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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:20241013T133215Z
UID:HarvardNT/1
DESCRIPTION:Title: Bounding torsion in class groups and families of local systems\nby Jacob Tsimerman (University of Toronto) as part of Harvard number th
eory seminar\n\n\nAbstract\n(joint w/ Arul Shankar) We discuss a new metho
d to bound 5-torsion in class groups of quadratic fields using the refined
BSD conjecture for elliptic curves. The most natural “trivial” bound
on the n-torsion is to bound it by the size of the entire class group\, fo
r which one has a global class number formula. We explain how to make sens
e of the n-torsion of a class group intrinsically as a selmer group of a G
alois module. We may then similarly bound its size by the Tate-Shafarevich
group of an appropriate elliptic curve\, which we can bound using the BSD
conjecture. This fits into a general paradigm where one bounds selmer gro
ups of finite Galois modules by embedding into global objects\, and using
class number formulas. If time permits\, we explain how the function field
picture yields unconditional results and suggests further generalizations
.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Kriz (MIT)
DTSTART;VALUE=DATE-TIME:20200415T190000Z
DTEND;VALUE=DATE-TIME:20200415T201500Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/2
DESCRIPTION:Title: Converse theorems for supersingular CM elliptic curves\nby Da
niel Kriz (MIT) as part of Harvard number theory seminar\n\nAbstract: TBA\
n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Weinstein (BU)
DTSTART;VALUE=DATE-TIME:20200422T190000Z
DTEND;VALUE=DATE-TIME:20200422T201500Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/3
DESCRIPTION:Title: Modularity for self-products of elliptic curves over function fie
lds\nby Jared Weinstein (BU) as part of Harvard number theory seminar\
n\nAbstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Newton (Kings College London)
DTSTART;VALUE=DATE-TIME:20200506T190000Z
DTEND;VALUE=DATE-TIME:20200506T201500Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/4
DESCRIPTION:Title: Symmetric power functoriality for modular forms\nby James New
ton (Kings College London) as part of Harvard number theory seminar\n\nAbs
tract: TBA\n
LOCATION:
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:20241013T133215Z
UID:HarvardNT/5
DESCRIPTION:Title: Prismatic Dieudonne theory\nby Arthur-Cesar Le Bras (CNRS/Par
is-13) as part of Harvard number theory seminar\n\nAbstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Preston Wake (Michigan State)
DTSTART;VALUE=DATE-TIME:20200520T190000Z
DTEND;VALUE=DATE-TIME:20200520T201500Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/6
DESCRIPTION:Title: Tame derivatives and the Eisenstein ideal\nby Preston Wake (M
ichigan State) as part of Harvard number theory seminar\n\nAbstract: TBA\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Landesman (Stanford University)
DTSTART;VALUE=DATE-TIME:20201104T200000Z
DTEND;VALUE=DATE-TIME:20201104T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/7
DESCRIPTION:Title: A geometric approach to the Cohen-Lenstra heuristics\nby Aaro
n Landesman (Stanford University) as part of Harvard number theory seminar
\n\n\nAbstract\nFor any positive integer $n$\,\nwe explain why the total n
umber of order $n$ elements\nin class groups of quadratic fields of discri
minant\nhaving absolute value at most $X$ is $O_n(X^{5/4})$.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karol Koziol (University of Michigan)
DTSTART;VALUE=DATE-TIME:20201028T190000Z
DTEND;VALUE=DATE-TIME:20201028T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/8
DESCRIPTION:Title: Supersingular representations of $p$-adic reductive groups\nb
y Karol Koziol (University of Michigan) as part of Harvard number theory s
eminar\n\n\nAbstract\nThe local Langlands conjectures predict that (packet
s of) irreducible complex representations of $p$-adic reductive groups (su
ch as $\\mathrm{GL}_n(\\mathbb{Q}_p)$\, $\\mathrm{GSp}_{2n}(\\mathbb{Q}_p)
$\, etc.) should be parametrized by certain representations of the Weil-De
ligne group. A special role in this hypothetical correspondence is held
by the supercuspidal representations\, which generically are expected to c
orrespond to irreducible objects on the Galois side\, and which serve as b
uilding blocks for all irreducible representations. Motivated by recent
advances in the mod-$p$ local Langlands program (i.e.\, with mod-$p$ coeff
icients instead of complex coefficients)\, I will give an overview of what
is known about supersingular representations of $p$-adic reductive groups
\, which are the "mod-$p$ coefficients" analogs of supercuspidal represent
ations. This is joint work with Florian Herzig and Marie-France Vigneras
.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arul Shankar (University of Toronto)
DTSTART;VALUE=DATE-TIME:20201202T200000Z
DTEND;VALUE=DATE-TIME:20201202T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/9
DESCRIPTION:Title: The 2-torsion subgroups of the class groups in families of cubic
fields\nby Arul Shankar (University of Toronto) as part of Harvard num
ber theory seminar\n\n\nAbstract\nThe Cohen--Lenstra--Martinet conjectures
have been verified in\nonly two cases. Davenport--Heilbronn compute the a
verage size of the\n3-torsion subgroups in the class group of quadratic fi
elds and Bhargava\ncomputes the average size of the 2-torsion subgroups in
the class groups of\ncubic fields. The values computed in the above two r
esults are remarkably\nstable. In particular\, work of Bhargava--Varma sho
ws that they do not\nchange if one instead averages over the family of qua
dratic or cubic fields\nsatisfying any finite set of splitting conditions.
\n\nHowever for certain "thin" families of cubic fields\, namely\, familie
s of\nmonogenic and n-monogenic cubic fields\, the story is very different
. In\nthis talk\, we will determine the average size of the 2-torsion subg
roups of\nthe class groups of fields in these thin families. Surprisingly\
, these\nvalues differ from the Cohen--Lenstra--Martinet predictions! We w
ill also\nprovide an explanation for this difference in terms of the Tamag
awa numbers\nof naturally arising reductive groups. This is joint work wit
h Manjul\nBhargava and Jon Hanke.\n
LOCATION:
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:20241013T133215Z
UID:HarvardNT/10
DESCRIPTION:Title: Hermite interpolation and counting number fields\nby Jean-Ma
rc Couveignes (University of Bordeaux) as part of Harvard number theory se
minar\n\n\nAbstract\nThere are several ways to specify a number\nfield. On
e can provide the minimal polynomial\nof a primitive element\, the multipl
ication\ntable of a $\\bf Q$-basis\, the traces of a large enough\nfamily
of elements\, etc.\nFrom any way of specifying a number field\none can h
ope 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 the number\nof isomorphism classes of number fields of degree $n$\nan
d discriminant bounded by $H$ is equivalent to $c(n)H$\nwhen $n\\geqslant
2$ is fixed and $H$ tends to infinity.\nSuch an estimate has been proved f
or $n=3$\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}
$\nusing Minkowski's theorem.\nEllenberg et Venkatesh have proved that the
exponent of\n$H$ in $N_n(H)$ is less than sub-exponential in $\\log (n)$.
