$ is equal\, as an algebraic cor
respondence over the special fiber\, to the sum of Frobenius and Verschebu
ng. Blasius and Rogawski proposed a generalization of this result for gene
ral Shimura varieties with good reduction at $p$\, and conjectured that th
e Frobenius satisfies a certain Hecke polynomial. I will talk about a rece
nt proof of this conjecture for Shimura varieties of Hodge type\, assuming
a technical condition on the unramified sigma-conjugacy classes in the as
sociated Kottwitz set.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Loeffler (University of Warwick)
DTSTART;VALUE=DATE-TIME:20201014T190000Z
DTEND;VALUE=DATE-TIME:20201014T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/15
DESCRIPTION:Title: The Bloch--Kato conjecture for GSp(4)\nby David Loeffler (Universit
y of Warwick) as part of Harvard number theory seminar\n\n\nAbstract\nThe
Bloch--Kato conjecture predicts that the dimension of the Selmer group of
a global Galois representation is equal to the order of vanishing of its L
-function. In this talk\, I will focus on the 4-dimensional Galois represe
ntations arising from cohomological automorphic representations of GSp(4)
(i.e. from genus two Siegel modular forms). I will show that if the L-func
tion is non-vanishing at some critical value\, then the corresponding Selm
er group is zero\, under a long list of technical hypotheses. The proof of
this theorem relies on an Euler system\, a p-adic L-function\, and a reci
procity law connecting those together. I will also survey work in progress
aiming to extend this result to some other classes of automorphic represe
ntations.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiuya Wang (Duke University)
DTSTART;VALUE=DATE-TIME:20200930T190000Z
DTEND;VALUE=DATE-TIME:20200930T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/16
DESCRIPTION:Title: Pointwise Bound for $\\ell$-torsion of Class Groups\nby Jiuya Wang
(Duke University) as part of Harvard number theory seminar\n\n\nAbstract\n
$\\ell$-torsion conjecture states that $\\ell$-torsion of the class group
$|\\text{Cl}_K[\\ell]|$ for every number field $K$ is bounded by $\\text{D
isc}(K)^{\\epsilon}$. It follows from a classical result of Brauer-Siegel\
, or even earlier result of Minkowski that the class number $|\\text{Cl}_K
|$ of a number field $K$ are always bounded by $\\text{Disc}(K)^{1/2+\\eps
ilon}$\, therefore we obtain a trivial bound $\\text{Disc}(K)^{1/2+\\epsil
on}$ on $|\\text{Cl}_K[\\ell]|$. We will talk about results on this conjec
ture\, and recent works on breaking the trivial bound for $\\ell$-torsion
of class groups in some cases based on a work of Ellenberg-Venkatesh.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jessica Fintzen (Cambridge/Duke/IAS)
DTSTART;VALUE=DATE-TIME:20200916T190000Z
DTEND;VALUE=DATE-TIME:20200916T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/17
DESCRIPTION:Title: Representations of p-adic groups and applications\nby Jessica Fintz
en (Cambridge/Duke/IAS) as part of Harvard number theory seminar\n\n\nAbst
ract\nThe Langlands program is a far-reaching collection of conjectures th
at relate different areas of mathematics including number theory and repre
sentation theory. A fundamental problem on the representation theory side
of the Langlands 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 em
phasis on recent progress.\n\nI will also outline how new results about th
e representation theory of p-adic groups can be used to obtain congruences
between arbitrary automorphic forms and automorphic forms which are super
cuspidal at p\, which is joint work with Sug Woo Shin. This simplifies ear
lier constructions of attaching Galois representations to automorphic repr
esentations\, i.e. the global Langlands correspondence\, for general linea
r groups. Moreover\, our results apply to general p-adic groups and have t
herefore the potential to become widely applicable beyond the case of the
general linear group.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kaisa Matomäki (University of Turku)
DTSTART;VALUE=DATE-TIME:20200923T140000Z
DTEND;VALUE=DATE-TIME:20200923T150000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/18
DESCRIPTION:Title: Multiplicative functions in short intervals revisited\nby Kaisa Mat
omäki (University of Turku) as part of Harvard number theory seminar\n\n\
nAbstract\nA few years ago Maksym Radziwill and I showed that the average
of a multiplicative function in almost all very short intervals $[x\, x+h]
$ is close to its average on a long interval $[x\, 2x]$. This result has s
ince been utilized in many applications.\nI will talk about recent work\,
where Radziwill and I revisit the problem and generalise our result to fun
ctions which vanish often as well as prove a power-saving upper bound for
the number of exceptional intervals (i.e. we show that there are $O(X/h^\\
kappa)$ exceptional $x \\in [X\, 2X]$).\nWe apply this result for instance
to studying gaps between norm forms of an arbitrary number field.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ziyang Gao (CNRS/IMJ-PRG)
DTSTART;VALUE=DATE-TIME:20201007T190000Z
DTEND;VALUE=DATE-TIME:20201007T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/19
DESCRIPTION:Title: Bounding the number of rational points on curves\nby Ziyang Gao (CN
RS/IMJ-PRG) as part of Harvard number theory seminar\n\n\nAbstract\nMazur
conjectured\, after Faltings’s proof of the Mordell conjecture\, that th
e number of rational points on a curve of genus g at least 2 defined over
a number field of degree d is bounded in terms of g\, d and the Mordell-We
il rank. In particular the height of the curve is not involved. In this ta
lk I will explain how to prove this conjecture and some generalizations. I
will focus on how functional transcendence and unlikely intersections are
applied in the proof. If time permits\, I will talk about how the depende
nce on d can be furthermore removed if we moreover assume the relative Bog
omolov conjecture. This is joint work with Vesselin Dimitrov and Philipp H
abegger.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niki Myrto Mavraki (Harvard University)
DTSTART;VALUE=DATE-TIME:20210127T200000Z
DTEND;VALUE=DATE-TIME:20210127T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/20
DESCRIPTION:Title: Arithmetic dynamics of random polynomials\nby Niki Myrto Mavraki (H
arvard University) as part of Harvard number theory seminar\n\n\nAbstract\
nWe begin with an introduction to arithmetic dynamics and heights\nattache
d to rational maps. We then introduce a dynamical version of Lang's\nconje
cture concerning the minimal canonical height of non-torsion rational\npoi
nts in elliptic curves (due to Silverman) as well as a conjectural\nanalog
ue of Mazur/Merel's theorem on uniform bounds of rational torsion\npoints
in elliptic curves (due to Morton-Silverman). It is likely that the\ntwo c
onjectures are harder in the dynamical setting due to the lack of\nstructu
re coming from a group law. We describe joint work with Pierre Le\nBoudec
in which we establish statistical versions of these conjectures for\npolyn
omial maps.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Timo Richarz (TU Darmstadt)
DTSTART;VALUE=DATE-TIME:20210407T190000Z
DTEND;VALUE=DATE-TIME:20210407T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/21
DESCRIPTION:Title: The motivic Satake equivalence\nby Timo Richarz (TU Darmstadt) as p
art of Harvard number theory seminar\n\n\nAbstract\nThe geometric Satake e
quivalence due to Lusztig\, Drinfeld\, Ginzburg\, Mirković and Vilonen is
an indispensable tool in the Langlands program. Versions of this equivale
nce are known for different cohomology theories such as Betti cohomology o
r algebraic D-modules over characteristic zero fields and $\\ell$-adic coh
omology over arbitrary fields. In this talk\, I explain how to apply the t
heory of motivic complexes as developed by Voevodsky\, Ayoub\, Cisinski-D
église and many others to the construction of a motivic Satake equivalenc
e. Under suitable cycle class maps\, it recovers the classical equivalence
. As dual group\, one obtains a certain extension of the Langlands dual gr
oup by a one dimensional torus. A key step in the proof is the constructio
n of intersection motives on affine Grassmannians. A direct consequence of
their existence is an unconditional construction of IC-Chow groups of mod
uli stacks of shtukas. My hope is to obtain on the long run independence-o
f-$\\ell$ results in the work of V. Lafforgue on the Langlands corresponde
nce for function fields. This is ongoing joint work with Jakob Scholbach f
rom Münster.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Scholze (University of Bonn)
DTSTART;VALUE=DATE-TIME:20210203T200000Z
DTEND;VALUE=DATE-TIME:20210203T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/22
DESCRIPTION:Title: Analytic geometry\nby Peter Scholze (University of Bonn) as part of
Harvard number theory seminar\n\n\nAbstract\nWe will outline a definition
of analytic spaces that relates\nto complex- or rigid-analytic varieties
in the same way that schemes\nrelate to algebraic varieties over a field.
Joint with Dustin Clausen.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christian Johansson (Chalmers/Gothenburg)
DTSTART;VALUE=DATE-TIME:20210224T200000Z
DTEND;VALUE=DATE-TIME:20210224T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/23
DESCRIPTION:Title: On the Calegari--Emerton conjectures for abelian type Shimura varieties
\nby Christian Johansson (Chalmers/Gothenburg) as part of Harvard numb
er theory seminar\n\n\nAbstract\nEmerton's completed cohomology gives\, at
present\, the most general notion of a space of p-adic automorphic forms.
Important properties of completed cohomology\, such as its 'size'\, is pr
edicted by a conjecture of Calegari and Emerton\, which may be viewed as a
non-abelian generalization of the Leopoldt conjecture. I will discuss the
proof of many new cases of this conjecture\, using a mixture of technique
s from p-adic and real geometry. This is joint work with David Hansen.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Pollack (UCSD)
DTSTART;VALUE=DATE-TIME:20210317T190000Z
DTEND;VALUE=DATE-TIME:20210317T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/24
DESCRIPTION:Title: Modular forms on G_2\nby Aaron Pollack (UCSD) as part of Harvard nu
mber theory seminar\n\n\nAbstract\nFollowing work of Gross-Wallach\, Gan-G
ross-Savin defined what are called "modular forms" on the split exceptiona
l group G_2. These are a special class of automorphic forms on G_2. I'l
l review their definition\, and give an update about what is known about t
hem. Results include a construction of cuspidal modular forms with all al
gebraic Fourier coefficients\, and the exact functional equation of the co
mpleted standard L-function of certain cusp forms. The results on L-funct
ions are joint with Fatma Cicek\, Giuliana Davidoff\, Sarah Dijols\, Traja
n Hammonds\, and Manami Roy.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Litt (University of Georgia)
DTSTART;VALUE=DATE-TIME:20210324T190000Z
DTEND;VALUE=DATE-TIME:20210324T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/25
DESCRIPTION:Title: Single-valued Hodge\, p-adic^2\, and tropical integration\nby Danie
l Litt (University of Georgia) as part of Harvard number theory seminar\n\
n\nAbstract\nI'll discuss 4 different types of integration -- one in the\n
complex setting\, one in the tropical setting\, and two in the p-adic\nset
ting\, and the relationships between them. In particular\, we explain how\
nto compute Vologodsky's "single-valued" iterated integrals on curves of b
ad\nreduction in terms of Berkovich integrals\, and how to give a single-v
alued\nintegration theory on complex varieties. Time permitting\, I'll exp
lain some\npotential arithmetic applications. This is a report on joint wo
rk in\nprogress with Sasha Shmakov (in the complex setting) and Eric Katz
(in the\np-adic setting).\n
LOCATION:https://researchseminars.org/talk/HarvardNT/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:François Charles (Université Paris-Sud)
DTSTART;VALUE=DATE-TIME:20210414T190000Z
DTEND;VALUE=DATE-TIME:20210414T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/26
DESCRIPTION:Title: Arithmetic curves lying in compact subsets of affine schemes\nby Fr
ançois Charles (Université Paris-Sud) as part of Harvard number theory s
eminar\n\n\nAbstract\nWe will describe the notion of affine schemes and th
eir modifications in the context of Arakelov geometry. Using geometry of n
umbers in infinite rank\, we will study their cohomological properties. Co
ncretely\, given an affine scheme X over Z and a compact subset K of the s
et of complex points of X\, we will investigate the geometry of those prop
er arithmetic curves in X whose complex points lie in K. This is joint wor
k with Jean-Benoît Bost.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bhargav Bhatt (University of Michigan)
DTSTART;VALUE=DATE-TIME:20210421T190000Z
DTEND;VALUE=DATE-TIME:20210421T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/27
DESCRIPTION:Title: The absolute prismatic site\nby Bhargav Bhatt (University of Michig
an) as part of Harvard number theory seminar\n\n\nAbstract\nThe absolute p
rismatic site of a p-adic formal scheme carries interesting\narithmetic an
d geometric information attached to the formal scheme. In this\ntalk\, aft
er recalling the definition of this site\, I will discuss an\nalgebro-geom
etric (stacky) approach to absolute prismatic cohomology and\nits concomit
ant structures (joint with Lurie\, and partially due\nindependently to Dri
nfeld). As a geometric application\, I'll explain\nDrinfeld's refinement o
f the Deligne-Illusie theorem on Hodge-to-de Rham\ndegeneration. On the ar
ithmetic side\, I'll describe a new classification of\ncrystalline represe
ntations of the Galois group of a local field in terms\nof F-crystals on t
he site (joint with Scholze).\n
LOCATION:https://researchseminars.org/talk/HarvardNT/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Keerthi Madapusi Pera (Boston College)
DTSTART;VALUE=DATE-TIME:20210310T200000Z
DTEND;VALUE=DATE-TIME:20210310T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/28
DESCRIPTION:Title: Existence of CM lifts for points on Shimura varieties\nby Keerthi M
adapusi Pera (Boston College) as part of Harvard number theory seminar\n\n
\nAbstract\nI'll explain a very simple proof of the fact that K3 surfaces
of\nfinite height admit (many) CM lifts\, a result due independently to\nI
to-Ito-Koshikawa and Z. Yang\, which was used by the former to prove the\n
Tate conjecture for products of K3s. This will be done directly showing\nt
hat the deformation ring of a polarized K3 surface of finite height admits
\nas a quotient that of its Brauer group. The method applies more generall
y\nto many isogeny classes of points on Shimura varieties of abelian type.