\nI will explain how Hermite interpolation (a theorem\nof Alexander and Hi
rschowitz) and geometry of numbers\ncombine to produce short models for nu
mber fields\nand sharper bounds for $N_n(H)$.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Cass (Harvard University)
DTSTART;VALUE=DATE-TIME:20200909T190000Z
DTEND;VALUE=DATE-TIME:20200909T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/11
DESCRIPTION:Title: A mod p geometric Satake isomorphism\nby Robert Cass (Harvar
d University) as part of Harvard number theory seminar\n\n\nAbstract\nWe a
pply methods from geometric representation theory toward the mod p\nLangla
nds program.\nMore specifically\, we explain a mod p version of the geomet
ric Satake\nisomorphism\, which gives a sheaf-theoretic description of the
spherical mod\np Hecke algebra. In our setup the mod p Satake category is
not controlled\nby the dual group but rather a certain affine monoid sche
me. Along the way\nwe will discuss some new results about the F-singularit
ies of affine\nSchubert varieties. Time permitting\, we will explain how t
o geometrically\nconstruct central elements in the Iwahori mod p Hecke alg
ebra by adapting a\nmethod due to Gaitsgory.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zijian Yao (CNRS/Harvard)
DTSTART;VALUE=DATE-TIME:20201111T200000Z
DTEND;VALUE=DATE-TIME:20201111T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/12
DESCRIPTION:Title: Frobenius and the Hodge numbers of the generic fiber\nby Zij
ian Yao (CNRS/Harvard) as part of Harvard number theory seminar\n\n\nAbstr
act\nFor a smooth proper (formal) scheme $\\mathfrak{X}$ defined over a va
luation\nring of mixed characteristic\, the crystalline cohomology H of it
s special\nfiber has the structure of an F-crystal\, to which one can atta
ch a Newton\npolygon and a Hodge polygon that describe the ''slopes of the
Frobenius\naction on H''. The shape of these polygons are constrained by
the geometry\nof $\\mathfrak{X}$ -- in particular by the Hodge numbers of
both the special\nfiber and the generic fiber of $\\mathfrak{X}$. One inst
ance of such\nconstraints is given by a beautiful conjecture of Katz (now
a theorem of\nMazur\, Ogus\, Nygaard etc.)\, another constraint comes from
the notion of\n"weakly admissible" Galois representations.\n\nIn this tal
k\, I will discuss some results regarding the shape of the\nFrobenius acti
on on the F-crystal H and the Hodge numbers of the generic\nfiber of $\\ma
thfrak{X}$\, along with generalizations in several directions.\nIn partic
ular\, we give a new proof of the fact that the Newton polygon of\nthe spe
cial fiber of $\\mathfrak{X}$ lies on or above the Hodge polygon of\nits g
eneric fiber\, without appealing to Galois representations. A new\ningredi
ent that appears is (a generalized version of) the Nygaard\nfiltration of
the prismatic/Ainf cohomology\, developed by Bhatt\, Morrow and\nScholze.\
n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elena Mantovan (Caltech)
DTSTART;VALUE=DATE-TIME:20201021T190000Z
DTEND;VALUE=DATE-TIME:20201021T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/13
DESCRIPTION:Title: p-adic differential operators on automorphic forms\, and mod p G
alois representations\nby Elena Mantovan (Caltech) as part of Harvard
number theory seminar\n\n\nAbstract\nIn this talk\, we will discuss a geom
etric construction of p-adic analogues of Maass--Shimura differential oper
ators on automorphic forms on Shimura varieties of PEL type A or C (that i
s\, unitary or symplectic)\, at p an unramified prime. Maass--Shimura oper
ators are smooth weight raising differential operators used in the study o
f special values of L-functions\, and in the arithmetic setting for the co
nstruction of p-adic L-functions. In this talk\, we will focus in particu
lar on the case of unitary groups of arbitrary signature\, when new phenom
ena arise for p non split. We will also discuss an application to the st
udy of modular mod p Galois representations. This talk is based on joint w
ork with Ellen Eischen (in the unitary case for p non split)\, and with Ei
schen\, Flanders\, Ghitza\, and Mc Andrew (in the other cases).\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Si Ying Lee (Harvard University)
DTSTART;VALUE=DATE-TIME:20201118T200000Z
DTEND;VALUE=DATE-TIME:20201118T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/14
DESCRIPTION:Title: Eichler-Shimura relations for Hodge type Shimura varieties\n
by Si Ying Lee (Harvard University) as part of Harvard number theory semin
ar\n\n\nAbstract\nThe well-known classical Eichler-Shimura relation for mo
dular curves asserts that the Hecke operator $T_p$ is equal\, as an algebr
aic correspondence over the special fiber\, to the sum of Frobenius and Ve
rschebung. Blasius and Rogawski proposed a generalization of this result f
or general Shimura varieties with good reduction at $p$\, and conjectured
that the Frobenius satisfies a certain Hecke polynomial. I will talk about
a recent proof of this conjecture for Shimura varieties of Hodge type\, a
ssuming a technical condition on the unramified sigma-conjugacy classes in
the associated Kottwitz set.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Loeffler (University of Warwick)
DTSTART;VALUE=DATE-TIME:20201014T190000Z
DTEND;VALUE=DATE-TIME:20201014T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/15
DESCRIPTION:Title: The Bloch--Kato conjecture for GSp(4)\nby David Loeffler (Un
iversity of Warwick) as part of Harvard number theory seminar\n\n\nAbstrac
t\nThe Bloch--Kato conjecture predicts that the dimension of the Selmer gr
oup of a global Galois representation is equal to the order of vanishing o
f its L-function. In this talk\, I will focus on the 4-dimensional Galois
representations arising from cohomological automorphic representations of
GSp(4) (i.e. from genus two Siegel modular forms). I will show that if the
L-function is non-vanishing at some critical value\, then the correspondi
ng Selmer group is zero\, under a long list of technical hypotheses. The p
roof of this theorem relies on an Euler system\, a p-adic L-function\, and
a reciprocity law connecting those together. I will also survey work in p
rogress aiming to extend this result to some other classes of automorphic
representations.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiuya Wang (Duke University)
DTSTART;VALUE=DATE-TIME:20200930T190000Z
DTEND;VALUE=DATE-TIME:20200930T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/16
DESCRIPTION:Title: Pointwise Bound for $\\ell$-torsion of Class Groups\nby Jiuy
a Wang (Duke University) as part of Harvard number theory seminar\n\n\nAbs
tract\n$\\ell$-torsion conjecture states that $\\ell$-torsion of the class
group $|\\text{Cl}_K[\\ell]|$ for every number field $K$ is bounded by $\
\text{Disc}(K)^{\\epsilon}$. It follows from a classical result of Brauer-
Siegel\, or even earlier result of Minkowski that the class number $|\\tex
t{Cl}_K|$ of a number field $K$ are always bounded by $\\text{Disc}(K)^{1/
2+\\epsilon}$\, therefore we obtain a trivial bound $\\text{Disc}(K)^{1/2+
\\epsilon}$ on $|\\text{Cl}_K[\\ell]|$. We will talk about results on this
conjecture\, and recent works on breaking the trivial bound for $\\ell$-t
orsion of class groups in some cases based on a work of Ellenberg-Venkates
h.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jessica Fintzen (Cambridge/Duke/IAS)
DTSTART;VALUE=DATE-TIME:20200916T190000Z
DTEND;VALUE=DATE-TIME:20200916T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/17
DESCRIPTION:Title: Representations of p-adic groups and applications\nby Jessic
a Fintzen (Cambridge/Duke/IAS) as part of Harvard number theory seminar\n\
n\nAbstract\nThe Langlands program is a far-reaching collection of conject
ures that relate different areas of mathematics including number theory an
d representation theory. A fundamental problem on the representation theor
y side of the Langlands program is the construction of all (irreducible\,
smooth\, complex) representations of p-adic groups.\n\nI will provide an o
verview of our understanding of the representations of p-adic groups\, wit
h an emphasis on recent progress.\n\nI will also outline how new results a
bout the representation theory of p-adic groups can be used to obtain cong
ruences between arbitrary automorphic forms and automorphic forms which ar
e supercuspidal at p\, which is joint work with Sug Woo Shin. This simplif
ies earlier constructions of attaching Galois representations to automorph
ic representations\, i.e. the global Langlands correspondence\, for genera
l linear groups. Moreover\, our results apply to general p-adic groups and
have therefore the potential to become widely applicable beyond the case
of the general linear group.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kaisa Matomäki (University of Turku)
DTSTART;VALUE=DATE-TIME:20200923T140000Z
DTEND;VALUE=DATE-TIME:20200923T150000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/18
DESCRIPTION:Title: Multiplicative functions in short intervals revisited\nby Ka
isa Matomäki (University of Turku) as part of Harvard number theory semin
ar\n\n\nAbstract\nA few years ago Maksym Radziwill and I showed that the a
verage 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 resul
t has since been utilized in many applications.\nI will talk about recent
work\, where Radziwill and I revisit the problem and generalise our result
to functions which vanish often as well as prove a power-saving upper bou
nd 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 i
nstance to studying gaps between norm forms of an arbitrary number field.\
n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ziyang Gao (CNRS/IMJ-PRG)
DTSTART;VALUE=DATE-TIME:20201007T190000Z
DTEND;VALUE=DATE-TIME:20201007T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/19
DESCRIPTION:Title: Bounding the number of rational points on curves\nby Ziyang
Gao (CNRS/IMJ-PRG) as part of Harvard number theory seminar\n\n\nAbstract\
nMazur conjectured\, after Faltings’s proof of the Mordell conjecture\,
that the number of rational points on a curve of genus g at least 2 define
d over a number field of degree d is bounded in terms of g\, d and the Mor
dell-Weil rank. In particular the height of the curve is not involved. In
this talk I will explain how to prove this conjecture and some generalizat
ions. I will focus on how functional transcendence and unlikely intersecti
ons are applied in the proof. If time permits\, I will talk about how the
dependence on d can be furthermore removed if we moreover assume the relat
ive Bogomolov conjecture. This is joint work with Vesselin Dimitrov and Ph
ilipp Habegger.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niki Myrto Mavraki (Harvard University)
DTSTART;VALUE=DATE-TIME:20210127T200000Z
DTEND;VALUE=DATE-TIME:20210127T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/20
DESCRIPTION:Title: Arithmetic dynamics of random polynomials\nby Niki Myrto Mav
raki (Harvard University) as part of Harvard number theory seminar\n\n\nAb
stract\nWe begin with an introduction to arithmetic dynamics and heights\n
attached to rational maps. We then introduce a dynamical version of Lang's
\nconjecture concerning the minimal canonical height of non-torsion ration
al\npoints in elliptic curves (due to Silverman) as well as a conjectural\
nanalogue of Mazur/Merel's theorem on uniform bounds of rational torsion\n
points in elliptic curves (due to Morton-Silverman). It is likely that the
\ntwo conjectures are harder in the dynamical setting due to the lack of\n
structure coming from a group law. We describe joint work with Pierre Le\n
Boudec in which we establish statistical versions of these conjectures for
\npolynomial maps.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Timo Richarz (TU Darmstadt)
DTSTART;VALUE=DATE-TIME:20210407T190000Z
DTEND;VALUE=DATE-TIME:20210407T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/21
DESCRIPTION:Title: The motivic Satake equivalence\nby Timo Richarz (TU Darmstad
t) as part of Harvard number theory seminar\n\n\nAbstract\nThe geometric S
atake equivalence due to Lusztig\, Drinfeld\, Ginzburg\, Mirković and Vil
onen is an indispensable tool in the Langlands program. Versions of this e
quivalence are known for different cohomology theories such as Betti cohom
ology or algebraic D-modules over characteristic zero fields and $\\ell$-a
dic cohomology over arbitrary fields. In this talk\, I explain how to appl
y the theory of motivic complexes as developed by Voevodsky\, Ayoub\, Cisi
nski-Déglise and many others to the construction of a motivic Satake equi
valence. Under suitable cycle class maps\, it recovers the classical equiv
alence. As dual group\, one obtains a certain extension of the Langlands d
ual group by a one dimensional torus. A key step in the proof is the const
ruction of intersection motives on affine Grassmannians. A direct conseque
nce of their existence is an unconditional construction of IC-Chow groups
of moduli stacks of shtukas. My hope is to obtain on the long run independ
ence-of-$\\ell$ results in the work of V. Lafforgue on the Langlands corre
spondence for function fields. This is ongoing joint work with Jakob Schol
bach from Münster.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Scholze (University of Bonn)
DTSTART;VALUE=DATE-TIME:20210203T200000Z
DTEND;VALUE=DATE-TIME:20210203T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
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 def
inition of analytic spaces that relates\nto complex- or rigid-analytic var
ieties in the same way that schemes\nrelate to algebraic varieties over a
field. Joint with Dustin Clausen.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christian Johansson (Chalmers/Gothenburg)
DTSTART;VALUE=DATE-TIME:20210224T200000Z
DTEND;VALUE=DATE-TIME:20210224T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/23
DESCRIPTION:Title: On the Calegari--Emerton conjectures for abelian type Shimura va
rieties\nby Christian Johansson (Chalmers/Gothenburg) as part of Harva
rd number theory seminar\n\n\nAbstract\nEmerton's completed cohomology giv
es\, at present\, the most general notion of a space of p-adic automorphic
forms. Important properties of completed cohomology\, such as its 'size'\
, is predicted by a conjecture of Calegari and Emerton\, which may be view
ed as a non-abelian generalization of the Leopoldt conjecture. I will disc
uss the proof of many new cases of this conjecture\, using a mixture of te
chniques from p-adic and real geometry. This is joint work with David Hans
en.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Pollack (UCSD)
DTSTART;VALUE=DATE-TIME:20210317T190000Z
DTEND;VALUE=DATE-TIME:20210317T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/24
DESCRIPTION:Title: Modular forms on G_2\nby Aaron Pollack (UCSD) as part of Har
vard number theory seminar\n\n\nAbstract\nFollowing work of Gross-Wallach\
, Gan-Gross-Savin defined what are called "modular forms" on the split exc
eptional group G_2. These are a special class of automorphic forms on G_2
. I'll review their definition\, and give an update about what is known
about them. Results include a construction of cuspidal modular forms with
all algebraic Fourier coefficients\, and the exact functional equation of
the completed standard L-function of certain cusp forms. The results on
L-functions are joint with Fatma Cicek\, Giuliana Davidoff\, Sarah Dijols\
, Trajan Hammonds\, and Manami Roy.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Litt (University of Georgia)
DTSTART;VALUE=DATE-TIME:20210324T190000Z
DTEND;VALUE=DATE-TIME:20210324T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/25
DESCRIPTION:Title: Single-valued Hodge\, p-adic^2\, and tropical integration\nb
y Daniel Litt (University of Georgia) as part of Harvard number theory sem
inar\n\n\nAbstract\nI'll discuss 4 different types of integration -- one i
n the\ncomplex setting\, one in the tropical setting\, and two in the p-ad
ic\nsetting\, and the relationships between them. In particular\, we expla
in how\nto compute Vologodsky's "single-valued" iterated integrals on curv
es of bad\nreduction in terms of Berkovich integrals\, and how to give a s
ingle-valued\nintegration theory on complex varieties. Time permitting\, I
'll explain some\npotential arithmetic applications. This is a report on j
oint work in\nprogress with Sasha Shmakov (in the complex setting) and Eri
c Katz (in the\np-adic setting).\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:François Charles (Université Paris-Sud)
DTSTART;VALUE=DATE-TIME:20210414T190000Z
DTEND;VALUE=DATE-TIME:20210414T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/26
DESCRIPTION:Title: Arithmetic curves lying in compact subsets of affine schemes
\nby François Charles (Université Paris-Sud) as part of Harvard number t
heory seminar\n\n\nAbstract\nWe will describe the notion of affine schemes
and their modifications in the context of Arakelov geometry. Using geomet
ry of numbers in infinite rank\, we will study their cohomological propert
ies. Concretely\, given an affine scheme X over Z and a compact subset K o
f the set of complex points of X\, we will investigate the geometry of tho
se proper arithmetic curves in X whose complex points lie in K. This is jo
int work with Jean-Benoît Bost.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bhargav Bhatt (University of Michigan)
DTSTART;VALUE=DATE-TIME:20210421T190000Z
DTEND;VALUE=DATE-TIME:20210421T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/27
DESCRIPTION:Title: The absolute prismatic site\nby Bhargav Bhatt (University of
Michigan) as part of Harvard number theory seminar\n\n\nAbstract\nThe abs
olute prismatic site of a p-adic formal scheme carries interesting\narithm
etic and geometric information attached to the formal scheme. In this\ntal
k\, after recalling the definition of this site\, I will discuss an\nalgeb
ro-geometric (stacky) approach to absolute prismatic cohomology and\nits c
oncomitant structures (joint with Lurie\, and partially due\nindependently
to Drinfeld). As a geometric application\, I'll explain\nDrinfeld's refin
ement of the Deligne-Illusie theorem on Hodge-to-de Rham\ndegeneration. On
the arithmetic side\, I'll describe a new classification of\ncrystalline
representations of the Galois group of a local field in terms\nof F-crysta
ls on the site (joint with Scholze).\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Keerthi Madapusi Pera (Boston College)
DTSTART;VALUE=DATE-TIME:20210310T200000Z
DTEND;VALUE=DATE-TIME:20210310T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/28
DESCRIPTION:Title: Existence of CM lifts for points on Shimura varieties\nby Ke
erthi Madapusi Pera (Boston College) as part of Harvard number theory semi
nar\n\n\nAbstract\nI'll explain a very simple proof of the fact that K3 su
rfaces of\nfinite height admit (many) CM lifts\, a result due independentl
y to\nIto-Ito-Koshikawa and Z. Yang\, which was used by the former to prov
e the\nTate conjecture for products of K3s. This will be done directly sho
wing\nthat the deformation ring of a polarized K3 surface of finite height
admits\nas a quotient that of its Brauer group. The method applies more g
enerally\nto many isogeny classes of points on Shimura varieties of abelia
n type.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laura DeMarco (Harvard University)
DTSTART;VALUE=DATE-TIME:20210210T200000Z
DTEND;VALUE=DATE-TIME:20210210T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/29
DESCRIPTION:Title: Elliptic surfaces\, bifurcations\, and equidistribution\nby
Laura DeMarco (Harvard University) as part of Harvard number theory semina
r\n\n\nAbstract\nIn joint work with Myrto Mavraki\, we studied the arithme
tic intersection of\nsections of elliptic surfaces\, defined over number f
ields. I will describe\nour results and formulate some related open quest
ions about families of\nmaps (dynamical systems) on P^1 over a base curve.
\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frank Calegari (University of Chicago)
DTSTART;VALUE=DATE-TIME:20210428T190000Z
DTEND;VALUE=DATE-TIME:20210428T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/30
DESCRIPTION:Title: From Ramanujan to K-theory\nby Frank Calegari (University of
Chicago) as part of Harvard number theory seminar\n\n\nAbstract\nThe Roge
rs-Ramanujan identity is an equality between a certain “q-series” (giv
en as an infinite sum) and a certain modular form (given as an infinite pr
oduct). Motivated by ideas from physics\, Nahm formulated a necessary cond
ition for when such q-hypergeometric series were modular. Perhaps surprisi
ngly\, this turns out to be related to algebraic K-theory. We discuss a pr
oof of this conjecture. This is joint work with Stavros Garoufalidis and D
on Zagier.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lars Kühne (University of Copenhagen)
DTSTART;VALUE=DATE-TIME:20210303T200000Z
DTEND;VALUE=DATE-TIME:20210303T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/31
DESCRIPTION:Title: Equidistribution and Uniformity in Families of Curves\nby La
rs Kühne (University of Copenhagen) as part of Harvard number theory semi
nar\n\n\nAbstract\nIn the talk\, I will present an equidistribution result
for families of (non-degenerate) subvarieties in a (general) family of ab
elian varieties. This extends a result of DeMarco and Mavraki for curves i
n fibered products of elliptic surfaces. Using this result\, one can deduc
e a uniform version of the classical Bogomolov conjecture for curves embed
ded in their Jacobians\, namely that the number of torsion points lying on
them is uniformly bounded in the genus of the curve. This has been previo
usly only known in a few select cases by work of David--Philippon and DeMa
rco--Krieger--Ye. Finally\, one can obtain a rather uniform version of the
Mordell-Lang conjecture as well by complementing a result of Dimitrov--Ga
o--Habegger: The number of rational points on a smooth algebraic curve def
ined over a number field can be bounded solely in terms of its genus and t
he Mordell-Weil rank of its Jacobian. Again\, this was previously known on
ly under additional assumptions (Stoll\, Katz--Rabinoff--Zureick-Brown).\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ziquan Yang (Harvard University)
DTSTART;VALUE=DATE-TIME:20210217T200000Z
DTEND;VALUE=DATE-TIME:20210217T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/32
DESCRIPTION:Title: Twisted derived equivalences and the Tate conjecture for K3 squa
res\nby Ziquan Yang (Harvard University) as part of Harvard number the
ory seminar\n\n\nAbstract\nThere is a long standing connection between the
Tate conjecture in codimension 1 and finiteness properties\, which first
appeared in Tate's seminal work on the endomorphisms of abelian varieties.