\n
LOCATION:https://researchseminars.org/talk/HarvardNT/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laura DeMarco (Harvard University)
DTSTART;VALUE=DATE-TIME:20210210T200000Z
DTEND;VALUE=DATE-TIME:20210210T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/29
DESCRIPTION:Title: Elliptic surfaces\, bifurcations\, and equidistribution\nby Laura D
eMarco (Harvard University) as part of Harvard number theory seminar\n\n\n
Abstract\nIn joint work with Myrto Mavraki\, we studied the arithmetic int
ersection of\nsections of elliptic surfaces\, defined over number fields.
I will describe\nour results and formulate some related open questions ab
out families of\nmaps (dynamical systems) on P^1 over a base curve.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frank Calegari (University of Chicago)
DTSTART;VALUE=DATE-TIME:20210428T190000Z
DTEND;VALUE=DATE-TIME:20210428T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/30
DESCRIPTION:Title: From Ramanujan to K-theory\nby Frank Calegari (University of Chicag
o) as part of Harvard number theory seminar\n\n\nAbstract\nThe Rogers-Rama
nujan identity is an equality between a certain “q-series” (given as a
n infinite sum) and a certain modular form (given as an infinite product).
Motivated by ideas from physics\, Nahm formulated a necessary condition f
or when such q-hypergeometric series were modular. Perhaps surprisingly\,
this turns out to be related to algebraic K-theory. We discuss a proof of
this conjecture. This is joint work with Stavros Garoufalidis and Don Zagi
er.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lars Kühne (University of Copenhagen)
DTSTART;VALUE=DATE-TIME:20210303T200000Z
DTEND;VALUE=DATE-TIME:20210303T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/31
DESCRIPTION:Title: Equidistribution and Uniformity in Families of Curves\nby Lars Küh
ne (University of Copenhagen) as part of Harvard number theory seminar\n\n
\nAbstract\nIn the talk\, I will present an equidistribution result for fa
milies of (non-degenerate) subvarieties in a (general) family of abelian v
arieties. This extends a result of DeMarco and Mavraki for curves in fiber
ed products of elliptic surfaces. Using this result\, one can deduce a uni
form version of the classical Bogomolov conjecture for curves embedded in
their Jacobians\, namely that the number of torsion points lying on them i
s uniformly bounded in the genus of the curve. This has been previously on
ly known in a few select cases by work of David--Philippon and DeMarco--Kr
ieger--Ye. Finally\, one can obtain a rather uniform version of the Mordel
l-Lang conjecture as well by complementing a result of Dimitrov--Gao--Habe
gger: The number of rational points on a smooth algebraic curve defined ov
er a number field can be bounded solely in terms of its genus and the Mord
ell-Weil rank of its Jacobian. Again\, this was previously known only unde
r additional assumptions (Stoll\, Katz--Rabinoff--Zureick-Brown).\n
LOCATION:https://researchseminars.org/talk/HarvardNT/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ziquan Yang (Harvard University)
DTSTART;VALUE=DATE-TIME:20210217T200000Z
DTEND;VALUE=DATE-TIME:20210217T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/32
DESCRIPTION:Title: Twisted derived equivalences and the Tate conjecture for K3 squares
\nby Ziquan Yang (Harvard University) as part of Harvard number theory sem
inar\n\n\nAbstract\nThere is a long standing connection between the Tate c
onjecture in codimension 1 and finiteness properties\, which first appeare
d in Tate's seminal work on the endomorphisms of abelian varieties. I will
explain how one can possibly extend this connection to codimension 2 cycl
es\, using the theory of Brauer groups\, moduli of twisted sheaves\, and t
wisted derived equivalences\, and prove the Tate conjecture for K3 squares
. This recovers an earlier result of Ito-Ito-Kashikawa\, which was establi
shed via a CM lifting theory\, and moreover provides a recipe of construct
ing all the cycles on these varieties by purely geometric methods.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Melanie Matchett Wood (Harvard University)
DTSTART;VALUE=DATE-TIME:20210908T190000Z
DTEND;VALUE=DATE-TIME:20210908T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/33
DESCRIPTION:Title: The average size of 3-torsion in class groups of 2-extensions\nby M
elanie Matchett Wood (Harvard University) as part of Harvard number theory
seminar\n\nLecture held in Room 507 in the Science Center.\n\nAbstract\nT
he p-torsion in the class group of a number field K is conjectured to\nbe
small: of size at most $|\\operatorname{Disc} K|^\\varepsilon$\, and to ha
ve constant\naverage size in families with a given Galois closure group (w
hen p\ndoesn't divide the order of the group). In general\, the best uppe
r\nbound we have is $|\\operatorname{Disc} K|^{1/2+\\varepsilon}$\, and pr
eviously the only two\ncases known with constant average were for 3-torsio
n in quadratic\nfields (Davenport and Heilbronn\, 1971) and 2-torsion in n
on-Galois\ncubic fields (Bhargava\, 2005). We prove that the 3-torsion is
\nconstant on average for fields with Galois closure group any 2-group\nwi
th a transposition\, including\, e.g. quartic $D_4$ fields. We will\ndisc
uss the main inputs into the proof with an eye towards giving an\nintroduc
tion to the tools in the area. This is joint work with Robert\nLemke Oliv
er and Jiuya Wang.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Salim Tayou (Harvard University)
DTSTART;VALUE=DATE-TIME:20210929T190000Z
DTEND;VALUE=DATE-TIME:20210929T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/34
DESCRIPTION:Title: Density of arithmetic Hodge loci\nby Salim Tayou (Harvard Universit
y) as part of Harvard number theory seminar\n\nLecture held in Room 507 in
the Science Center.\n\nAbstract\nI will explain a conjecture on density o
f arithmetic Hodge loci which includes and generalizes several recent dens
ity results of these loci in arithmetic geometry. This conjecture has also
analogues over functions fields that I will survey. As a particular insta
nce\, I will outline the proof of the following result: a K3 surface over
a number field admits infinitely many specializations where its Picard ran
k jumps. This last result is joint work with Ananth Shankar\, Arul Shankar
and Yunqing Tang.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean Kieffer (Harvard University)
DTSTART;VALUE=DATE-TIME:20211006T190000Z
DTEND;VALUE=DATE-TIME:20211006T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/35
DESCRIPTION:Title: Higher-dimensional modular equations and point counting on abelian surf
aces\nby Jean Kieffer (Harvard University) as part of Harvard number t
heory seminar\n\nLecture held in Room 507 in the Science Center.\n\nAbstra
ct\nGiven a prime number l\, the elliptic modular polynomial of level l is
an explicit equation for the locus of elliptic curves related by an l-iso
geny. These polynomials have a large number of algorithmic applications: i
n particular\, they are an essential ingredient in the celebrated SEA algo
rithm for counting points on elliptic curves over finite fields of large c
haracteristic.\n\nMore generally\, modular equations describe the locus of
isogenous abelian varieties in certain moduli spaces called PEL Shimura v
arieties. We will present upper bounds on the size of modular equations in
terms of their level\, and outline a general strategy to compute an isoge
ny A->A' given a point (A\,A') where the modular equations are satisfied.
This generalizes well-known properties of elliptic modular polynomials to
higher dimensions.\n\nThe isogeny algorithm is made fully explicit for Jac
obians of genus 2 curves. In combination with efficient evaluation methods
for modular equations in genus 2 via complex approximations\, this gives
rise to point counting algorithms for (Jacobians of) genus 2 curves whose\
nasymptotic costs\, under standard heuristics\, improve on previous result
s.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Shusterman (Harvard University)
DTSTART;VALUE=DATE-TIME:20210915T190000Z
DTEND;VALUE=DATE-TIME:20210915T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/36
DESCRIPTION:Title: Finitely Presented Groups in Arithmetic Geometry\nby Mark Shusterma
n (Harvard University) as part of Harvard number theory seminar\n\nLecture
held in Room 507 in the Science Center.\n\nAbstract\nWe discuss the probl
em of determining the number of generators and relations of several profin
ite groups of arithmetic and geometric origin. \nThese include etale funda
mental groups of smooth projective varieties\, absolute Galois groups of l
ocal fields\, and Galois groups of maximal unramified extensions of number
fields. The results are based on a cohomological presentability criterion
of Lubotzky\, and draw inspiration from well-known facts about three-dime
nsional manifolds\, as in arithmetic topology. \n\nThe talk is based on
a joint work with Esnault and Srinivas\, on a joint work with Jarden\, and
on work of Yuan Liu.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Petrov (Harvard University)
DTSTART;VALUE=DATE-TIME:20210922T190000Z
DTEND;VALUE=DATE-TIME:20210922T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/37
DESCRIPTION:Title: Galois action on the pro-algebraic completion of the fundamental group<
/a>\nby Alexander Petrov (Harvard University) as part of Harvard number th
eory seminar\n\nLecture held in Room 507 in the Science Center.\n\nAbstrac
t\nGiven a variety over a number field\, its geometric etale\nfundamental
group comes equipped with an action of the Galois group. This\ninduces a G
alois action on the pro-algebraic completion of the etale\nfundamental gro
up and hence the ring of functions on that pro-algebraic\ncompletion provi
des a supply of Galois representations.\n\nIt turns out that any finite-di
mensional p-adic Galois representation\ncontained in the ring of functions
on the pro-algebraic completion of the\nfundamental group of a smooth var
iety satisfies the assumptions of the\nFontaine-Mazur conjecture: it is de
Rham at places above p and is a. e.\nunramified.\n\nConversely\, we will
show that every semi-simple representation of the\nGalois group of a numbe
r field coming from algebraic geometry (that is\,\nappearing as a subquoti
ent of the etale cohomology of an algebraic variety)\ncan be established a
s a subquotient of the ring of functions on the\npro-algebraic completion
of the fundamental group of the projective line\nwith 3 punctures.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Levent Alpöge (Harvard University)
DTSTART;VALUE=DATE-TIME:20211020T190000Z
DTEND;VALUE=DATE-TIME:20211020T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/38
DESCRIPTION:Title: Effective height bounds for odd-degree totally real points on some curv
es\nby Levent Alpöge (Harvard University) as part of Harvard number t
heory seminar\n\nLecture held in Room 507 in the Science Center.\n\nAbstra
ct\nI will give a finite-time algorithm that\, on input (g\,K\,S) with g >
0\, K a totally real number field of odd degree\, and S a finite set of p
laces of K\, outputs the finitely many g-dimensional abelian varieties A/K
which are of $\\operatorname{GL}_2$-type over K and have good reduction o
utside S.\n\nThe point of this is to effectively compute the S-integral K-
points on a Hilbert modular variety\, and the point of that is to be able
to compute all K-rational points on complete curves inside such varieties.