I will explain how one can possibly extend this connection to codimension
2 cycles\, using the theory of Brauer groups\, moduli of twisted sheaves\
, and twisted derived equivalences\, and prove the Tate conjecture for K3
squares. This recovers an earlier result of Ito-Ito-Kashikawa\, which was
established via a CM lifting theory\, and moreover provides a recipe of co
nstructing all the cycles on these varieties by purely geometric methods.\
n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Melanie Matchett Wood (Harvard University)
DTSTART;VALUE=DATE-TIME:20210908T190000Z
DTEND;VALUE=DATE-TIME:20210908T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/33
DESCRIPTION:Title: The average size of 3-torsion in class groups of 2-extensions\nby Melanie Matchett Wood (Harvard University) as part of Harvard number
theory seminar\n\nLecture held in Room 507 in the Science Center.\n\nAbst
ract\nThe p-torsion in the class group of a number field K is conjectured
to\nbe small: of size at most $|\\operatorname{Disc} K|^\\varepsilon$\, an
d to have constant\naverage size in families with a given Galois closure g
roup (when p\ndoesn't divide the order of the group). In general\, the be
st upper\nbound we have is $|\\operatorname{Disc} K|^{1/2+\\varepsilon}$\,
and previously the only two\ncases known with constant average were for 3
-torsion in quadratic\nfields (Davenport and Heilbronn\, 1971) and 2-torsi
on in non-Galois\ncubic fields (Bhargava\, 2005). We prove that the 3-tor
sion is\nconstant on average for fields with Galois closure group any 2-gr
oup\nwith a transposition\, including\, e.g. quartic $D_4$ fields. We wil
l\ndiscuss the main inputs into the proof with an eye towards giving an\ni
ntroduction to the tools in the area. This is joint work with Robert\nLem
ke Oliver and Jiuya Wang.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Salim Tayou (Harvard University)
DTSTART;VALUE=DATE-TIME:20210929T190000Z
DTEND;VALUE=DATE-TIME:20210929T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/34
DESCRIPTION:Title: Density of arithmetic Hodge loci\nby Salim Tayou (Harvard Un
iversity) 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 de
nsity of arithmetic Hodge loci which includes and generalizes several rece
nt density results of these loci in arithmetic geometry. This conjecture h
as also analogues over functions fields that I will survey. As a particula
r instance\, I will outline the proof of the following result: a K3 surfac
e over a number field admits infinitely many specializations where its Pic
ard rank jumps. This last result is joint work with Ananth Shankar\, Arul
Shankar and Yunqing Tang.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean Kieffer (Harvard University)
DTSTART;VALUE=DATE-TIME:20211006T190000Z
DTEND;VALUE=DATE-TIME:20211006T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/35
DESCRIPTION:Title: Higher-dimensional modular equations and point counting on abeli
an surfaces\nby Jean Kieffer (Harvard University) as part of Harvard n
umber theory seminar\n\nLecture held in Room 507 in the Science Center.\n\
nAbstract\nGiven a prime number l\, the elliptic modular polynomial of lev
el l is an explicit equation for the locus of elliptic curves related by a
n l-isogeny. These polynomials have a large number of algorithmic applicat
ions: in particular\, they are an essential ingredient in the celebrated S
EA algorithm for counting points on elliptic curves over finite fields of
large characteristic.\n\nMore generally\, modular equations describe the l
ocus of isogenous abelian varieties in certain moduli spaces called PEL Sh
imura varieties. We will present upper bounds on the size of modular equat
ions in terms of their level\, and outline a general strategy to compute a
n isogeny A->A' given a point (A\,A') where the modular equations are sati
sfied. This generalizes well-known properties of elliptic modular polynomi
als to higher dimensions.\n\nThe isogeny algorithm is made fully explicit
for Jacobians 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
results.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Shusterman (Harvard University)
DTSTART;VALUE=DATE-TIME:20210915T190000Z
DTEND;VALUE=DATE-TIME:20210915T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/36
DESCRIPTION:Title: Finitely Presented Groups in Arithmetic Geometry\nby Mark Sh
usterman (Harvard University) as part of Harvard number theory seminar\n\n
Lecture held in Room 507 in the Science Center.\n\nAbstract\nWe discuss th
e problem of determining the number of generators and relations of several
profinite groups of arithmetic and geometric origin. \nThese include etal
e fundamental groups of smooth projective varieties\, absolute Galois grou
ps of local fields\, and Galois groups of maximal unramified extensions of
number fields. The results are based on a cohomological presentability cr
iterion of Lubotzky\, and draw inspiration from well-known facts about thr
ee-dimensional manifolds\, as in arithmetic topology. \n\nThe talk is ba
sed on a joint work with Esnault and Srinivas\, on a joint work with Jarde
n\, and on work of Yuan Liu.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Petrov (Harvard University)
DTSTART;VALUE=DATE-TIME:20210922T190000Z
DTEND;VALUE=DATE-TIME:20210922T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/37
DESCRIPTION:Title: Galois action on the pro-algebraic completion of the fundamental
group\nby Alexander Petrov (Harvard University) as part of Harvard nu
mber theory seminar\n\nLecture held in Room 507 in the Science Center.\n\n
Abstract\nGiven a variety over a number field\, its geometric etale\nfunda
mental group comes equipped with an action of the Galois group. This\nindu
ces a Galois action on the pro-algebraic completion of the etale\nfundamen
tal group and hence the ring of functions on that pro-algebraic\ncompletio
n provides a supply of Galois representations.\n\nIt turns out that any fi
nite-dimensional p-adic Galois representation\ncontained in the ring of fu
nctions on the pro-algebraic completion of the\nfundamental group of a smo
oth variety satisfies the assumptions of the\nFontaine-Mazur conjecture: i
t is de Rham at places above p and is a. e.\nunramified.\n\nConversely\, w
e will show that every semi-simple representation of the\nGalois group of
a number field coming from algebraic geometry (that is\,\nappearing as a s
ubquotient of the etale cohomology of an algebraic variety)\ncan be establ
ished as a subquotient of the ring of functions on the\npro-algebraic comp
letion of the fundamental group of the projective line\nwith 3 punctures.\
n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Levent Alpöge (Harvard University)
DTSTART;VALUE=DATE-TIME:20211020T190000Z
DTEND;VALUE=DATE-TIME:20211020T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/38
DESCRIPTION:Title: Effective height bounds for odd-degree totally real points on so
me curves\nby Levent Alpöge (Harvard University) as part of Harvard n
umber theory seminar\n\nLecture held in Room 507 in the Science Center.\n\
nAbstract\nI will give a finite-time algorithm that\, on input (g\,K\,S) w
ith g > 0\, K a totally real number field of odd degree\, and S a finite s
et of places of K\, outputs the finitely many g-dimensional abelian variet
ies A/K which are of $\\operatorname{GL}_2$-type over K and have good redu
ction outside S.\n\nThe point of this is to effectively compute the S-inte
gral K-points on a Hilbert modular variety\, and the point of that is to b
e able to compute all K-rational points on complete curves inside such var
ieties.\n\nThis gives effective height bounds for rational points on infin
itely many curves and (for each curve) over infinitely many number fields.
For example 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:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wei Zhang (MIT)
DTSTART;VALUE=DATE-TIME:20211027T190000Z
DTEND;VALUE=DATE-TIME:20211027T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/39
DESCRIPTION:Title: p-adic Heights of the arithmetic diagonal cycles\nby Wei Zha
ng (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 Daniel 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
associated to cuspidal automorphic representations. In the good ordinary c
ase we are able to prove the conjecture\, at least when the ramifications
are mild at inert primes. We deduce some application to the p-adic version
of the Bloch-Kato conjecture.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhiwei Yun (MIT)
DTSTART;VALUE=DATE-TIME:20211103T190000Z
DTEND;VALUE=DATE-TIME:20211103T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/40
DESCRIPTION:Title: Special cycles for unitary Shtukas and modularity\nby Zhiwei
Yun (MIT) as part of Harvard number theory seminar\n\nLecture held in Roo
m 507 in the Science Center.\n\nAbstract\nWe define a generating series of
algebraic cycles on the moduli\nstack of unitary Shtukas and conjecture t
hat it is a Chow-group valued\nautomorphic form. This is a function field
analogue of the special cycles\ndefined by Kudla and Rapoport\, but with a
n extra degree of freedom namely\nthe number of legs of the Shtukas. I wil
l talk about several pieces of\nevidence for the conjecture. This is joint
work with Tony Feng and Wei\nZhang.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Feng (MIT)
DTSTART;VALUE=DATE-TIME:20211110T200000Z
DTEND;VALUE=DATE-TIME:20211110T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/41
DESCRIPTION:Title: The Galois action on symplectic K-theory\nby Tony Feng (MIT)
as part of Harvard number theory seminar\n\nLecture held in Room 507 in t
he Science Center.\n\nAbstract\nA phenomenon underlying many remarkable re
sults in number theory is the natural Galois action on the cohomology of s
ymplectic groups of integers. In joint work with Soren Galatius and Akshay
Venkatesh\, we define a symplectic variant of algebraic K-theory\, which
carries a natural Galois action for similar reasons. We compute this Galoi
s action and characterize it in terms of a universality property\, in the
spirit of the Langlands philosophy.\n
LOCATION:
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:20241013T133215Z
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\nL
ecture held in Room 507 in the Science Center.\n\nAbstract\nThe étale coh
omology of varieties over Q enjoys a Galois action. In the\ncase of Hilber
t modular varieties\, Nekovář-Scholl observed that this Galois\naction o
n the level of cohomology extends to a much larger profinite group:\nthe p
lectic group. They conjectured that this extension holds even on the\nleve
l of complexes\, as well as for more general Shimura varieties.\n\nWe pres
ent a proof of the analogue of this conjecture for local Shimura\nvarietie
s. This includes (the generic fibers of) Lubin–Tate spaces\,\nDrinfeld u
pper half spaces\, and more generally Rapoport–Zink spaces. The\nproof c
rucially uses Scholze's theory of diamonds.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Howard (Boston College)
DTSTART;VALUE=DATE-TIME:20211117T200000Z
DTEND;VALUE=DATE-TIME:20211117T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/43
DESCRIPTION:Title: Arithmetic volumes of unitary Shimura varieties\nby Benjamin
Howard (Boston College) as part of Harvard number theory seminar\n\nLectu
re held in Room 507 in the Science Center.\n\nAbstract\nThe integral model
of a GU(n-1\,1) Shimura variety carries a natural metrized line bundle of
modular forms. Viewing this metrized line bundle as a class in the codim
ension one arithmetic Chow group\, one can define its arithmetic volume as
an iterated self-intersection. We will show that this volume can be expr
essed in terms of logarithmic derivatives of Dirichlet L-functions at inte
ger points\, and explain the connection with the arithmetic Siegel-Weil co
njecture of Kudla-Rapoport. This is joint work with Jan Bruinier.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Kisin (Harvard University)
DTSTART;VALUE=DATE-TIME:20211201T200000Z
DTEND;VALUE=DATE-TIME:20211201T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/44
DESCRIPTION:Title: Mod p points on Shimura varieties\nby Mark Kisin (Harvard Un
iversity) 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 Shim
ura varieties was originally\nmotivated by the study of the Hasse-Weil zet
a function for Shimura\nvarieties.\nIt involves some rather subtle problem
s which test just how much we know\nabout motives over finite fields. In t
his talk I will explain some recent\nresults\, and\napplications\, and wha
t still remains conjectural.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Betts (Harvard University)
DTSTART;VALUE=DATE-TIME:20220209T200000Z
DTEND;VALUE=DATE-TIME:20220209T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/45
DESCRIPTION:Title: Galois sections and the method of Lawrence--Venkatesh\nby Al
exander Betts (Harvard University) as part of Harvard number theory semina
r\n\nLecture held in Room 507 in the Science Center.\n\nAbstract\nGrothend
ieck's Section Conjecture posits that the set of rational\npoints on a smo
oth projective curve Y of genus at least two should be equal\nto a certain
"section set" defined purely in terms of the etale fundamental\ngroup of
Y. In this talk\, I will preview some upcoming work with Jakob Stix\nin wh
ich we prove a partial finiteness result for this section set\, thereby\ng
iving an unconditional verification of a prediction of the Section\nConjec
ture for a general curve Y. We do this by adapting the recent p-adic\nproo
f of the Mordell Conjecture due to Brian Lawrence and Akshay Venkatesh.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fabian Gundlach (Harvard University)
DTSTART;VALUE=DATE-TIME:20220202T200000Z
DTEND;VALUE=DATE-TIME:20220202T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/46
DESCRIPTION:Title: Counting quaternionic extensions\nby Fabian Gundlach (Harvar
d University) as part of Harvard number theory seminar\n\nLecture held in
Room 507 in the Science Center.\n\nAbstract\nConsider the set of Galois ex
tensions $L$ of $\\mathbb Q$ whose Galois group is the quaternion group. F
or large $X$\, Klüners counted extensions with $|\\mathrm{disc}(L)| <= X$
. We discuss asymptotics when bounding invariants other than the discrimin
ant.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Naomi Sweeting (Harvard University)
DTSTART;VALUE=DATE-TIME:20220216T200000Z
DTEND;VALUE=DATE-TIME:20220216T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/47
DESCRIPTION:Title: Kolyvagin's conjecture\, bipartite Euler systems\, and higher co
ngruences of modular forms\nby Naomi Sweeting (Harvard University) as
part of Harvard number theory seminar\n\nLecture held in Room 507 in the S
cience Center.\n\nAbstract\nFor an elliptic curve E\, Kolyvagin used Heeg
ner points to construct\nspecial Galois cohomology classes valued in the t
orsion points of E. Under\nthe conjecture that not all of these classes va
nish\, he showed that they\nencode the Selmer rank of E. I will explain a
proof of new cases of this\nconjecture that builds on prior work of Wei Zh
ang. The proof naturally\nleads to a generalization of Kolyvagin's work in
a complimentary "definite"\nsetting\, where Heegner points are replaced b
y special values of a\nquaternionic modular form. I'll also explain an "ul
trapatching" formalism\nwhich simplifies the Selmer group arguments requir
ed for the proof.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Landesman (Harvard University)
DTSTART;VALUE=DATE-TIME:20220223T200000Z
DTEND;VALUE=DATE-TIME:20220223T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/48
DESCRIPTION:Title: Geometric local systems on very general curves\nby Aaron Lan
desman (Harvard University) as part of Harvard number theory seminar\n\nLe
cture held 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
character variety\nparameterizing complex rank r local systems.\nIn joint
work with Daniel Litt\, we show these conjectures fail to hold when\nX is
a sufficiently general curve of genus $g$ and $r < 2\\sqrt{g+1}$\nby show
ing that any such local system coming from geometry is in fact\nisotrivial
.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Myrto Mavraki (Harvard University)
DTSTART;VALUE=DATE-TIME:20220302T200000Z
DTEND;VALUE=DATE-TIME:20220302T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/49
DESCRIPTION:Title: Towards uniformity in the dynamical Bogomolov conjecture\nby
Myrto Mavraki (Harvard University) as part of Harvard number theory semin
ar\n\nLecture held in Room 507 in the Science Center.\n\nAbstract\nInspire
d by an analogy between torsion and preperiodic points\,\nZhang has propos
ed a dynamical generalization of the classical\nManin-Mumford and Bogomolo
v conjectures. A special case of these\nconjectures\, for `split' maps\, h
as recently been established by Nguyen\,\nGhioca and Ye. In particular\, t
hey show that two rational maps have at most\nfinitely many common preperi
odic points\, unless they are `related'. Recent\nbreakthroughs by Dimitrov
\, Gao\, Habegger and Kühne have established that\nthe classical Bogomolo
v conjecture holds uniformly across curves of given\ngenus.\nIn this talk
we discuss uniform versions of the dynamical Bogomolov\nconjecture across
1-parameter families of split maps and curves. To this\nend\, we establish
instances of a 'relative dynamical Bogomolov conjecture'.\nThis is joint
work with Harry Schmidt (University of Basel).\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Pollack (Boston University)
DTSTART;VALUE=DATE-TIME:20220427T190000Z
DTEND;VALUE=DATE-TIME:20220427T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/50
DESCRIPTION:Title: Slopes of modular forms and reductions of crystalline representa
tions\nby Robert Pollack (Boston University) as part of Harvard number
theory seminar\n\nLecture held in Room 507 in the Science Center.\n\nAbst
ract\nThe ghost conjecture predicts slopes of modular forms whose\nresidua
l representation is locally reducible. In this talk\, we'll examine\nloca
lly irreducible representations and discuss recent progress on\nformulatin
g a conjecture in this case. It's a lot trickier and the story\nremains i
ncomplete\, but we will discuss how an irregular ghost conjecture\nis inti
mately related to reductions of crystalline representations.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jennifer Balakrishnan (Boston University)
DTSTART;VALUE=DATE-TIME:20220420T190000Z
DTEND;VALUE=DATE-TIME:20220420T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/51
DESCRIPTION:Title: Quadratic Chabauty for modular curves\nby Jennifer Balakrish
nan (Boston University) as part of Harvard number theory seminar\n\nLectur
e held in Room 507 in the Science Center.\n\nAbstract\nAbstract: We descri
be how p-adic height pairings can be used to\ndetermine the set of rationa
l points on curves\, in the spirit of Kim's\nnonabelian Chabauty program.
In particular\, 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:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Allechar Serrano López
DTSTART;VALUE=DATE-TIME:20220309T200000Z
DTEND;VALUE=DATE-TIME:20220309T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/52
DESCRIPTION:Title: Counting fields generated by points on plane curves\nby Alle
char Serrano López as part of Harvard number theory seminar\n\nLecture he
ld in Room 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 o
f $C(\\overline{\\mathbb{Q}})$? Can we further\ncount the number of such e
xtensions with a specified Galois group?\nAsymptotic lower bounds for thes
e quantities have been found for elliptic\ncurves by Lemke Oliver and Thor
ne\, for hyperelliptic curves by Keyes\, and\nfor superelliptic curves by
Beneish and Keyes. We discuss similar\nasymptotic lower bounds that hold f
or all smooth plane curves $C$.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Will Sawin (Columbia University)
DTSTART;VALUE=DATE-TIME:20220323T190000Z
DTEND;VALUE=DATE-TIME:20220323T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/53
DESCRIPTION:Title: A visit to 3-manifolds in the quest to understand random Galois
groups\nby Will Sawin (Columbia University) as part of Harvard number
theory seminar\n\nLecture held in Room 507 in the Science Center.\n\nAbstr
act\nCohen and Lenstra gave a conjectural distribution for the class group
of a random quadratic number field. Since the class group is the Galois g
roup of the maximum abelian unramified extension\, a natural generalizatio
n would be to give a conjecture for the distribution of the Galois group o
f the maximal unramified extension. Previous work has produced a plausible
conjecture in special cases\, with the most general being recent work of
Liu\, Wood\, and Zurieck-Brown.\n\nThere is a deep analogy between number
fields and 3-manifolds. Thus\, an analogous question would be to describe
the distribution of the profinite completion of the fundamental group of a
random 3-manifold. In this talk\, I will explain how Melanie Wood and I a
nswered this question for a model of random 3-manifolds defined by Dunfiel
d and Thurston\, and how the techniques we used should allow us\, in futur
e work\, to give a more general conjecture in the number field case.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eric Urban (Columbia University)
DTSTART;VALUE=DATE-TIME:20220413T190000Z
DTEND;VALUE=DATE-TIME:20220413T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/54
DESCRIPTION:Title: Euler systems and the p-adic Langlands correspondence\nby Er
ic Urban (Columbia University) as part of Harvard number theory seminar\n\
nLecture held in Room 507 in the Science Center.\n\nAbstract\nAbout 2 year
s ago\, I have given a new construction of the Euler system of cyclotomic
units via Eisenstein congruences in which the p-adic Langlands correspond
ence for $\\GL_2(\\Q_p)$ plays a central role. In this talk\, I want to ex
plain how 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:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yunqing Tang (Princeton University)
DTSTART;VALUE=DATE-TIME:20220330T190000Z
DTEND;VALUE=DATE-TIME:20220330T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/55
DESCRIPTION:Title: The unbounded denominators conjecture\nby Yunqing Tang (Prin
ceton University) as part of Harvard number theory seminar\n\nLecture held
in Room 507 in the Science Center.\n\nAbstract\nThe unbounded denominator
s conjecture\, first raised by Atkin and Swinnerton-Dyer\, asserts that a
modular form for a finite index subgroup of $\\SL_2(\\mathbb Z)$ whose Fou