\n\nThis gives effective height bounds for rational points on infinitely m
any curves and (for each curve) over infinitely many number fields. For ex
ample one gets effective height points for odd-degree totally real points
on $x^6 + 4y^3 = 1$\, by using the hypergeometric family associated to the
arithmetic triangle group $\\Delta(3\,6\,6)$.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wei Zhang (MIT)
DTSTART;VALUE=DATE-TIME:20211027T190000Z
DTEND;VALUE=DATE-TIME:20211027T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/39
DESCRIPTION:Title: p-adic Heights of the arithmetic diagonal cycles\nby Wei Zhang (MIT
) as part of Harvard number theory seminar\n\nLecture held in Room 507 in
the Science Center.\n\nAbstract\nThis is a work in progress joint with Da
niel Disegni. We formulate a p-adic analogue of the Arithmetic Gan--Gross-
-Prasad conjecture for unitary groups\, relating the p-adic height pairing
of the arithmetic diagonal cycles to the first central derivative (along
the cyclotomic direction) of a p-adic Rankin—Selberg L-function associa
ted to cuspidal automorphic representations. In the good ordinary case we
are able to prove the conjecture\, at least when the ramifications are mil
d at inert primes. We deduce some application to the p-adic version of the
Bloch-Kato conjecture.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhiwei Yun (MIT)
DTSTART;VALUE=DATE-TIME:20211103T190000Z
DTEND;VALUE=DATE-TIME:20211103T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/40
DESCRIPTION:Title: Special cycles for unitary Shtukas and modularity\nby Zhiwei Yun (M
IT) as part of Harvard number theory seminar\n\nLecture held in Room 507 i
n the Science Center.\n\nAbstract\nWe define a generating series of algebr
aic cycles on the moduli\nstack of unitary Shtukas and conjecture that it
is a Chow-group valued\nautomorphic form. This is a function field analogu
e of the special cycles\ndefined by Kudla and Rapoport\, but with an extra
degree of freedom namely\nthe number of legs of the Shtukas. I will talk
about several pieces of\nevidence for the conjecture. This is joint work w
ith Tony Feng and Wei\nZhang.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Feng (MIT)
DTSTART;VALUE=DATE-TIME:20211110T200000Z
DTEND;VALUE=DATE-TIME:20211110T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/41
DESCRIPTION:Title: The Galois action on symplectic K-theory\nby Tony Feng (MIT) as par
t of Harvard number theory seminar\n\nLecture held in Room 507 in the Scie
nce Center.\n\nAbstract\nA phenomenon underlying many remarkable results i
n number theory is the natural Galois action on the cohomology of symplect
ic groups of integers. In joint work with Soren Galatius and Akshay Venkat
esh\, we define a symplectic variant of algebraic K-theory\, which carries
a natural Galois action for similar reasons. We compute this Galois actio
n and characterize it in terms of a universality property\, in the spirit
of the Langlands philosophy.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Siyan Daniel Li-Huerta (Harvard University)
DTSTART;VALUE=DATE-TIME:20211013T190000Z
DTEND;VALUE=DATE-TIME:20211013T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/42
DESCRIPTION:Title: The plectic conjecture over local fields\nby Siyan Daniel Li-Huerta
(Harvard University) as part of Harvard number theory seminar\n\nLecture
held in Room 507 in the Science Center.\n\nAbstract\nThe étale cohomology
of varieties over Q enjoys a Galois action. In the\ncase of Hilbert modul
ar varieties\, Nekovář-Scholl observed that this Galois\naction on the l
evel of cohomology extends to a much larger profinite group:\nthe plectic
group. They conjectured that this extension holds even on the\nlevel of co
mplexes\, as well as for more general Shimura varieties.\n\nWe present a p
roof of the analogue of this conjecture for local Shimura\nvarieties. This
includes (the generic fibers of) Lubin–Tate spaces\,\nDrinfeld upper ha
lf spaces\, and more generally Rapoport–Zink spaces. The\nproof cruciall
y uses Scholze's theory of diamonds.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Howard (Boston College)
DTSTART;VALUE=DATE-TIME:20211117T200000Z
DTEND;VALUE=DATE-TIME:20211117T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/43
DESCRIPTION:Title: Arithmetic volumes of unitary Shimura varieties\nby Benjamin Howard
(Boston College) as part of Harvard number theory seminar\n\nLecture held
in Room 507 in the Science Center.\n\nAbstract\nThe integral model of a G
U(n-1\,1) Shimura variety carries a natural metrized line bundle of modula
r forms. Viewing this metrized line bundle as a class in the codimension
one arithmetic Chow group\, one can define its arithmetic volume as an ite
rated self-intersection. We will show that this volume can be expressed i
n terms of logarithmic derivatives of Dirichlet L-functions at integer poi
nts\, and explain the connection with the arithmetic Siegel-Weil conjectur
e of Kudla-Rapoport. This is joint work with Jan Bruinier.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Kisin (Harvard University)
DTSTART;VALUE=DATE-TIME:20211201T200000Z
DTEND;VALUE=DATE-TIME:20211201T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/44
DESCRIPTION:Title: Mod p points on Shimura varieties\nby Mark Kisin (Harvard Universit
y) as part of Harvard number theory seminar\n\nLecture held in Room 507 in
the Science Center.\n\nAbstract\nThe study of mod p points on Shimura var
ieties was originally\nmotivated by the study of the Hasse-Weil zeta funct
ion for Shimura\nvarieties.\nIt involves some rather subtle problems which
test just how much we know\nabout motives over finite fields. In this tal
k I will explain some recent\nresults\, and\napplications\, and what still
remains conjectural.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Betts (Harvard University)
DTSTART;VALUE=DATE-TIME:20220209T200000Z
DTEND;VALUE=DATE-TIME:20220209T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/45
DESCRIPTION:Title: Galois sections and the method of Lawrence--Venkatesh\nby Alexander
Betts (Harvard University) as part of Harvard number theory seminar\n\nLe
cture held in Room 507 in the Science Center.\n\nAbstract\nGrothendieck's
Section Conjecture posits that the set of rational\npoints on a smooth pro
jective curve Y of genus at least two should be equal\nto a certain "secti
on set" defined purely in terms of the etale fundamental\ngroup of Y. In t
his talk\, I will preview some upcoming work with Jakob Stix\nin which we
prove a partial finiteness result for this section set\, thereby\ngiving a
n unconditional verification of a prediction of the Section\nConjecture fo
r a general curve Y. We do this by adapting the recent p-adic\nproof of th
e Mordell Conjecture due to Brian Lawrence and Akshay Venkatesh.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fabian Gundlach (Harvard University)
DTSTART;VALUE=DATE-TIME:20220202T200000Z
DTEND;VALUE=DATE-TIME:20220202T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/46
DESCRIPTION:Title: Counting quaternionic extensions\nby Fabian Gundlach (Harvard Unive
rsity) as part of Harvard number theory seminar\n\nLecture held in Room 50
7 in the Science Center.\n\nAbstract\nConsider the set of Galois extension
s $L$ of $\\mathbb Q$ whose Galois group is the quaternion group. For larg
e $X$\, Klüners counted extensions with $|\\mathrm{disc}(L)| <= X$. We di
scuss asymptotics when bounding invariants other than the discriminant.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Naomi Sweeting (Harvard University)
DTSTART;VALUE=DATE-TIME:20220216T200000Z
DTEND;VALUE=DATE-TIME:20220216T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/47
DESCRIPTION:Title: Kolyvagin's conjecture\, bipartite Euler systems\, and higher congruenc
es of modular forms\nby Naomi Sweeting (Harvard University) as part of
Harvard number theory seminar\n\nLecture held in Room 507 in the Science
Center.\n\nAbstract\nFor an elliptic curve E\, Kolyvagin used Heegner poi
nts to construct\nspecial Galois cohomology classes valued in the torsion
points of E. Under\nthe conjecture that not all of these classes vanish\,
he showed that they\nencode the Selmer rank of E. I will explain a proof o
f new cases of this\nconjecture that builds on prior work of Wei Zhang. Th
e proof naturally\nleads to a generalization of Kolyvagin's work in a comp
limentary "definite"\nsetting\, where Heegner points are replaced by speci
al values of a\nquaternionic modular form. I'll also explain an "ultrapatc
hing" formalism\nwhich simplifies the Selmer group arguments required for
the proof.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Landesman (Harvard University)
DTSTART;VALUE=DATE-TIME:20220223T200000Z
DTEND;VALUE=DATE-TIME:20220223T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/48
DESCRIPTION:Title: Geometric local systems on very general curves\nby Aaron Landesman
(Harvard University) as part of Harvard number theory seminar\n\nLecture h
eld in Room 507 in the Science Center.\n\nAbstract\nConjectures of Esnault
-Kerz and Budur-Wang state\nthat the locus of rank r complex local systems
on a complex variety\nof geometric origin are Zariski dense in the charac
ter variety\nparameterizing complex rank r local systems.\nIn joint work w
ith Daniel Litt\, we show these conjectures fail to hold when\nX is a suff
iciently general curve of genus $g$ and $r < 2\\sqrt{g+1}$\nby showing tha
t any such local system coming from geometry is in fact\nisotrivial.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Myrto Mavraki (Harvard University)
DTSTART;VALUE=DATE-TIME:20220302T200000Z
DTEND;VALUE=DATE-TIME:20220302T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/49
DESCRIPTION:Title: Towards uniformity in the dynamical Bogomolov conjecture\nby Myrto
Mavraki (Harvard University) as part of Harvard number theory seminar\n\nL
ecture held in Room 507 in the Science Center.\n\nAbstract\nInspired by an
analogy between torsion and preperiodic points\,\nZhang has proposed a dy
namical generalization of the classical\nManin-Mumford and Bogomolov conje
ctures. A special case of these\nconjectures\, for `split' maps\, has rece
ntly been established by Nguyen\,\nGhioca and Ye. In particular\, they sho
w that two rational maps have at most\nfinitely many common preperiodic po
ints\, unless they are `related'. Recent\nbreakthroughs by Dimitrov\, Gao\
, Habegger and Kühne have established that\nthe classical Bogomolov conje
cture holds uniformly across curves of given\ngenus.\nIn this talk we disc
uss uniform versions of the dynamical Bogomolov\nconjecture across 1-param
eter families of split maps and curves. To this\nend\, we establish instan
ces of a 'relative dynamical Bogomolov conjecture'.\nThis is joint work wi
th Harry Schmidt (University of Basel).\n
LOCATION:https://researchseminars.org/talk/HarvardNT/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Pollack (Boston University)
DTSTART;VALUE=DATE-TIME:20220427T190000Z
DTEND;VALUE=DATE-TIME:20220427T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/50
DESCRIPTION:Title: Slopes of modular forms and reductions of crystalline representations\nby Robert Pollack (Boston University) as part of Harvard number theory
seminar\n\nLecture held in Room 507 in the Science Center.\n\nAbstract\nT
he ghost conjecture predicts slopes of modular forms whose\nresidual repre
sentation is locally reducible. In this talk\, we'll examine\nlocally irr
educible representations and discuss recent progress on\nformulating a con
jecture in this case. It's a lot trickier and the story\nremains incomple
te\, but we will discuss how an irregular ghost conjecture\nis intimately
related to reductions of crystalline representations.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jennifer Balakrishnan (Boston University)
DTSTART;VALUE=DATE-TIME:20220420T190000Z
DTEND;VALUE=DATE-TIME:20220420T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/51
DESCRIPTION:Title: Quadratic Chabauty for modular curves\nby Jennifer Balakrishnan (Bo
ston University) as part of Harvard number theory seminar\n\nLecture held
in Room 507 in the Science Center.\n\nAbstract\nAbstract: We describe how
p-adic height pairings can be used to\ndetermine the set of rational point
s on curves\, in the spirit of Kim's\nnonabelian Chabauty program. In part
icular\, we discuss what aspects of\nthe quadratic Chabauty method can be
made practical for certain\nmodular curves. This is joint work with Netan
Dogra\, Steffen Mueller\,\nJan Tuitman\, and Jan Vonk.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Allechar Serrano López
DTSTART;VALUE=DATE-TIME:20220309T200000Z
DTEND;VALUE=DATE-TIME:20220309T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/52
DESCRIPTION:Title: Counting fields generated by points on plane curves\nby Allechar Se
rrano López as part of Harvard number theory seminar\n\nLecture held in R
oom 507 in the Science Center.\n\nAbstract\nFor a smooth projective curve
$C/\\mathbb{Q}$\, how many field\nextensions of $\\mathbb{Q}$ -- of given
degree and bounded discriminant ---\narise from adjoining a point of $C(\\
overline{\\mathbb{Q}})$? Can we further\ncount the number of such extensio
ns with a specified Galois group?\nAsymptotic lower bounds for these quant
ities have been found for elliptic\ncurves by Lemke Oliver and Thorne\, fo
r hyperelliptic curves by Keyes\, and\nfor superelliptic curves by Beneish
and Keyes. We discuss similar\nasymptotic lower bounds that hold for all
smooth plane curves $C$.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Will Sawin (Columbia University)
DTSTART;VALUE=DATE-TIME:20220323T190000Z
DTEND;VALUE=DATE-TIME:20220323T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/53
DESCRIPTION:Title: A visit to 3-manifolds in the quest to understand random Galois groups<
/a>\nby Will Sawin (Columbia University) as part of Harvard number theory
seminar\n\nLecture held in Room 507 in the Science Center.\n\nAbstract\nCo
hen and Lenstra gave a conjectural distribution for the class group of a r
andom quadratic number field. Since the class group is the Galois group of
the maximum abelian unramified extension\, a natural generalization would
be to give a conjecture for the distribution of the Galois group of the m
aximal unramified extension. Previous work has produced a plausible conjec
ture in special cases\, with the most general being recent work of Liu\, W