rier coefficients have bounded denominators must be a modular form for som
e congruence subgroup. In this talk\, we will give a sketch of the proof o
f this conjecture based on a new arithmetic algebraization theorem. This i
s joint work with Frank Calegari and Vesselin Dimitrov.\n
LOCATION:
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:20241013T133215Z
UID:HarvardNT/56
DESCRIPTION:Title: Non-archimedean and tropical geometry\, algebraic groups\, modul
i spaces of matroids\, and the field with one element\nby Matt Baker (
Georgia Institute of Technology) as part of Harvard number theory seminar\
n\nLecture held in Room 507 in the Science Center.\n\nAbstract\nI will giv
e an introduction to Oliver Lorscheid’s theory of\nordered blueprints
– one of the more successful approaches to “the field of\none element
” – and sketch its relationship to Berkovich spaces\, tropical\ngeomet
ry\, Tits models for algebraic groups\, and moduli spaces of matroids.\nTh
e basic idea for the latter two applications is quite simple: given a\nsch
eme over **Z** defined by equations with coefficients in {0\,1\,-1}\, t
here\nis a corresponding “blue model” whose **K**-points (where **
K** is the Krasner\nhyperfield) sometimes correspond to interesting comb
inatorial structures.\nFor example\, taking **K**-points of a suitable
blue model for a split\nreductive group scheme G over **Z** gives the W
eyl group of G\, and\ntaking **K**-points\nof a suitable blue model for
the Grassmannian G(r\,n) gives the set of\nmatroids of rank r on {1\,…\
,n}. Similarly\, the Berkovich analytification of\na scheme X over a value
d field K coincides\, as a topological space\, with\nthe set of **T**-p
oints of X\, considered as an ordered blue scheme over K.\nHere **T** i
s the tropical hyperfield\, and **T**-points are defined using the\nobs
ervation that a (height 1) valuation on K is nothing other than a\nhomomor
phism to **T**.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bianca Viray (University of Washington)
DTSTART;VALUE=DATE-TIME:20220406T190000Z
DTEND;VALUE=DATE-TIME:20220406T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/57
DESCRIPTION:Title: Isolated points on modular curves\nby Bianca Viray (Universi
ty of Washington) as part of Harvard number theory seminar\n\nLecture held
in Room 507 in the Science Center.\n\nAbstract\nLet C be an algebraic cur
ve over a number field. Faltings's theorem on\nrational points on subvarie
ties of abelian varieties implies that all\nalgebraic points on C arise in
algebraic families\, with finitely many\nexceptions. These exceptions ar
e known as isolated points. We study how\nisolated points behave under mor
phisms and then specialize to the case of\nmodular curves. We show that i
solated points on X_1(n) push down to\nisolated points on a modular curve
whose level 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:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akshay Venkatesh (IAS)
DTSTART;VALUE=DATE-TIME:20220914T190000Z
DTEND;VALUE=DATE-TIME:20220914T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/60
DESCRIPTION:Title: Symplectic Reidemeister torsion and symplectic $L$-functions
\nby Akshay Venkatesh (IAS) as part of Harvard number theory seminar\n\nLe
cture held in Room 507 in the Science Center.\n\nAbstract\nMany of the qua
ntities appearing in the conjecture of Birch and Swinnerton-Dyer look susp
iciously like squares. Motivated by this and related examples\, we may ask
if the central value of an $L$-function "of symplectic type" admits a pre
ferred square root.\n\nThe answer is no: there's an interesting cohomologi
cal obstruction. More formally\, in the everywhere unramified situation ov
er a function field\, I will describe an explicit cohomological formula fo
r the $L$-function modulo squares. This is based on a purely topological r
esult about $3$-manifolds. If time permits I'll speculate on generalizatio
ns. This is based on joint work with Amina Abdurrahman.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Koymans (University of Michigan)
DTSTART;VALUE=DATE-TIME:20220921T190000Z
DTEND;VALUE=DATE-TIME:20220921T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/61
DESCRIPTION:Title: The negative Pell equation and applications\nby Peter Koyman
s (University of Michigan) as part of Harvard number theory seminar\n\nLec
ture held in Room 507 in the Science Center.\n\nAbstract\nIn this talk we
will study 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
concerned with the following question: as we vary over squarefree integers
$D$\, how often is $C_D$ soluble? Stevenhagen conjectured an asymptotic f
ormula for such $D$. Fouvry and Klüners gave upper and lower bounds of th
e correct order of magnitude. We will discuss a proof of Stevenhagen's con
jecture\, and potential applications of the new proof techniques. This is
joint work with Carlo Pagano.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ziquan Yang (UW Madison)
DTSTART;VALUE=DATE-TIME:20220928T190000Z
DTEND;VALUE=DATE-TIME:20220928T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/62
DESCRIPTION:Title: The Tate conjecture for $h^{2\, 0} = 1$ varieties over finite fi
elds\nby Ziquan Yang (UW Madison) as part of Harvard number theory sem
inar\n\nLecture held in Room 507 in the Science Center.\n\nAbstract\nThe p
ast decade has witnessed a great advancement on the Tate conjecture for va
rieties with Hodge number $h^{2\, 0} = 1$. Charles\, Madapusi-Pera and Mau
lik completely settled the conjecture for K3 surfaces over finite fields\,
and Moonen proved the Mumford-Tate (and hence also Tate) conjecture for m
ore or less arbitrary $h^{2\, 0} = 1$ varieties in characteristic $0$.\n\n
In this talk\, I will explain that the Tate conjecture is true for mod $p$
reductions of complex projective $h^{2\, 0} = 1$ varieties when $p$ is bi
g enough\, under a mild assumption on moduli. By refining this general res
ult\, we prove that in characteristic $p$ at least $5$ the BSD conjecture
holds for 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 conjecture over finite fields for a class of surfaces of general type
and a class of Fano varieties. The overall philosophy is that the connect
ion between the Tate conjecture over finite fields and the Lefschetz $(1\,
1)$-theorem over the complex numbers is very robust for $h^{2\, 0} = 1$ v
arieties\, and works well beyond the hyperkähler world.\n\nThis is based
on joint work with Paul Hamacher and Xiaolei Zhao.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Li-Huerta (Harvard)
DTSTART;VALUE=DATE-TIME:20221005T190000Z
DTEND;VALUE=DATE-TIME:20221005T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/63
DESCRIPTION:Title: Local-global compatibility over function fields\nby Daniel L
i-Huerta (Harvard) as part of Harvard number theory seminar\n\nLecture hel
d in Room 507 in the Science Center.\n\nAbstract\nThe Langlands program pr
edicts a relationship between automorphic representations of a reductive g
roup $G$ and Galois representations valued in its $L$-group. For general $
G$ over a global function field\, the automorphic-to-Galois direction has
been constructed by V. Lafforgue. More recently\, for general $G$ over a n
onarchimedean local field\, a similar correspondence has been constructed
by Fargues–Scholze.\n\nWe present a proof that the V. Lafforgue and Farg
ues–Scholze correspondences are compatible\, generalizing local-global c
ompatibility from class field theory. As a consequence\, the correspondenc
es of Genestier–Lafforgue and Fargues–Scholze agree\, which answers a
question of Fargues–Scholze\, Hansen\, Harris\, and Kaletha.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hélène Esnault (Freie Universität Berlin)
DTSTART;VALUE=DATE-TIME:20221012T190000Z
DTEND;VALUE=DATE-TIME:20221012T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/64
DESCRIPTION:Title: Integrality properties of the Betti moduli space\nby Hélèn
e Esnault (Freie Universität Berlin) as part of Harvard number theory sem
inar\n\nLecture held in Room 507 in the Science Center.\n\nAbstract\nWe st
udy them\, in particular showing on a smooth complex quasi-projective vari
ety the existence of $\\ell$-adic absolutely irreducible local systems fo
r 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 de Jong\, relying in part on earlier work with Michael Groechenig.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jef Laga (Princeton)
DTSTART;VALUE=DATE-TIME:20221019T190000Z
DTEND;VALUE=DATE-TIME:20221019T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/65
DESCRIPTION:Title: Arithmetic statistics via graded Lie algebras\nby Jef Laga (
Princeton) as part of Harvard number theory seminar\n\nLecture held in Roo
m 507 in the Science Center.\n\nAbstract\nI will explain how various resul
ts in arithmetic statistics by Bhargava\, Gross\, Shankar and others on $2
$-Selmer groups of Jacobians of (hyper)elliptic curves can be organised an
d reproved using the theory of graded Lie algebras\, following earlier wor
k of Thorne. This gives a uniform proof of these results and yields new th
eorems for certain families of non-hyperelliptic curves. I will also menti
on some applications to rational points on certain families of curves.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shai Haran (Technion)
DTSTART;VALUE=DATE-TIME:20221026T190000Z
DTEND;VALUE=DATE-TIME:20221026T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
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 Room 507 in the Science Center.\n\nAbstract\nThe usual language of alg
ebraic geometry is not appropriate for arithmetical geometry: addition is
singular at the real prime. We developed two languages that overcome this
problem: one replace s rings by the collection of “vectors” or by bi-o
perads\, and another based on “matrices” or props. Once one understand
s the delicate commutativity condition one can proceed following Grothendi
eck's footsteps exactly. The props\, when viewed up to conjugation\, give
us new commutative rings with Frobenius endomorphisms.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Spencer Leslie (Boston College)
DTSTART;VALUE=DATE-TIME:20221102T190000Z
DTEND;VALUE=DATE-TIME:20221102T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/67
DESCRIPTION:Title: Endoscopy for symmetric varieties\nby Spencer Leslie (Boston
College) as part of Harvard number theory seminar\n\nLecture held in Room
507 in the Science Center.\n\nAbstract\nRelative trace formulas are centr
al tools in the study of relative functoriality. In many cases of interest
\, basic stability problems have not previously been addressed. In this ta
lk\, I discuss a theory of endoscopy in the context of symmetric varieties
with the global goal of stabilizing the associated relative trace formula
. I outline how\, using the dual group of the symmetric variety\, one can
give a good notion of endoscopic symmetric variety and conjecture a matchi
ng of relative orbital integrals in order to stabilize the relative trace
formula\, which can be proved in some cases. Time permitting\, I will expl
ain my proof of these conjectures in the case of unitary Friedberg–Jacqu
et periods.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gyujin Oh (Columbia)