ood\, and Zurieck-Brown.\n\nThere is a deep analogy between number fields
and 3-manifolds. Thus\, an analogous question would be to describe the dis
tribution of the profinite completion of the fundamental group of a random
3-manifold. In this talk\, I will explain how Melanie Wood and I answered
this question for a model of random 3-manifolds defined by Dunfield and T
hurston\, and how the techniques we used should allow us\, in future work\
, to give a more general conjecture in the number field case.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eric Urban (Columbia University)
DTSTART;VALUE=DATE-TIME:20220413T190000Z
DTEND;VALUE=DATE-TIME:20220413T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/54
DESCRIPTION:Title: Euler systems and the p-adic Langlands correspondence\nby Eric Urba
n (Columbia University) as part of Harvard number theory seminar\n\nLectur
e held in Room 507 in the Science Center.\n\nAbstract\nAbout 2 years ago\,
I have given a new construction of the Euler system of cyclotomic units
via Eisenstein congruences in which the p-adic Langlands correspondence fo
r $\\GL_2(\\Q_p)$ plays a central role. In this talk\, I want to explain h
ow one can extend this method to obtain a large class of new Euler systems
attached to ordinary automorphic forms. This is a work in progress.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yunqing Tang (Princeton University)
DTSTART;VALUE=DATE-TIME:20220330T190000Z
DTEND;VALUE=DATE-TIME:20220330T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/55
DESCRIPTION:Title: The unbounded denominators conjecture\nby Yunqing Tang (Princeton U
niversity) as part of Harvard number theory seminar\n\nLecture held in Roo
m 507 in the Science Center.\n\nAbstract\nThe unbounded denominators conje
cture\, first raised by Atkin and Swinnerton-Dyer\, asserts that a modular
form for a finite index subgroup of $\\SL_2(\\mathbb Z)$ whose Fourier co
efficients have bounded denominators must be a modular form for some congr
uence subgroup. In this talk\, we will give a sketch of the proof of this
conjecture based on a new arithmetic algebraization theorem. This is joint
work with Frank Calegari and Vesselin Dimitrov.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matt Baker (Georgia Institute of Technology)
DTSTART;VALUE=DATE-TIME:20220504T190000Z
DTEND;VALUE=DATE-TIME:20220504T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/56
DESCRIPTION:Title: Non-archimedean and tropical geometry\, algebraic groups\, moduli space
s of matroids\, and the field with one element\nby Matt Baker (Georgia
Institute of Technology) as part of Harvard number theory seminar\n\nLect
ure held in Room 507 in the Science Center.\n\nAbstract\nI will give an in
troduction to Oliver Lorscheid’s theory of\nordered blueprints – one o
f the more successful approaches to “the field of\none element” – an
d sketch its relationship to Berkovich spaces\, tropical\ngeometry\, Tits
models for algebraic groups\, and moduli spaces of matroids.\nThe basic id
ea for the latter two applications is quite simple: given a\nscheme over <
b>Z defined by equations with coefficients in {0\,1\,-1}\, there\nis a
corresponding “blue model” whose **K**-points (where **K** is t
he Krasner\nhyperfield) sometimes correspond to interesting combinatorial
structures.\nFor example\, taking **K**-points of a suitable blue model
for a split\nreductive group scheme G over **Z** gives the Weyl group
of G\, and\ntaking **K**-points\nof a suitable blue model for the Grass
mannian G(r\,n) gives the set of\nmatroids of rank r on {1\,…\,n}. Simil
arly\, the Berkovich analytification of\na scheme X over a valued field K
coincides\, as a topological space\, with\nthe set of **T**-points of X
\, considered as an ordered blue scheme over K.\nHere **T** is the trop
ical hyperfield\, and **T**-points are defined using the\nobservation t
hat a (height 1) valuation on K is nothing other than a\nhomomorphism to <
b>T.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bianca Viray (University of Washington)
DTSTART;VALUE=DATE-TIME:20220406T190000Z
DTEND;VALUE=DATE-TIME:20220406T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/57
DESCRIPTION:Title: Isolated points on modular curves\nby Bianca Viray (University of W
ashington) as part of Harvard number theory seminar\n\nLecture held in Roo
m 507 in the Science Center.\n\nAbstract\nLet C be an algebraic curve over
a number field. Faltings's theorem on\nrational points on subvarieties of
abelian varieties implies that all\nalgebraic points on C arise in algebr
aic families\, with finitely many\nexceptions. These exceptions are known
as isolated points. We study how\nisolated points behave under morphisms
and then specialize to the case of\nmodular curves. We show that isolated
points on X_1(n) push down to\nisolated points on a modular curve whose l
evel is bounded by a constant\nthat depends only on the j-invariant of the
isolated point. This is joint\nwork with A. Bourdon\, O. Ejder\, Y. Liu\
, and F. Odumodu.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akshay Venkatesh (IAS)
DTSTART;VALUE=DATE-TIME:20220914T190000Z
DTEND;VALUE=DATE-TIME:20220914T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/60
DESCRIPTION:Title: Symplectic Reidemeister torsion and symplectic $L$-functions\nby Ak
shay Venkatesh (IAS) as part of Harvard number theory seminar\n\nLecture h
eld in Room 507 in the Science Center.\n\nAbstract\nMany of the quantities
appearing in the conjecture of Birch and Swinnerton-Dyer look suspiciousl
y like squares. Motivated by this and related examples\, we may ask if the
central value of an $L$-function "of symplectic type" admits a preferred
square root.\n\nThe answer is no: there's an interesting cohomological obs
truction. More formally\, in the everywhere unramified situation over a fu
nction field\, I will describe an explicit cohomological formula for the $
L$-function modulo squares. This is based on a purely topological result a
bout $3$-manifolds. If time permits I'll speculate on generalizations. Thi
s is based on joint work with Amina Abdurrahman.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Koymans (University of Michigan)
DTSTART;VALUE=DATE-TIME:20220921T190000Z
DTEND;VALUE=DATE-TIME:20220921T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/61
DESCRIPTION:Title: The negative Pell equation and applications\nby Peter Koymans (Univ
ersity of Michigan) as part of Harvard number theory seminar\n\nLecture he
ld in Room 507 in the Science Center.\n\nAbstract\nIn this talk we will st
udy the negative Pell equation\, which is the conic $C_D : x^2 - D y^2 =
-1$ to be solved in integers $x\, y \\in \\mathbb{Z}$. We shall be concern
ed with the following question: as we vary over squarefree integers $D$\,
how often is $C_D$ soluble? Stevenhagen conjectured an asymptotic formula
for such $D$. Fouvry and Klüners gave upper and lower bounds of the corre
ct order of magnitude. We will discuss a proof of Stevenhagen's conjecture
\, and potential applications of the new proof techniques. This is joint w
ork with Carlo Pagano.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ziquan Yang (UW Madison)
DTSTART;VALUE=DATE-TIME:20220928T190000Z
DTEND;VALUE=DATE-TIME:20220928T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/62
DESCRIPTION:Title: The Tate conjecture for $h^{2\, 0} = 1$ varieties over finite fields\nby Ziquan Yang (UW Madison) as part of Harvard number theory seminar\n\
nLecture held in Room 507 in the Science Center.\n\nAbstract\nThe past dec
ade has witnessed a great advancement on the Tate conjecture for varieties
with Hodge number $h^{2\, 0} = 1$. Charles\, Madapusi-Pera and Maulik com
pletely settled the conjecture for K3 surfaces over finite fields\, and Mo
onen proved the Mumford-Tate (and hence also Tate) conjecture for more or
less arbitrary $h^{2\, 0} = 1$ varieties in characteristic $0$.\n\nIn this
talk\, I will explain that the Tate conjecture is true for mod $p$ reduct
ions of complex projective $h^{2\, 0} = 1$ varieties when $p$ is big enoug
h\, under a mild assumption on moduli. By refining this general result\, w
e prove that in characteristic $p$ at least $5$ the BSD conjecture holds f
or a height $1$ elliptic curve $E$ over a function field of genus $1$\, as
long as $E$ is subject to the generic condition that all singular fibers
in its minimal compactification are irreducible. We also prove the Tate co
njecture over finite fields for a class of surfaces of general type and a
class of Fano varieties. The overall philosophy is that the connection bet
ween the Tate conjecture over finite fields and the Lefschetz $(1\, 1)$-th
eorem over the complex numbers is very robust for $h^{2\, 0} = 1$ varietie
s\, and works well beyond the hyperkähler world.\n\nThis is based on join
t work with Paul Hamacher and Xiaolei Zhao.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Li-Huerta (Harvard)
DTSTART;VALUE=DATE-TIME:20221005T190000Z
DTEND;VALUE=DATE-TIME:20221005T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/63
DESCRIPTION:Title: Local-global compatibility over function fields\nby Daniel Li-Huert
a (Harvard) as part of Harvard number theory seminar\n\nLecture held in Ro
om 507 in the Science Center.\n\nAbstract\nThe Langlands program predicts
a relationship between automorphic representations of a reductive group $G
$ and Galois representations valued in its $L$-group. For general $G$ over
a global function field\, the automorphic-to-Galois direction has been co
nstructed by V. Lafforgue. More recently\, for general $G$ over a nonarchi
medean local field\, a similar correspondence has been constructed by Farg
ues–Scholze.\n\nWe present a proof that the V. Lafforgue and Fargues–S
cholze correspondences are compatible\, generalizing local-global compatib
ility from class field theory. As a consequence\, the correspondences of G
enestier–Lafforgue and Fargues–Scholze agree\, which answers a questio
n of Fargues–Scholze\, Hansen\, Harris\, and Kaletha.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hélène Esnault (Freie Universität Berlin)
DTSTART;VALUE=DATE-TIME:20221012T190000Z
DTEND;VALUE=DATE-TIME:20221012T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/64
DESCRIPTION:Title: Integrality properties of the Betti moduli space\nby Hélène Esnau
lt (Freie Universität Berlin) as part of Harvard number theory seminar\n\
nLecture held in Room 507 in the Science Center.\n\nAbstract\nWe study the
m\, in particular showing on a smooth complex quasi-projective variety the
existence of $\\ell$-adic absolutely irreducible local systems for all $
\\ell$ the moment there is a complex irreducible topological local system
. The proof is purely arithmetic.\n\nThis is work in progress with Johan d
e Jong\, relying in part on earlier work with Michael Groechenig.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jef Laga (Princeton)
DTSTART;VALUE=DATE-TIME:20221019T190000Z
DTEND;VALUE=DATE-TIME:20221019T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/65
DESCRIPTION:Title: Arithmetic statistics via graded Lie algebras\nby Jef Laga (Princet
on) as part of Harvard number theory seminar\n\nLecture held in Room 507 i
n the Science Center.\n\nAbstract\nI will explain how various results in a
rithmetic statistics by Bhargava\, Gross\, Shankar and others on $2$-Selme
r groups of Jacobians of (hyper)elliptic curves can be organised and repro
ved using the theory of graded Lie algebras\, following earlier work of Th
orne. This gives a uniform proof of these results and yields new theorems
for certain families of non-hyperelliptic curves. I will also mention some
applications to rational points on certain families of curves.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shai Haran (Technion)
DTSTART;VALUE=DATE-TIME:20221026T190000Z
DTEND;VALUE=DATE-TIME:20221026T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/66
DESCRIPTION:Title: Non additive geometry and Frobenius correspondences\nby Shai Haran
(Technion) as part of Harvard number theory seminar\n\nLecture held in Roo
m 507 in the Science Center.\n\nAbstract\nThe usual language of algebraic
geometry is not appropriate for arithmetical geometry: addition is singula
r at the real prime. We developed two languages that overcome this problem
: one replace s rings by the collection of “vectors” or by bi-operads\
, and another based on “matrices” or props. Once one understands the d
elicate commutativity condition one can proceed following Grothendieck's f
ootsteps exactly. The props\, when viewed up to conjugation\, give us new
commutative rings with Frobenius endomorphisms.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Spencer Leslie (Boston College)
DTSTART;VALUE=DATE-TIME:20221102T190000Z
DTEND;VALUE=DATE-TIME:20221102T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/67
DESCRIPTION:Title: Endoscopy for symmetric varieties\nby Spencer Leslie (Boston Colleg
e) as part of Harvard number theory seminar\n\nLecture held in Room 507 in
the Science Center.\n\nAbstract\nRelative trace formulas are central tool
s in the study of relative functoriality. In many cases of interest\, basi
c stability problems have not previously been addressed. In this talk\, I
discuss a theory of endoscopy in the context of symmetric varieties with t
he global goal of stabilizing the associated relative trace formula. I out
line how\, using the dual group of the symmetric variety\, one can give a
good notion of endoscopic symmetric variety and conjecture a matching of r
elative orbital integrals in order to stabilize the relative trace formula
\, which can be proved in some cases. Time permitting\, I will explain my
proof of these conjectures in the case of unitary Friedberg–Jacquet peri
ods.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gyujin Oh (Columbia)
DTSTART;VALUE=DATE-TIME:20221109T200000Z
DTEND;VALUE=DATE-TIME:20221109T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/68
DESCRIPTION:Title: Cohomological degree-shifting operators on Shimura varieties\nby Gy
ujin Oh (Columbia) as part of Harvard number theory seminar\n\nLecture hel
d in Room 507 in the Science Center.\n\nAbstract\nAn automorphic form can
appear in multiple degrees of the cohomology of arithmetic manifolds\, and
this happens mostly when the arithmetic manifolds are not algebraic. This
phenomenon is a part of the "derived" structures of the Langlands program
\, suggested by Venkatesh. However\, even over algebraic arithmetic manifo
lds\, certain automorphic forms like weight-one elliptic modular forms pos
sess a derived structure. In this talk\, we discuss this idea over Shimura
varieties. A part of the story is the construction of archimedean/p-adic
"derived" operators on the cohomology of Shimura varieties\, using complex
/p-adic Hodge theory.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tasho Kaletha (University of Michigan)
DTSTART;VALUE=DATE-TIME:20221116T200000Z
DTEND;VALUE=DATE-TIME:20221116T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/69
DESCRIPTION:Title: Covers of reductive groups and functoriality\nby Tasho Kaletha (Uni
versity of Michigan) as part of Harvard number theory seminar\n\nLecture h
eld in Room 507 in the Science Center.\n\nAbstract\nTo a connected reducti
ve group $G$ over a local field $F$ we define a compact topological group
$\\tilde\\pi_1(G)$ and an extension $G(F)_\\infty$ of $G(F)$ by $\\tilde\\
pi_1(G)$. From any character $x$ of $\\tilde\\pi_1(G)$ of order $n$ we obt
ain an $n$-fold cover $G(F)_x$ of the topological group $G(F)$. We also de
fine an $L$-group for $G(F)_x$\, which is a usually non-split extension of
the Galois group by the dual group of G\, and deduce from the linear case
a refined local Langlands correspondence between genuine representations
of $G(F)_x$ and $L$-parameters valued in this $L$-group.\n\nThis construct
ion is motivated by Langlands functoriality. We show that a subgroup of th
e $L$-group of $G$ of a certain kind naturally lead to a smaller quasi-spl
it group $H$ and a double cover of $H(F)$. Genuine representations of this
double cover are expected to be in functorial relationship with represent
ations of $G(F)$. We will present two concrete applications of this\, one
that gives a characterization of the local Langlands correspondence for su
percuspidal $L$-parameters when $p$ is sufficiently large\, and one to the
theory of endoscopy.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abhishek Oswal (Caltech)
DTSTART;VALUE=DATE-TIME:20221130T200000Z
DTEND;VALUE=DATE-TIME:20221130T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/70
DESCRIPTION:Title: A $p$-adic analogue of an algebraization theorem of Borel\nby Abhis
hek Oswal (Caltech) as part of Harvard number theory seminar\n\nLecture he
ld in Room 507 in the Science Center.\n\nAbstract\nLet $S$ be a Shimura va
riety such that the connected components of the set of complex points $S(\
\mathbb{C})$ are of the form $D/\\Gamma$\, where $\\Gamma$ is a torsion-fr
ee arithmetic group acting on the Hermitian symmetric domain $D$. Borel pr
oved that any holomorphic map from any complex algebraic variety into $S(\
\mathbb{C})$ is an algebraic map. In this talk I shall describe ongoing jo
int work with Ananth Shankar and Xinwen Zhu\, where we prove a $p$-adic an
alogue of this result of Borel for compact Shimura varieties of abelian ty
pe.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Pagano (Concordia)
DTSTART;VALUE=DATE-TIME:20221207T200000Z
DTEND;VALUE=DATE-TIME:20221207T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/71
DESCRIPTION:Title: Malle's conjecture for nilpotent groups\nby Carlo Pagano (Concordia
) as part of Harvard number theory seminar\n\nLecture held in Room 507 in
the Science Center.\n\nAbstract\nMalle's conjecture is a quantitative vers
ion of the Galois inverse problem. Namely\, fixing some ramification invar
iant of number fields (discriminant\, product of ramified primes\, etc)\,
for a finite group $G$ one seeks an asymptotic formula for the number of $
G$-extensions (of a given number field) having bounded ramification invari
ant. In this talk I will overview past and ongoing joint work with Peter K
oymans focusing on the case of nilpotent groups.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ari Shnidman (Hebrew University of Jerusalem)
DTSTART;VALUE=DATE-TIME:20230201T200000Z
DTEND;VALUE=DATE-TIME:20230201T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/72
DESCRIPTION:Title: Bielliptic Picard curves\nby Ari Shnidman (Hebrew University of Jer
usalem) as part of Harvard number theory seminar\n\nLecture held in Room 5
07 in the Science Center.\n\nAbstract\nI'll describe the geometry and arit
hmetic of the curves $y^3 = x^4 + ax^2 + b$. The Jacobians of these curves
factor as a product of an elliptic curve and an abelian surface $A$. The
latter is an example of a "false elliptic curve"\, i.e. an abelian surface
with quaternionic multiplication. I'll explain how to see this from the
geometry of the curve\, and then I'll give some results on the Mordell–W
eil groups $A(\\mathbb{Q})$. This is based on joint work with Laga and Lag
a–Schembri–Voight.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Weinstein (Boston University)
DTSTART;VALUE=DATE-TIME:20230208T200000Z
DTEND;VALUE=DATE-TIME:20230208T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/73
DESCRIPTION:Title: Higher modularity of elliptic curves\nby Jared Weinstein (Boston Un
iversity) as part of Harvard number theory seminar\n\nLecture held in Room
507 in the Science Center.\n\nAbstract\nElliptic curves $E$ over the rati
onal numbers are modular: this means there is a nonconstant map from a mod
ular curve to $E$. When instead the coefficients of $E$ belong to a functi
on field\, it still makes sense to talk about the modularity of $E$ (and t
his is known)\, but one can also extend the idea further and ask whether $
E$ is '$r$-modular' for $r=2\,3\\ldots$. To define this generalization\, t
he modular curve gets replaced with Drinfeld's concept of a 'shtuka space'
. The $r$-modularity of $E$ is predicted by Tate's conjecture. In joint wo
rk with Adam Logan\, we give some classes of elliptic curves $E$ which are
$2$- and $3$-modular.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Levent Alpöge (Harvard)
DTSTART;VALUE=DATE-TIME:20230215T200000Z
DTEND;VALUE=DATE-TIME:20230215T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/74
DESCRIPTION:Title: Integers which are(n’t) the sum of two cubes\nby Levent Alpöge (
Harvard) as part of Harvard number theory seminar\n\nLecture held in Room
507 in the Science Center.\n\nAbstract\nFermat identified the integers whi
ch are a sum of two squares\, integral or rational: they are exactly those
integers which have all primes congruent to 3 (mod 4) occurring to an eve
n power in their prime factorization — a condition satisfied by 0% of in
tegers!\n\nWhat about the integers which are a sum of two cubes? 0% are a
sum of two integral cubes\, but...\n\nMain Theorem:\n\n1. A positive propo
rtion of integers aren’t the sum of two rational cubes\,\n\n2. and also
a positive proportion are!\n\n(Joint with Manjul Bhargava and Ari Shnidman
.)\n
LOCATION:https://researchseminars.org/talk/HarvardNT/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yujie Xu (MIT)
DTSTART;VALUE=DATE-TIME:20230222T200000Z
DTEND;VALUE=DATE-TIME:20230222T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/75
DESCRIPTION:Title: Hecke algebras for $p$-adic groups and the explicit local Langlands cor
respondence for $\\mathrm{G}_2$\nby Yujie Xu (MIT) as part of Harvard
number theory seminar\n\nLecture held in Room 507 in the Science Center.\n
\nAbstract\nI will talk about my recent joint work with Aubert where we pr
ove the local Langlands conjecture for $\\mathrm{G}_2$ (explicitly). This
uses our earlier results on Hecke algebras attached to Bernstein component
s of reductive $p$-adic groups\, as well as an expected property on cuspid
al support\, along with a list of characterizing properties. In particular
\, we obtain "mixed" $L$-packets containing $F$-singular supercuspidals an
d non-supercuspidals.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashvin Swaminathan (Harvard)
DTSTART;VALUE=DATE-TIME:20230301T200000Z
DTEND;VALUE=DATE-TIME:20230301T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/76
DESCRIPTION:Title: Counting integral points on symmetric varieties\, and applications to a
rithmetic statistics\nby Ashvin Swaminathan (Harvard) as part of Harva
rd number theory seminar\n\nLecture held in Room 507 in the Science Center
.\n\nAbstract\nOver the past few decades\, significant progress has been m
ade in arithmetic statistics by the following two-step process: (1) parame
trize arithmetic objects of interest in terms of the integral orbits of a
representation of a group $G$ acting on a vector space $V$\; and (2) use g
eometry-of-numbers methods to count the orbits of $G(\\mathbb{Z})$ on $V(\
\mathbb{Z})$. But it often happens that the arithmetic objects of interest
correspond to orbits that lie on a proper subvariety of $V$. In such case
s\, geometry-of-numbers methods do not suffice to obtain precise asymptoti
cs\, and more sophisticated point-counting techniques are required. In thi
s talk\, we explain how the Eskin–McMullen method for counting integral
points on symmetric varieties can be used to study the distribution of $2$
-class groups in certain thin families of cubic number fields.\n\n(Joint w
ith Iman Setayesh\, Arul Shankar\, and Artane Siad)\n
LOCATION:https://researchseminars.org/talk/HarvardNT/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Shusterman (Harvard)
DTSTART;VALUE=DATE-TIME:20230308T200000Z
DTEND;VALUE=DATE-TIME:20230308T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/77
DESCRIPTION:Title: The different of a branched cover of $3$-manifolds is a square\nby
Mark Shusterman (Harvard) as part of Harvard number theory seminar\n\nLect
ure held in Room 507 in the Science Center.\n\nAbstract\nHecke has shown t
hat the different ideal of a number field is a square in the class group.
In joint work with Will Sawin we obtain an analogous result for closed $3$
-manifolds saying that the branch divisor of a covering is a square in the
first homology group.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lillian Pierce (Duke)
DTSTART;VALUE=DATE-TIME:20230322T190000Z
DTEND;VALUE=DATE-TIME:20230322T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/78
DESCRIPTION:Title: A polynomial sieve: beyond separation of variables\nby Lillian Pier
ce (Duke) as part of Harvard number theory seminar\n\nLecture held in Room
507 in the Science Center.\n\nAbstract\nMany problems in number theory ca
n be framed as questions about counting solutions to a Diophantine equatio
n (say\, within a certain “box”). If there are very few\, or very many
variables\, certain methods gain an advantage\, but sometimes there is ex
tra structure that can be exploited as well. For example: let $f$ be a giv
en polynomial with integer coefficients in $n$ variables. How many values
of $f$ are a perfect square? A perfect cube? Or\, more generally\, a value
of a different polynomial of interest\, say $g(y)$? These questions arise
in a variety of specific applications\, and also in the context of a gene
ral conjecture of Serre on counting points in thin sets. We will describe
how sieve methods can exploit this type of structure\, and explain how a n
ew polynomial sieve method allows greater flexibility\, so that the variab
les in the polynomials $f$ and $g$ can “mix.” This is joint work with
Dante Bonolis.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wei Ho (Princeton / IAS)
DTSTART;VALUE=DATE-TIME:20230329T190000Z
DTEND;VALUE=DATE-TIME:20230329T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/79
DESCRIPTION:Title: Selmer averages in families of elliptic curves and applications\nby
Wei Ho (Princeton / IAS) as part of Harvard number theory seminar\n\nLect
ure held in Room 507 in the Science Center.\n\nAbstract\nOrbits of many co
regular representations of algebraic groups are closely linked to moduli s
paces of genus one curves with extra data. We may use these orbit parametr
izations to compute the average size of Selmer groups of elliptic curves i
n certain families\, e.g.\, with marked points\, thus obtaining upper boun
ds for the average ranks of the elliptic curves in these families. (This i
s joint work with Manjul Bhargava.) We will also describe some other appli
cations and related work (some joint with collaborators\, including Levent
Alpöge\, Manjul Bhargava\, Tom Fisher\, Jennifer Park).\n
LOCATION:https://researchseminars.org/talk/HarvardNT/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rachel Newton (King's College London)
DTSTART;VALUE=DATE-TIME:20230405T190000Z
DTEND;VALUE=DATE-TIME:20230405T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/80
DESCRIPTION:Title: Evaluating the wild Brauer group\nby Rachel Newton (King's College
London) as part of Harvard number theory seminar\n\nLecture held in Room 5
07 in the Science Center.\n\nAbstract\nThe local-global approach to the st
udy of rational points on varieties over number fields begins by embedding
the set of rational points on a variety $X$ into the set of its adelic po
ints. The Brauer–Manin pairing cuts out a subset of the adelic points\,
called the Brauer–Manin set\, that contains the rational points. If the
set of adelic points is non-empty but the Brauer–Manin set is empty then
we say there's a Brauer–Manin obstruction to the existence of rational
points on $X$. Computing the Brauer–Manin pairing involves evaluating el
ements of the Brauer group of $X$ at local points. If an element of the Br
auer group has order coprime to $p$\, then its evaluation at a $p$-adic po
int factors via reduction of the point modulo $p$. For elements of order a
power of $p$\, this is no longer the case: in order to compute the evalua
tion map one must know the point to a higher $p$-adic precision. Classifyi
ng Brauer group elements according to the precision required to evaluate t
hem at $p$-adic points gives a filtration which we describe using work of
Kato. Applications of our work include addressing Swinnerton-Dyer's questi
on about which places can play a role in the Brauer–Manin obstruction. T
his is joint work with Martin Bright.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Harris (Columbia)
DTSTART;VALUE=DATE-TIME:20230412T190000Z
DTEND;VALUE=DATE-TIME:20230412T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/81
DESCRIPTION:Title: Square root $p$-adic $L$-functions\nby Michael Harris (Columbia) as
part of Harvard number theory seminar\n\nLecture held in Room 507 in the
Science Center.\n\nAbstract\nThe Ichino–Ikeda conjecture\, and its gener
alization to unitary groups by N. Harris\, gives explicit formulas for cen
tral critical values of a large class of Rankin–Selberg tensor products.