DTSTART;VALUE=DATE-TIME:20221109T200000Z
DTEND;VALUE=DATE-TIME:20221109T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/68
DESCRIPTION:Title: Cohomological degree-shifting operators on Shimura varieties
\nby Gyujin Oh (Columbia) as part of Harvard number theory seminar\n\nLect
ure held in Room 507 in the Science Center.\n\nAbstract\nAn automorphic fo
rm can appear in multiple degrees of the cohomology of arithmetic manifold
s\, and this happens mostly when the arithmetic manifolds are not algebrai
c. This phenomenon is a part of the "derived" structures of the Langlands
program\, suggested by Venkatesh. However\, even over algebraic arithmetic
manifolds\, certain automorphic forms like weight-one elliptic modular fo
rms possess 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:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tasho Kaletha (University of Michigan)
DTSTART;VALUE=DATE-TIME:20221116T200000Z
DTEND;VALUE=DATE-TIME:20221116T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/69
DESCRIPTION:Title: Covers of reductive groups and functoriality\nby Tasho Kalet
ha (University of Michigan) as part of Harvard number theory seminar\n\nLe
cture held in Room 507 in the Science Center.\n\nAbstract\nTo a connected
reductive 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 obtain an $n$-fold cover $G(F)_x$ of the topological group $G(F)$. We
also define an $L$-group for $G(F)_x$\, which is a usually non-split exten
sion of the Galois group by the dual group of G\, and deduce from the line
ar case a refined local Langlands correspondence between genuine represent
ations of $G(F)_x$ and $L$-parameters valued in this $L$-group.\n\nThis co
nstruction is motivated by Langlands functoriality. We show that a subgrou
p of the $L$-group of $G$ of a certain kind naturally lead to a smaller qu
asi-split group $H$ and a double cover of $H(F)$. Genuine representations
of this double cover are expected to be in functorial relationship with re
presentations of $G(F)$. We will present two concrete applications of this
\, one that gives a characterization of the local Langlands correspondence
for supercuspidal $L$-parameters when $p$ is sufficiently large\, and one
to the theory of endoscopy.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abhishek Oswal (Caltech)
DTSTART;VALUE=DATE-TIME:20221130T200000Z
DTEND;VALUE=DATE-TIME:20221130T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/70
DESCRIPTION:Title: A $p$-adic analogue of an algebraization theorem of Borel\nb
y Abhishek Oswal (Caltech) as part of Harvard number theory seminar\n\nLec
ture held in Room 507 in the Science Center.\n\nAbstract\nLet $S$ be a Shi
mura variety such that the connected components of the set of complex poin
ts $S(\\mathbb{C})$ are of the form $D/\\Gamma$\, where $\\Gamma$ is a tor
sion-free arithmetic group acting on the Hermitian symmetric domain $D$. B
orel proved that any holomorphic map from any complex algebraic variety in
to $S(\\mathbb{C})$ is an algebraic map. In this talk I shall describe ong
oing joint work with Ananth Shankar and Xinwen Zhu\, where we prove a $p$-
adic analogue of this result of Borel for compact Shimura varieties of abe
lian type.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Pagano (Concordia)
DTSTART;VALUE=DATE-TIME:20221207T200000Z
DTEND;VALUE=DATE-TIME:20221207T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/71
DESCRIPTION:Title: Malle's conjecture for nilpotent groups\nby Carlo Pagano (Co
ncordia) as part of Harvard number theory seminar\n\nLecture held in Room
507 in the Science Center.\n\nAbstract\nMalle's conjecture is a quantitati
ve version of the Galois inverse problem. Namely\, fixing some ramificatio
n invariant of number fields (discriminant\, product of ramified primes\,
etc)\, for a finite group $G$ one seeks an asymptotic formula for the numb
er of $G$-extensions (of a given number field) having bounded ramification
invariant. In this talk I will overview past and ongoing joint work with
Peter Koymans focusing on the case of nilpotent groups.\n
LOCATION:
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:20241013T133215Z
UID:HarvardNT/72
DESCRIPTION:Title: Bielliptic Picard curves\nby Ari Shnidman (Hebrew University
of Jerusalem) as part of Harvard number theory seminar\n\nLecture held in
Room 507 in the Science Center.\n\nAbstract\nI'll describe the geometry a
nd arithmetic 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 fr
om the geometry of the curve\, and then I'll give some results on the Mord
ell–Weil groups $A(\\mathbb{Q})$. This is based on joint work with Laga
and Laga–Schembri–Voight.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Weinstein (Boston University)
DTSTART;VALUE=DATE-TIME:20230208T200000Z
DTEND;VALUE=DATE-TIME:20230208T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/73
DESCRIPTION:Title: Higher modularity of elliptic curves\nby Jared Weinstein (Bo
ston University) as part of Harvard number theory seminar\n\nLecture held
in Room 507 in the Science Center.\n\nAbstract\nElliptic curves $E$ over t
he rational numbers are modular: this means there is a nonconstant map fro
m a modular curve to $E$. When instead the coefficients of $E$ belong to a
function field\, it still makes sense to talk about the modularity of $E$
(and this is known)\, but one can also extend the idea further and ask wh
ether $E$ is '$r$-modular' for $r=2\,3\\ldots$. To define this generalizat
ion\, the modular curve gets replaced with Drinfeld's concept of a 'shtuka
space'. The $r$-modularity of $E$ is predicted by Tate's conjecture. In j
oint work with Adam Logan\, we give some classes of elliptic curves $E$ wh
ich are $2$- and $3$-modular.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Levent Alpöge (Harvard)
DTSTART;VALUE=DATE-TIME:20230215T200000Z
DTEND;VALUE=DATE-TIME:20230215T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/74
DESCRIPTION:Title: Integers which are(n’t) the sum of two cubes\nby Levent Al
pöge (Harvard) as part of Harvard number theory seminar\n\nLecture held i
n Room 507 in the Science Center.\n\nAbstract\nFermat identified the integ
ers which are a sum of two squares\, integral or rational: they are exactl
y those integers which have all primes congruent to 3 (mod 4) occurring to
an even power in their prime factorization — a condition satisfied by 0
% of integers!\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 positiv
e proportion of integers aren’t the sum of two rational cubes\,\n\n2. an
d also a positive proportion are!\n\n(Joint with Manjul Bhargava and Ari S
hnidman.)\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yujie Xu (MIT)
DTSTART;VALUE=DATE-TIME:20230222T200000Z
DTEND;VALUE=DATE-TIME:20230222T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/75
DESCRIPTION:Title: Hecke algebras for $p$-adic groups and the explicit local Langla
nds correspondence for $\\mathrm{G}_2$\nby Yujie Xu (MIT) as part of H
arvard number theory seminar\n\nLecture held in Room 507 in the Science Ce
nter.\n\nAbstract\nI will talk about my recent joint work with Aubert wher
e we prove the local Langlands conjecture for $\\mathrm{G}_2$ (explicitly)
. This uses our earlier results on Hecke algebras attached to Bernstein co
mponents of reductive $p$-adic groups\, as well as an expected property on
cuspidal support\, along with a list of characterizing properties. In par
ticular\, we obtain "mixed" $L$-packets containing $F$-singular supercuspi
dals and non-supercuspidals.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashvin Swaminathan (Harvard)
DTSTART;VALUE=DATE-TIME:20230301T200000Z
DTEND;VALUE=DATE-TIME:20230301T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/76
DESCRIPTION:Title: Counting integral points on symmetric varieties\, and applicatio
ns to arithmetic statistics\nby Ashvin Swaminathan (Harvard) as part o
f Harvard number theory seminar\n\nLecture held in Room 507 in the Science
Center.\n\nAbstract\nOver the past few decades\, significant progress has
been made in arithmetic statistics by the following two-step process: (1)
parametrize arithmetic objects of interest in terms of the integral orbit
s of a representation of a group $G$ acting on a vector space $V$\; and (2
) use geometry-of-numbers methods to count the orbits of $G(\\mathbb{Z})$
on $V(\\mathbb{Z})$. But it often happens that the arithmetic objects of i
nterest correspond to orbits that lie on a proper subvariety of $V$. In su
ch cases\, geometry-of-numbers methods do not suffice to obtain precise as
ymptotics\, and more sophisticated point-counting techniques are required.
In this talk\, we explain how the Eskin–McMullen method for counting in
tegral 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 with Iman Setayesh\, Arul Shankar\, and Artane Siad)\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Shusterman (Harvard)
DTSTART;VALUE=DATE-TIME:20230308T200000Z
DTEND;VALUE=DATE-TIME:20230308T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
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\nLecture held in Room 507 in the Science Center.\n\nAbstract\nHecke has
shown that 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 clo
sed $3$-manifolds saying that the branch divisor of a covering is a square
in the first homology group.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lillian Pierce (Duke)
DTSTART;VALUE=DATE-TIME:20230322T190000Z
DTEND;VALUE=DATE-TIME:20230322T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/78
DESCRIPTION:Title: A polynomial sieve: beyond separation of variables\nby Lilli
an Pierce (Duke) as part of Harvard number theory seminar\n\nLecture held
in Room 507 in the Science Center.\n\nAbstract\nMany problems in number th
eory can be framed as questions about counting solutions to a Diophantine
equation (say\, within a certain “box”). If there are very few\, or ve
ry many variables\, certain methods gain an advantage\, but sometimes ther
e is extra structure that can be exploited as well. For example: let $f$ b
e a given 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 question
s arise in a variety of specific applications\, and also in the context of
a general conjecture of Serre on counting points in thin sets. We will de
scribe how sieve methods can exploit this type of structure\, and explain
how a new polynomial sieve method allows greater flexibility\, so that the
variables in the polynomials $f$ and $g$ can “mix.” This is joint wor
k with Dante Bonolis.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wei Ho (Princeton / IAS)
DTSTART;VALUE=DATE-TIME:20230329T190000Z
DTEND;VALUE=DATE-TIME:20230329T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/79
DESCRIPTION:Title: Selmer averages in families of elliptic curves and applications<
/a>\nby Wei Ho (Princeton / IAS) as part of Harvard number theory seminar\
n\nLecture held in Room 507 in the Science Center.\n\nAbstract\nOrbits of
many coregular representations of algebraic groups are closely linked to m
oduli spaces of genus one curves with extra data. We may use these orbit p
arametrizations to compute the average size of Selmer groups of elliptic c
urves in certain families\, e.g.\, with marked points\, thus obtaining upp
er bounds for the average ranks of the elliptic curves in these families.