The version for unitary groups is now a theorem\, and expresses the centr
al critical value of $L$-functions of the form $L(s\,\\Pi \\times \\Pi')$
in terms of squares of automorphic periods on unitary groups. Here $\\Pi
\\times \\Pi'$ is an automorphic representation of $\\mathrm{GL}(n\,F)\\ti
mes\\mathrm{GL}(n-1\,F)$ that descends to an automorphic representation of
$\\mathrm{U}(V) \\times \\mathrm{U}(V')$\, where $V$ and $V'$ are hermiti
an spaces over $F$\, with respect to a Galois involution $c$ of $F$\, of d
imension $n$ and $n-1$\, respectively.\n\nI will report on the constructio
n of a $p$-adic interpolation of the automorphic period — in other words
\, of the square root of the central values of the $L$-functions — when
$\\Pi'$ varies in a Hida family. The construction is based on the theory o
f $p$-adic differential operators due to Eischen\, Fintzen\, Mantovan\, an
d Varma. Most aspects of the construction should generalize to higher Hida
theory. I will explain the archimedean theory of the expected generalizat
ion\, which is the subject of work in progress with Speh and Kobayashi.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Keerthi Madapusi (Boston College)
DTSTART;VALUE=DATE-TIME:20230419T190000Z
DTEND;VALUE=DATE-TIME:20230419T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/82
DESCRIPTION:Title: Derived cycles on Shimura varieties\nby Keerthi Madapusi (Boston Co
llege) as part of Harvard number theory seminar\n\nLecture held in Room 50
7 in the Science Center.\n\nAbstract\nI will show how methods from derived
algebraic geometry can be used to give a uniform definition of generating
series of cycles on integral models of Shimura varieties of Hodge or even
abelian type. Following conjectures of Kudla\, these series are expected
to converge to half-integer weight automorphic forms on split unitary grou
ps\, and certain ‘easy’ consequences of this expectation turn out to b
e indeed easy given the derived perspective.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tomer Schlank (Hebrew University of Jerusalem)
DTSTART;VALUE=DATE-TIME:20230426T190000Z
DTEND;VALUE=DATE-TIME:20230426T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/83
DESCRIPTION:Title: Knots Invariants and Arithmetic Statistics\nby Tomer Schlank (Hebre
w University of Jerusalem) as part of Harvard number theory seminar\n\nLec
ture held in Room 507 in the Science Center.\n\nAbstract\nThe Grothendieck
school introduced étale topology to attach algebraic-topological invaria
nts such as cohomology to varieties and schemes. Although the original mot
ivations came from studying varieties over fields\, interesting phenomena
such as Artin–Verdier duality also arise when considering the spectra of
integer rings in number fields and related schemes. A deep insight\, due
to B. Mazur\, is that through the lens of étale topology\, spectra of int
eger rings behave as $3$-dimensional manifolds while prime ideals correspo
nd to knots in these manifolds. This knots and primes analogy provides a d
ictionary between knot theory and number theory\, giving some surprising a
nalogies. For example\, this theory relates the linking number to the Lege
ndre symbol and the Alexander polynomial to Iwasawa theory. In this talk\
, we shall start by describing some of the classical ideas in this theory.
I shall then proceed by describing how via this theory\, giving a random
model on knots and links can be used to predict the statistical behavior o
f arithmetic functions. This is joint work with Ariel Davis.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bjorn Poonen (MIT)
DTSTART;VALUE=DATE-TIME:20231018T190000Z
DTEND;VALUE=DATE-TIME:20231018T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/84
DESCRIPTION:Title: Integral points on curves via Baker's method and finite étale covers\nby Bjorn Poonen (MIT) as part of Harvard number theory seminar\n\nLect
ure held in Science Center Room 507.\n\nAbstract\nWe prove results in the
direction of showing that for some affine\ncurves\, Baker's method applied
to finite étale covers is insufficient to\ndetermine the integral points
. This is joint work with Aaron Landesman.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Zureick-Brown (Amherst College)
DTSTART;VALUE=DATE-TIME:20231108T200000Z
DTEND;VALUE=DATE-TIME:20231108T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/85
DESCRIPTION:Title: $\\ell$-adic images of Galois for elliptic curves over $\\mathbb{Q}$\nby David Zureick-Brown (Amherst College) as part of Harvard number theo
ry seminar\n\nLecture held in Science Center Room 507.\n\nAbstract\nI will
discuss recent joint work with Jeremy Rouse and Drew Sutherland on Mazur
’s “Program B” — the classification of the possible “images of G
alois” associated to an elliptic curve (equivalently\, classification of
all rational points on certain modular curves $X_H$). The main result is
a provisional classification of the possible images of $\\ell$-adic Galois
representations associated to elliptic curves over $\\mathbb{Q}$ and is p
rovably complete barring the existence of unexpected rational points on mo
dular curves associated to the normalizers of non-split Cartan subgroups a
nd two additional genus 9 modular curves of level 49.\n\nI will also discu
ss the framework and various applications (for example: a very fast algori
thm to rigorously compute the $\\ell$-adic image of Galois of an elliptic
curve over $\\mathbb{Q}$)\, and then highlight several new ideas from the
joint work\, including techniques for computing models of modular curves a
nd novel arguments to determine their rational points\, a computational ap
proach that works directly with moduli and bypasses defining equations\, a
nd (with John Voight) a generalization of Kolyvagin’s theorem to the mod
ular curves we study.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Drew Sutherland (MIT)
DTSTART;VALUE=DATE-TIME:20231206T200000Z
DTEND;VALUE=DATE-TIME:20231206T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/86
DESCRIPTION:Title: L-functions from nothing\nby Drew Sutherland (MIT) as part of Harva
rd number theory seminar\n\nLecture held in Science Center Room 507.\n\nAb
stract\nI will report on joint work in progress with Andrew Booker on\nthe
practical implementation of an axiomatic approach to the enumeration\nof
arithmetic $L$-functions that lie in a certain subset of the Selberg\nclas
s that is expected to include all $L$-functions of abelian varieties.\nAs
in the work of Farmer\, Koutsoliotas\, and Lemurell\, our approach is\nbas
ed on the approximate functional equation. We obtain additional\nconstrai
nts by considering twists (and more general Rankin-Selberg\nconvolutions)
of our unknown $L$-function that yield a system of linear\nconstraints tha
t can be solved using the simplex method. This allows us\nto significantl
y extend the range of our computations for the family of\n$L$-functions as
sociated to abelian surfaces over $\\mathbb{Q}$. We also introduce a\nmet
hod for certifying the completeness of our enumeration.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Padmavathi Srinivasan (ICERM)
DTSTART;VALUE=DATE-TIME:20231025T190000Z
DTEND;VALUE=DATE-TIME:20231025T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/87
DESCRIPTION:Title: Towards a unified theory of canonical heights on abelian varieties\
nby Padmavathi Srinivasan (ICERM) as part of Harvard number theory seminar
\n\nLecture held in Science Center Room 507.\n\nAbstract\n$p$-adic heights
have been a rich source of explicit functions vanishing on rational point
s on a curve. In this talk\, we will outline a new construction of canonic
al $p$-adic heights on abelian varieties from $p$-adic adelic metrics\, us
ing $p$-adic Arakelov theory developed by Besser. This construction closel
y mirrors Zhang's construction of canonical real valued heights from real-
valued adelic metrics. We will use this new construction to give direct ex
planations (avoiding $p$-adic Hodge theory) of the key properties of heigh
t pairings needed for the quadratic Chabauty method for rational points. T
his is joint work with Amnon Besser and Steffen Mueller.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ananth Shankar (Northwestern University)
DTSTART;VALUE=DATE-TIME:20231115T200000Z
DTEND;VALUE=DATE-TIME:20231115T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/88
DESCRIPTION:Title: Semisimplicity and CM lifts\nby Ananth Shankar (Northwestern Univer
sity) as part of Harvard number theory seminar\n\nLecture held in Science
Center Room 507.\n\nAbstract\nConsider the setting of a smooth variety $S$
over $\\mathbb{F}_q$\, and an $\\ell$-adic local on $S$ which has finite
determinant and is geometrically irreducible. Work of Lafforgue proves tha
t such a local system must be pure\, and it is conjectured that the action
of Frobenius at closed points is semisimple. I will sketch a proof of thi
s conjecture in the setting of mod $p$ Shimura varieties\, and will deduce
applications to the existence of CM lifts of certain mod p points. If tim
e permits\, I will also address the question of integral canonical models
of Shimura varieties.\nThis is joint work with Ben Bakker and Jacob Tsimer
man.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tom Weston (UMass Amherst)
DTSTART;VALUE=DATE-TIME:20230927T190000Z
DTEND;VALUE=DATE-TIME:20230927T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/89
DESCRIPTION:Title: Diophantine Stability for Elliptic Curves\nby Tom Weston (UMass Amh
erst) as part of Harvard number theory seminar\n\nLecture held in Science
Center Room 507.\n\nAbstract\nWe prove\, for any prime $l$ greater than or
equal to 5\, that a density one set of rational elliptic curves are $l$-D
iophantine stable in the sense of Mazur and Rubin. This is joint work wit
h Anwesh Ray.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jun Yang (Harvard University)
DTSTART;VALUE=DATE-TIME:20231101T190000Z
DTEND;VALUE=DATE-TIME:20231101T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/90
DESCRIPTION:Title: The limit multiplicities and von Neumann dimensions\nby Jun Yang (H
arvard University) as part of Harvard number theory seminar\n\nLecture hel
d in Science Center Room 507.\n\nAbstract\nGiven an arithmetic subgroup $\
\Gamma$ in a semi-simple Lie group $G$\, the multiplicity of an irreducibl
e representation of $G$ in $L^2(\\Gamma\\backslash G)$ is unknown in gener
al.\nWe observe the multiplicity of any discrete series representation $\\
pi$ of $\\rm{SL}(2\,\\mathbb{R})$ in $L^2(\\Gamma(n)\\backslash \\rm{SL}(2
\,\\mathbb{R}))$ is close to the von Neumann dimension of $\\pi$ over the
group algebra of $\\Gamma(n)$.\nWe extend this result to other Lie groups
and bounded families of irreducible representations of them.\nBy applying
the trace formulas\, we show the multiplicities are exactly the von Neuman
n dimensions if we take certain towers of descending lattices in some Lie
groups.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Padmavathi Srinivasan (ICERM)
DTSTART;VALUE=DATE-TIME:20230913T190000Z
DTEND;VALUE=DATE-TIME:20230913T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/91
DESCRIPTION:Title: Towards a unified theory of canonical heights on abelian varieties\
nby Padmavathi Srinivasan (ICERM) as part of Harvard number theory seminar
\n\nLecture held in Science Center Room 507.\n\nAbstract\n$p$-adic heights
have been a rich source of explicit functions vanishing on rational point
s on a curve. In this talk\, we will outline a new construction of canonic
al $p$-adic heights on abelian varieties from $p$-adic adelic metrics\, us
ing $p$-adic Arakelov theory developed by Besser. This construction closel
y mirrors Zhang's construction of canonical real valued heights from real-
valued adelic metrics. We will use this new construction to give direct ex
planations (avoiding $p$-adic Hodge theory) of the key properties of heigh
t pairings needed for the quadratic Chabauty method for rational points. T
his is joint work with Amnon Besser and Steffen Mueller.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Lemke Oliver (Tufts University)
DTSTART;VALUE=DATE-TIME:20231129T200000Z
DTEND;VALUE=DATE-TIME:20231129T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/92
DESCRIPTION:Title: Faithful induction theorems and the Chebotarev density theorem\nby
Robert Lemke Oliver (Tufts University) as part of Harvard number theory se
minar\n\nLecture held in Science Center Room 507.\n\nAbstract\nThe Chebota
rev density theorem is a powerful and ubiquitous tool in number theory use
d to guarantee the existence of infinitely many primes satisfying splittin
g conditions in a Galois extension of number fields. In many applications
\, however\, it is necessary to know not just that there are many such pri
mes in the limit\, but to know that there are many such primes up to a giv
en finite point. This is the domain of so-called effective Chebotarev den
sity theorems. In forthcoming joint work with Alex Smith that extends pre
vious joint work of the author with Thorner and Zaman and earlier work of
Pierce\, Turnage-Butterbaugh\, and Wood\, we prove that in any family of i
rreducible complex Artin representations\, almost all are subject to a ver
y strong effective prime number theorem. This implies that almost all num
ber fields with a fixed Galois group are subject to a similarly strong eff
ective form of the Chebotarev density theorem. Under the hood\, the key r
esult is a new theorem in the character theory of finite groups that is si
milar in spirit to classical work of Artin and Brauer on inductions of one
-dimensional characters.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robin Zhang (MIT)
DTSTART;VALUE=DATE-TIME:20230920T190000Z
DTEND;VALUE=DATE-TIME:20230920T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/93
DESCRIPTION:Title: Harris–Venkatesh plus Stark\nby Robin Zhang (MIT) as part of Harv
ard number theory seminar\n\nLecture held in Science Center Room 507.\n\nA
bstract\nThe class number formula describes the behavior of the Dedekind z
eta function at $s=0$ and $s=1$. The Stark conjecture extends the class nu
mber formula\, describing the behavior of Artin $L$-functions and $p$-adic
$L$-functions at $s=0$ and $s=1$ in terms of units. The Harris–Venkates
h conjecture describes the residue of Stark units modulo $p$\, giving a mo
dular analogue to the Stark and Gross conjectures while also serving as th
e first verifiable part of the broader conjectures of Venkatesh\, Prasanna
\, and Galatius. In this talk\, I will draw an introductory picture\, form
ulate a unified conjecture combining Harris–Venkatesh and Stark for weig
ht one modular forms\, and describe the proof of this in the imaginary dih
edral case.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Betts (Harvard)
DTSTART;VALUE=DATE-TIME:20231011T190000Z
DTEND;VALUE=DATE-TIME:20231011T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/94
DESCRIPTION:Title: A relative Oda's criterion\nby Alex Betts (Harvard) as part of Harv
ard number theory seminar\n\nLecture held in Science Center Hall A.\n\nAbs
tract\nThe Neron--Ogg--Shafarevich criterion asserts that an abelian varie
ty over $\\mathbb{Q}_p$ has good reduction if and only if the Galois actio
n on its $\\mathbb{Z}_\\ell$-linear Tate module is unramified (for $\\ell$
different from $p$). In 1995\, Oda formulated and proved an analogue of t
he Neron--Ogg--Shafarevich criterion for smooth projective curves $X$ of g
enus at least two: $X$ has good reduction if and only if the outer Galois
action on its pro-$\\ell$ geometric fundamental group is unramified. In th
is talk\, I will explain a relative version of Oda's criterion\, due to my
self and Netan Dogra\, in which we answer the question of when the Galois
action on the pro-$\\ell$ torsor of paths between two points $x$ and $y$ i
s unramified in terms of the relative position of $x$ and $y$ on the reduc
tion of $X$. On the way\, we will touch on topics from mapping class group
s and the theory of electrical circuits\, and\, time permitting\, will out
line some consequences for the Chabauty--Kim method.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Naomi Sweeting (Harvard)
DTSTART;VALUE=DATE-TIME:20231004T190000Z
DTEND;VALUE=DATE-TIME:20231004T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/95
DESCRIPTION:Title: Tate Classes and Endoscopy for $\\operatorname{GSp}_4$\nby Naomi Sw
eeting (Harvard) as part of Harvard number theory seminar\n\nLecture held
in Science Center Room 507.\n\nAbstract\nWeissauer proved using the theory
of endoscopy that the Galois representations associated to classical modu
lar forms of weight two appear in the middle cohomology of both a modular
curve and a Siegel modular threefold. Correspondingly\, there are large f
amilies of Tate classes on the product of these two Shimura varieties\, an
d it is natural to ask whether one can construct algebraic cycles giving r
ise to these Tate classes. It turns out that a natural algebraic cycle gen
erates some\, but not all\, of the Tate classes: to be precise\, it genera
tes exactly the Tate classes which are associated to generic members of th
e endoscopic $L$-packets on $\\operatorname{GSp}_4$. In the non-generic ca
se\, one can at least show that all the Tate classes arise from Hodge cycl
es. For this talk\, I'll focus on the behavior of the algebraic cycle clas
s. NB: This talk is independent of the one in last week's number theorists
' seminar.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jordan Ellenberg (University of Wisconsin-Madison)
DTSTART;VALUE=DATE-TIME:20240207T200000Z
DTEND;VALUE=DATE-TIME:20240207T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/96
DESCRIPTION:Title: Variation of Selmer groups in quadratic twist families of abelian varie
ties over function fields\nby Jordan Ellenberg (University of Wisconsi
n-Madison) as part of Harvard number theory seminar\n\nLecture held in Sci
ence Center Room 507.\n\nAbstract\nA basic question in arithmetic statisti
cs is: what does the Selmer group of a random abelian variety look like?
This question is governed by the Poonen-Rains heuristics\, later generali
zed by Bhargava-Kane-Lenstra-Poonen-Rains\, which predict\, for instance\,
that the mod p Selmer group of an elliptic curve has size p+1 on average.
Results towards these heuristics have been very partial but have nonethe
less enabled major progress in studying the distribution of ranks of abeli
an varieties.\n\n \n\nWe will describe new work\, joint with Aaron Landesm
an\, which establishes a version of these heuristics for the mod n Selmer
group of a random quadratic twist of a fixed abelian variety over a global
function field. This allows us\, for instance\, to bound the probability
that a random quadratic twist of an abelian variety A over a global funct
ion field has rank at least 2. The method is very much in the spirit of e
arlier work with Venkatesh and Westerland which proved a version of the Co
hen-Lenstra heuristics over function fields by means of homological stabil
ization for Hurwitz spaces\; in other words\, the main argument is topolog
ical in nature. I will try to embed the talk in a general discussion of h
ow one gets from topological results to consequences in arithmetic statist
ics\, and what the prospects for further developments in this area look li
ke.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART;VALUE=DATE-TIME:20231122T200000Z
DTEND;VALUE=DATE-TIME:20231122T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/97
DESCRIPTION:by TBA as part of Harvard number theory seminar\n\nLecture hel
d in Science Center Room 507.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/HarvardNT/97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jennifer Balakrishnan (Boston University)
DTSTART;VALUE=DATE-TIME:20240424T190000Z
DTEND;VALUE=DATE-TIME:20240424T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/98
DESCRIPTION:Title: Shadow line distributions\nby Jennifer Balakrishnan (Boston Univers
ity) as part of Harvard number theory seminar\n\nLecture held in Science C
enter Room 507.\n\nAbstract\nLet $E/\\mathbb{Q}$ be an elliptic curve of a
nalytic rank $2$\, and let $p$\nbe an odd prime of good\, ordinary reducti
on such that the $p$-torsion of\n$E(\\mathbb{Q})$ is trivial. Let $K$ be a
n imaginary quadratic field satisfying the\nHeegner hypothesis for $E$ and
such that the analytic rank of the\ntwisted curve $E^K/\\mathbb{Q}$ is $1
$. Further suppose that $p$ splits in $\\mathcal{O}_K$. Under\nthese assum
ptions\, there is a $1$-dimensional $\\mathbb{Q}_p$-vector space attached\
nto the triple $(E\, p\, K)$\, known as the shadow line\, and it can be\nc
omputed using anticyclotomic $p$-adic heights. We describe the\ncomputatio
n of these heights and shadow lines. Furthermore\, fixing\npairs $(E\, p)
$ and varying $K$\, we present some data on the distribution\nof these sha
dow lines. This is joint work with Mirela Çiperiani\,\nBarry Mazur\, and
Karl Rubin.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuan Liu (University of Illinois Urbana-Champaign)
DTSTART;VALUE=DATE-TIME:20240417T190000Z
DTEND;VALUE=DATE-TIME:20240417T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/99
DESCRIPTION:Title: On the distribution of class groups — beyond Cohen-Lenstra and Gerth<
/a>\nby Yuan Liu (University of Illinois Urbana-Champaign) as part of Harv
ard number theory seminar\n\nLecture held in Science Center Room 507.\n\nA
bstract\nThe Cohen-Lenstra heuristic studies the distribution of the p-par
t of the class group of quadratic number fields for odd prime $p$. Gerth
’s conjecture regards the distribution of the $2$-part of the class grou
p of quadratic fields. The main difference between these conjectures is th
at while the (odd) $p$-part of the class group behaves completely “rando
mly”\, the $2$-part of the class group does not since the $2$-torsion of
the class group is controlled by the genus field. In this talk\, we will
discuss a new conjecture generalizing Cohen-Lenstra and Gerth’s conjectu
res. The techniques involve Galois cohomology and the embedding problem of
global fields.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/99/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefan Patrikis (The Ohio State University)
DTSTART;VALUE=DATE-TIME:20240214T200000Z
DTEND;VALUE=DATE-TIME:20240214T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/101
DESCRIPTION:Title: Compatibility of the canonical $l$-adic local systems on exceptional S
himura varieties\nby Stefan Patrikis (The Ohio State University) as pa
rt of Harvard number theory seminar\n\nLecture held in Science Center Room
507.\n\nAbstract\nLet $(G\, X)$ be a Shimura datum\, and let $K$ be a com
pact open subgroup of $G(\\mathbb{A}_f)$. One hopes that under mild assump
tions on $G$ and $K$\, the points of the Shimura variety $Sh_K(G\, X)$ par
ametrize a family of motives\; in abelian type this is well-understood\, b
ut in non-abelian type it is completely mysterious. I will discuss joint w
ork with Christian Klevdal showing that for exceptional Shimura varieties
the points (over number fields\, say) at least yield compatible systems of
l-adic representations\, which should be the l-adic realizations of the c
onjectural motives.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven Achenjang (MIT)
DTSTART;VALUE=DATE-TIME:20240221T200000Z
DTEND;VALUE=DATE-TIME:20240221T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/102
DESCRIPTION:Title: The Average Size of 2-Selmer Groups of Elliptic Curves over Function F
ields\nby Niven Achenjang (MIT) as part of Harvard number theory semin
ar\n\nLecture held in Science Center Room 507.\n\nAbstract\nGiven an ellip
tic curve $E$ over a global field $K$\, the abelian group $E(K)$ is finite
ly generated\, and so much effort has been put into trying to understand t
he behavior of $\\operatorname{rank}E(K)$\, as $E$ varies. Of note\, it is
a folklore conjecture that\, when all elliptic curves $E/K$ are ordered b
y a suitably defined height\, the average value of their ranks is exactly
$1/2$. One fruitful avenue for understanding the distribution of $\\operat
orname{rank}E(K)$ has been to first understand the distribution of the siz
es of Selmer groups of elliptic curves. In this direction\, various author
s (including Bhargava-Shankar\, Poonen-Rains\, and Bhargava-Kane-Lenstra-P
oonen-Rains) have made conjectures which predict\, for example\, that the
average size of the $n$-Selmer group of $E/K$ is equal to the sum of the d
ivisors of $n$. In this talk\, I will report on some recent work verifying
this average size prediction\, "up to small error term\," whenever $n=2$
and $K$ is any global *function* field. Results along these lines were pre
viously known whenever $K$ was a number field or function field of charact
eristic $\\ge 5$\, so the novelty of my work is that it applies even in "b
ad" characteristic.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/102/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Mundy (Princeton University)
DTSTART;VALUE=DATE-TIME:20240410T190000Z
DTEND;VALUE=DATE-TIME:20240410T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/103
DESCRIPTION:Title: Vanishing of Selmer groups for Siegel modular forms\nby Sam Mundy
(Princeton University) as part of Harvard number theory seminar\n\nLecture
held in Science Center Room 507.\n\nAbstract\nLet $\\pi$ be a cuspidal au
tomorphic representation of $\\mathrm{Sp}_{2n}$ over $\\mathbb{Q}$ which i
s holomorphic discrete series at infinity\, and $\\chi$ a Dirichlet charac
ter. Then one can attach to $\\pi$ an orthogonal $p$-adic Galois represent
ation $\\rho$ of dimension $2n+1$. Assume $\\rho$ is irreducible\, that $\
\pi$ is ordinary at $p$\, and that $p$ does not divide the conductor of $\
\chi$. I will describe work in progress which aims to prove that the Bloch
--Kato Selmer group attached to the twist of $\\rho$ by $\\chi$ vanishes\,
under some mild ramification assumptions on $\\pi$\; this is what is pred
icted by the Bloch--Kato conjectures.\n\n\nThe proof uses "ramified Eisens
tein congruences" by constructing $p$-adic families of Siegel cusp forms d
egenerating to Klingen Eisenstein series of nonclassical weight\, and usin
g these families to construct ramified Galois cohomology classes for the T
ate dual of the twist of $\\rho$ by $\\chi$.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shiva Chidambaram (MIT)
DTSTART;VALUE=DATE-TIME:20240228T200000Z
DTEND;VALUE=DATE-TIME:20240228T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/104
DESCRIPTION:Title: Computing Galois images of Picard curves\nby Shiva Chidambaram (MI
T) as part of Harvard number theory seminar\n\nLecture held in Science Cen
ter Room 507.\n\nAbstract\nLet $C$ be a genus $3$ curve whose Jacobian is
geometrically simple and has geometric endomorphism algebra equal to an im
aginary quadratic field. In particular\, consider Picard curves $y^3 = f_4
(x)$ where the geometric endomorphism algebra is $\\mathbb{Q}(\\zeta_3)$.
We study the associated mod-$\\ell$ Galois representations and their image
s. I will discuss an algorithm\, developed in ongoing joint work with Pip
Goodman\, to compute the set of primes $\\ell$ for which the images are no
t maximal. By running it on several datasets of Picard curves\, the larges
t non-maximal prime we obtain is $13$. This may be compared with genus 1\,
where Serre's uniformity question asks if the mod-$\\ell$ Galois image of
non-CM elliptic curves over $\\Q$ is maximal for all primes $\\ell > 37$.
\n
LOCATION:https://researchseminars.org/talk/HarvardNT/104/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Salim Tayou (Harvard University)
DTSTART;VALUE=DATE-TIME:20240501T190000Z
DTEND;VALUE=DATE-TIME:20240501T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/106
DESCRIPTION:Title: Modularity of special cycles in orthogonal and unitary Shimura varieti
es\nby Salim Tayou (Harvard University) as part of Harvard number theo
ry seminar\n\nLecture held in Science Center Room 507.\n\nAbstract\nSince
the work of Jacobi and Siegel\, it is well known that\nTheta series of qua
dratic lattices produce modular forms. In a vast\ngeneralization\, Kudla a
nd Millson have proved that the generating series\nof special cycles in or
thogonal and unitary Shimura varieties are\nmodular forms. In this talk\,
I will explain an extension of these\nresults to toroidal compactification
s where we prove that the generating\nseries of divisors is a mixed mock m
odular form. This recovers and\nrefines earlier results of Bruinier and Ze
mel. The results of this talk\nare joint work with Philip Engel and Franç
ois Greer.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frank Calegari (University of Chicago)
DTSTART;VALUE=DATE-TIME:20240327T190000Z
DTEND;VALUE=DATE-TIME:20240327T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/107
DESCRIPTION:Title: “everywhere unramified” objects in number theory and the cohomolog
y of $\\mathrm{GL}_n(\\mathbb{Z})$\nby Frank Calegari (University of C
hicago) as part of Harvard number theory seminar\n\nLecture held in Scienc
e Center Room 507.\n\nAbstract\nOne theme in number theory is to study obj
ects via their ramification: the discriminant of a number field\, the cond
uctor of an elliptic curve\, the level of a modular form\, and so on.\nThe
re is\, however\, some particular interest in understanding objects which
are “everywhere unramified” — and also understanding when such objec
ts don’t exist. Such non-existence results\nare often the starting point
for inductive arguments. For example\, Minkowski’s theorem that there a
re no unramified extensions of $\\mathbb{Q}$ can be used to prove the Kron
ecker-Weber theorem\, and the vanishing\nof a certain space of modular for
ms is the starting point for Wiles’ proof of Fermat’s Last Theorem. In
this talk\, I will begin by describing many such vanishing results both i
n arithmetic and in the\ntheory of automorphic forms\, and how they are re
lated by the Langlands program (sometimes only conjecturally). Then I will
descibe the construction of a new example of an automorphic form of level
one\nand “weight zero”. This construction also gives the first non-z
ero classes in the cohomology of $\\mathrm{GL}_n(\\mathbb{Z})$ (for some $
n$) that come from “cuspidal” modular forms (for $n > 0$).\n\nThis is
joint work with George Boxer and Toby Gee.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/107/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Weinstein (Boston University)
DTSTART;VALUE=DATE-TIME:20240306T200000Z
DTEND;VALUE=DATE-TIME:20240306T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/109
DESCRIPTION:Title: Integral Ax-Sen-Tate theory\nby Jared Weinstein (Boston University
) as part of Harvard number theory seminar\n\nLecture held in Science Cent
er Room 507.\n\nAbstract\nLet $K$ be a local field of mixed characteristic
\, let $G$ be the absolute Galois group of $K$\, and let $C$ be the comple
tion of an algebraic closure of $K$. The Ax-Sen-Tate theorem states that
the field of $G$-invariant elements in $C$ is $K$ itself: $H^0(G\,C)=K$.
Tate also proved statements about higher cohomology (with continuous cocy
cles): $H^1(G\,C)=K$ and $H^i(G\,C)=0$ for $i>1$. \n Let $O_C$ be the
ring of integers in $C$. Our main theorem is that the torsion subgroup o
f $H^i(G\,O_C)$ is killed by a constant which only depends on the residue
characteristic $p$ (in fact $p^6$ suffices). This is a part of a project
with coauthors Tobias Barthel\, Tomer Schlank\, and Nathaniel Stapleton.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/109/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sasha Petrov (MIT)
DTSTART;VALUE=DATE-TIME:20240911T190000Z
DTEND;VALUE=DATE-TIME:20240911T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/110
DESCRIPTION:Title: Characteristic classes of p-adic local systems\nby Sasha Petrov (M
IT) as part of Harvard number theory seminar\n\nLecture held in Science Ce
nter Room 507.\n\nAbstract\nGiven an étale Z_p-local system of rank n on
an algebraic variety X\, continuous cohomology classes of the group GL_n(Z
_p) give rise to classes in (absolute) étale cohomology of the variety wi
th coefficients in Q_p. These characteristic classes can be thought of as
p-adic analogs of Chern-Simons characteristic classes of vector bundles wi
th a flat connection.\n\nOn a smooth projective variety over complex numbe
rs\, Chern-Simons classes of all flat bundles are torsion in degrees >1 by
a theorem of Reznikov. But for varieties over non-closed fields the chara
cteristic classes of p-adic local systems turn out to often be non-zero ev
en rationally. When X is defined over a p-adic field\, characteristic clas
ses of a p-adic local system on it can be partially expressed in terms of
Hodge-theoretic invariants of the local system. This relation is establish
ed through considering an analog of Chern classes for vector bundles on th
e pro-étale site of X.\n\nThis is joint work with Lue Pan.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/110/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jit Wu Yap (Harvard University)
DTSTART;VALUE=DATE-TIME:20240918T190000Z
DTEND;VALUE=DATE-TIME:20240918T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/111
DESCRIPTION:Title: Quantitative Equidistribution of Small Points for Canonical Heights\nby Jit Wu Yap (Harvard University) as part of Harvard number theory sem
inar\n\nLecture held in Science Center Room 507.\n\nAbstract\nLet K be a n
umber field with algebraic closure L and A an abelian variety over K. Then
if (x_n) is a generic sequence of points of A(L) with Neron-Tate height t
ending to 0\, Szpiro-Ullmo-Zhang proved that the Galois orbits of x_n conv
erges weakly to the Haar measure of A. Yuan then generalized Szpiro-Ullmo-
Zhang's result to the setting of polarized endomorphisms on a projective v
ariety X defined over K. In this talk\, I will explain how to prove a quan
titative version of Yuan's result when X is assumed to be smooth. This was
previously only known when dim X = 1.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/111/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hélène Esnault (Freie Universität Berlin)
DTSTART;VALUE=DATE-TIME:20240925T190000Z
DTEND;VALUE=DATE-TIME:20240925T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/112
DESCRIPTION:Title: Diophantine Properties of the Betti Moduli Space\nby Hélène Esna
ult (Freie Universität Berlin) as part of Harvard number theory seminar\n
\nLecture held in Science Center Room 507.\n\nAbstract\nWe prove in partic
ular that when the Betti moduli space of a smooth quasi-projective variety
\nover the complex number with some quasi-unipotent monodromies at infinit
y. finite determinant\nis irreducible over the integers and over the compl
ex numbers\, then it possesses an integral point. \nA more general version
of the theorem yields a new obstruction for the finitely presented group
to be the topological fundamental group\nof a smooth complex quasi-project
ive variety. \n\n(Joint with J. de Jong\, based in part on joint work with
M. Groechenig).\n
LOCATION:https://researchseminars.org/talk/HarvardNT/112/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sanath Devalapurkar (Harvard University)
DTSTART;VALUE=DATE-TIME:20241002T190000Z
DTEND;VALUE=DATE-TIME:20241002T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/113
DESCRIPTION:Title: The image of J and p-adic geometry\nby Sanath Devalapurkar (Harvar
d University) as part of Harvard number theory seminar\n\nLecture held in
Science Center Room 507.\n\nAbstract\nFor a prime p\, Bhatt\, Lurie\, and
Drinfeld constructed the "prismatization" of a p-adic formal scheme\; this
is a stack which computes prismatic cohomology\, which is a "universal" c
ohomology theory for p-adic formal schemes. I will describe joint work wit
h Hahn\, Raksit\, and Yuan (building on work of Hahn-Raksit-Wilson)\, in w
hich we give a new construction of prismatization using the methods of hom
otopy theory (in particular\, the theory of topological Hochschild homolog
y\, aka THH). The case when R is Z_{p} turns out to be particularly intere
sting\, and I will discuss joint work with Raksit which describes a constr
uction of THH(Z_{p}) for odd primes p in terms of a very classical object
in homotopy theory called the "image-of-J spectrum" studied by Adams. This
plays the same role for prismatic cohomology as the usual commutative rin
g Z_{p} plays for crystalline cohomology. It gives an alternative perspect
ive on results of Bhatt and Lurie\, and is also related to Lurie’s "pris
matization of F_{1}".\n
LOCATION:https://researchseminars.org/talk/HarvardNT/113/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sameera Vemulapalli (Harvard University)
DTSTART;VALUE=DATE-TIME:20241009T190000Z
DTEND;VALUE=DATE-TIME:20241009T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/114
DESCRIPTION:Title: Steinitz classes of number fields and Tschirnhausen bundles of covers
of the projective line\nby Sameera Vemulapalli (Harvard University) as
part of Harvard number theory seminar\n\nLecture held in Science Center R
oom 507.\n\nAbstract\nGiven a number field extension $L/K$ of fixed degree
\, one may consider $\\mathcal{O}_L$ as an $\\mathcal{O}_K$-module. Which
modules arise this way? Analogously\, in the geometric setting\, a cover o
f the complex projective line by a smooth curve yields a vector bundle on
the projective line by pushforward of the structure sheaf\; which bundles
arise this way? In this talk\, I'll describe recent work with Vakil in whi
ch we use tools in arithmetic statistics (in particular\, binary forms) to
completely answer the first question and make progress towards the second
.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/114/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Artane Siad (Princeton University)
DTSTART;VALUE=DATE-TIME:20241016T190000Z
DTEND;VALUE=DATE-TIME:20241016T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/115
DESCRIPTION:Title: Spin structures\, quadratic maps\, and the missing class group heurist
ic\nby Artane Siad (Princeton University) as part of Harvard number th
eory seminar\n\nLecture held in Science Center Room 507.\n\nAbstract\nI wi
ll report on joint work in progress with Akshay Venkatesh where we propose
an arithmetic analogue of the association\, in topology\, of quadratic en
hancements to spin structures on closed oriented 2- and 3-manifolds: a cho
ice of spin structure provides\, respectively\, a quadratic refinement of
the mod 2 intersection form and of the linking pairing on the first torsio
n homology. This adds an entry to the number field/3-manifold analogy of M
umford\, Mazur\, and Manin and furnishes a conceptual explanation of anoma
lous class group statistics.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/115/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linus Hamann (Harvard University)
DTSTART;VALUE=DATE-TIME:20241023T190000Z
DTEND;VALUE=DATE-TIME:20241023T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/116
DESCRIPTION:Title: Shimura Varieties and Eigensheaves\nby Linus Hamann (Harvard Unive
rsity) as part of Harvard number theory seminar\n\nLecture held in Science
Center Room 507.\n\nAbstract\nThe cohomology of Shimura varieties is a fu
ndamental object of study in algebraic number theory by virtue of the fact
that it is the only known geometric realization of the global Langlands c
orrespondence over number fields. Usually\, the cohomology is computed thr
ough very delicate techniques involving the trace formula. However\, this
perspective has several limitations\, especially with regards to questions
concerning torsion. In this talk\, we will discuss a new paradigm for co
mputing the cohomology of Shimura varieties by decomposing certain sheaves
coming from Igusa varieties into Hecke eigensheaves on the moduli stack o
f G-bundles on the Fargues-Fontaine curve. Using this point of view\, we w
ill describe several conjectures on the torsion cohomology of Shimura vari
eties after localizing at suitably "generic" L-parameters\, as well as som
e known results in the case that the parameter factors through a maximal t
orus. Motivated by this\, we will sketch part of an emerging picture for d
escribing the cohomology beyond this generic locus by considering certain
"generalized eigensheaves" whose eigenvalues are spread out in multiple co
homological degrees based on the size of a certain Arthur SL_{2} in a way
that is reminiscent of Arthur's cohomological conjectures on the intersect
ion cohomology of Shimura Varieties. This is based on joint work with Lee\
, joint work in progress with Caraiani and Zhang\, and conversations with
Bertoloni-Meli and Koshikawa.\n
LOCATION:https://researchseminars.org/talk/HarvardNT/116/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wei Zhang (MIT)
DTSTART;VALUE=DATE-TIME:20241030T190000Z
DTEND;VALUE=DATE-TIME:20241030T200000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/117
DESCRIPTION:by Wei Zhang (MIT) as part of Harvard number theory seminar\n\
nLecture held in Science Center Room 507.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/HarvardNT/117/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sachi Hashimoto (Brown University)
DTSTART;VALUE=DATE-TIME:20241106T200000Z
DTEND;VALUE=DATE-TIME:20241106T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/118
DESCRIPTION:by Sachi Hashimoto (Brown University) as part of Harvard numbe
r theory seminar\n\nLecture held in Science Center Room 507.\nAbstract: TB
A\n
LOCATION:https://researchseminars.org/talk/HarvardNT/118/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Bertoloni Meli (Boston University)
DTSTART;VALUE=DATE-TIME:20241113T200000Z
DTEND;VALUE=DATE-TIME:20241113T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/119
DESCRIPTION:by Alexander Bertoloni Meli (Boston University) as part of Har
vard number theory seminar\n\nLecture held in Science Center Room 507.\nAb
stract: TBA\n
LOCATION:https://researchseminars.org/talk/HarvardNT/119/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nina Zubrilina (Harvard University)
DTSTART;VALUE=DATE-TIME:20241120T200000Z
DTEND;VALUE=DATE-TIME:20241120T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/120
DESCRIPTION:by Nina Zubrilina (Harvard University) as part of Harvard numb
er theory seminar\n\nLecture held in Science Center Room 507.\nAbstract: T
BA\n
LOCATION:https://researchseminars.org/talk/HarvardNT/120/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Smith (UCLA)
DTSTART;VALUE=DATE-TIME:20241204T200000Z
DTEND;VALUE=DATE-TIME:20241204T210000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080156Z
UID:HarvardNT/121
DESCRIPTION:by Alex Smith (UCLA) as part of Harvard number theory seminar\
n\nLecture held in Science Center Room 507.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/HarvardNT/121/
END:VEVENT
END:VCALENDAR