(This is joint work with Manjul Bhargava.) We will also describe some othe
r applications and related work (some joint with collaborators\, including
Levent Alpöge\, Manjul Bhargava\, Tom Fisher\, Jennifer Park).\n
LOCATION:
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:20241013T133215Z
UID:HarvardNT/80
DESCRIPTION:Title: Evaluating the wild Brauer group\nby Rachel Newton (King's C
ollege London) as part of Harvard number theory seminar\n\nLecture held in
Room 507 in the Science Center.\n\nAbstract\nThe local-global approach to
the study of rational points on varieties over number fields begins by em
bedding the set of rational points on a variety $X$ into the set of its ad
elic points. The Brauer–Manin pairing cuts out a subset of the adelic po
ints\, 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 emp
ty then we say there's a Brauer–Manin obstruction to the existence of ra
tional points on $X$. Computing the Brauer–Manin pairing involves evalua
ting elements of the Brauer group of $X$ at local points. If an element of
the Brauer group has order coprime to $p$\, then its evaluation at a $p$-
adic point factors via reduction of the point modulo $p$. For elements of
order a power of $p$\, this is no longer the case: in order to compute the
evaluation map one must know the point to a higher $p$-adic precision. Cl
assifying Brauer group elements according to the precision required to eva
luate them at $p$-adic points gives a filtration which we describe using w
ork of Kato. Applications of our work include addressing Swinnerton-Dyer's
question about which places can play a role in the Brauer–Manin obstruc
tion. This is joint work with Martin Bright.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Harris (Columbia)
DTSTART;VALUE=DATE-TIME:20230412T190000Z
DTEND;VALUE=DATE-TIME:20230412T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/81
DESCRIPTION:Title: Square root $p$-adic $L$-functions\nby Michael Harris (Colum
bia) as part of Harvard number theory seminar\n\nLecture held in Room 507
in the Science Center.\n\nAbstract\nThe Ichino–Ikeda conjecture\, and it
s generalization to unitary groups by N. Harris\, gives explicit formulas
for central critical values of a large class of Rankin–Selberg tensor pr
oducts. The version for unitary groups is now a theorem\, and expresses th
e central 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 $\\mathrm{GL}(n\
,F)\\times\\mathrm{GL}(n-1\,F)$ that descends to an automorphic representa
tion of $\\mathrm{U}(V) \\times \\mathrm{U}(V')$\, where $V$ and $V'$ are
hermitian spaces over $F$\, with respect to a Galois involution $c$ of $F$
\, of dimension $n$ and $n-1$\, respectively.\n\nI will report on the cons
truction of a $p$-adic interpolation of the automorphic period — in othe
r 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 of $p$-adic differential operators due to Eischen\, Fintzen\, Mant
ovan\, and Varma. Most aspects of the construction should generalize to hi
gher Hida theory. I will explain the archimedean theory of the expected ge
neralization\, which is the subject of work in progress with Speh and Koba
yashi.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Keerthi Madapusi (Boston College)
DTSTART;VALUE=DATE-TIME:20230419T190000Z
DTEND;VALUE=DATE-TIME:20230419T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/82
DESCRIPTION:Title: Derived cycles on Shimura varieties\nby Keerthi Madapusi (Bo
ston College) as part of Harvard number theory seminar\n\nLecture held in
Room 507 in the Science Center.\n\nAbstract\nI will show how methods from
derived algebraic geometry can be used to give a uniform definition of gen
erating series of cycles on integral models of Shimura varieties of Hodge
or even abelian type. Following conjectures of Kudla\, these series are ex
pected to converge to half-integer weight automorphic forms on split unita
ry groups\, and certain ‘easy’ consequences of this expectation turn o
ut to be indeed easy given the derived perspective.\n
LOCATION:
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:20241013T133215Z
UID:HarvardNT/83
DESCRIPTION:Title: Knots Invariants and Arithmetic Statistics\nby Tomer Schlank
(Hebrew University of Jerusalem) as part of Harvard number theory seminar
\n\nLecture held in Room 507 in the Science Center.\n\nAbstract\nThe Groth
endieck school introduced étale topology to attach algebraic-topological
invariants such as cohomology to varieties and schemes. Although the origi
nal motivations came from studying varieties over fields\, interesting phe
nomena such as Artin–Verdier duality also arise when considering the spe
ctra 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 integer rings behave as $3$-dimensional manifolds while prime ideals c
orrespond to knots in these manifolds. This knots and primes analogy provi
des a dictionary between knot theory and number theory\, giving some surpr
ising analogies. For example\, this theory relates the linking number to t
he Legendre symbol and the Alexander polynomial to Iwasawa theory. In thi
s 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 beh
avior of arithmetic functions. This is joint work with Ariel Davis.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bjorn Poonen (MIT)
DTSTART;VALUE=DATE-TIME:20231018T190000Z
DTEND;VALUE=DATE-TIME:20231018T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/84
DESCRIPTION:Title: Integral points on curves via Baker's method and finite étale c
overs\nby Bjorn Poonen (MIT) as part of Harvard number theory seminar\
n\nLecture 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:
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Zureick-Brown (Amherst College)
DTSTART;VALUE=DATE-TIME:20231108T200000Z
DTEND;VALUE=DATE-TIME:20231108T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
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 numb
er theory 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 “image
s of Galois” associated to an elliptic curve (equivalently\, classificat
ion of all rational points on certain modular curves $X_H$). The main resu
lt is a provisional classification of the possible images of $\\ell$-adic
Galois representations associated to elliptic curves over $\\mathbb{Q}$ an
d is provably complete barring the existence of unexpected rational points
on modular curves associated to the normalizers of non-split Cartan subgr
oups and two additional genus 9 modular curves of level 49.\n\nI will also
discuss the framework and various applications (for example: a very fast
algorithm to rigorously compute the $\\ell$-adic image of Galois of an ell
iptic curve over $\\mathbb{Q}$)\, and then highlight several new ideas fro
m the joint work\, including techniques for computing models of modular cu
rves and novel arguments to determine their rational points\, a computatio
nal approach that works directly with moduli and bypasses defining equatio
ns\, and (with John Voight) a generalization of Kolyvagin’s theorem to t
he modular curves we study.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Drew Sutherland (MIT)
DTSTART;VALUE=DATE-TIME:20231206T200000Z
DTEND;VALUE=DATE-TIME:20231206T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/86
DESCRIPTION:Title: L-functions from nothing\nby Drew Sutherland (MIT) as part o
f Harvard number theory seminar\n\nLecture held in Science Center Room 507
.\n\nAbstract\nI will report on joint work in progress with Andrew Booker
on\nthe practical implementation of an axiomatic approach to the enumerati
on\nof arithmetic $L$-functions that lie in a certain subset of the Selber
g\nclass that is expected to include all $L$-functions of abelian varietie
s.\nAs in the work of Farmer\, Koutsoliotas\, and Lemurell\, our approach
is\nbased on the approximate functional equation. We obtain additional\nc
onstraints by considering twists (and more general Rankin-Selberg\nconvolu
tions) of our unknown $L$-function that yield a system of linear\nconstrai
nts that can be solved using the simplex method. This allows us\nto signi
ficantly extend the range of our computations for the family of\n$L$-funct
ions associated to abelian surfaces over $\\mathbb{Q}$. We also introduce
a\nmethod for certifying the completeness of our enumeration.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Padmavathi Srinivasan (ICERM)
DTSTART;VALUE=DATE-TIME:20231025T190000Z
DTEND;VALUE=DATE-TIME:20231025T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/87
DESCRIPTION:Title: Towards a unified theory of canonical heights on abelian varieti
es\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 rationa
l points on a curve. In this talk\, we will outline a new construction of
canonical $p$-adic heights on abelian varieties from $p$-adic adelic metri
cs\, using $p$-adic Arakelov theory developed by Besser. This construction
closely mirrors Zhang's construction of canonical real valued heights fro
m real-valued adelic metrics. We will use this new construction to give di
rect explanations (avoiding $p$-adic Hodge theory) of the key properties o
f height pairings needed for the quadratic Chabauty method for rational po
ints. This is joint work with Amnon Besser and Steffen Mueller.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ananth Shankar (Northwestern University)
DTSTART;VALUE=DATE-TIME:20231115T200000Z
DTEND;VALUE=DATE-TIME:20231115T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/88
DESCRIPTION:Title: Semisimplicity and CM lifts\nby Ananth Shankar (Northwestern
University) as part of Harvard number theory seminar\n\nLecture held in S
cience Center Room 507.\n\nAbstract\nConsider the setting of a smooth vari
ety $S$ over $\\mathbb{F}_q$\, and an $\\ell$-adic local on $S$ which has
finite determinant and is geometrically irreducible. Work of Lafforgue pro
ves that 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 this 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 time permits\, I will also address the question of integral canonical
models of Shimura varieties.\nThis is joint work with Ben Bakker and Jacob
Tsimerman.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tom Weston (UMass Amherst)
DTSTART;VALUE=DATE-TIME:20230927T190000Z
DTEND;VALUE=DATE-TIME:20230927T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/89
DESCRIPTION:Title: Diophantine Stability for Elliptic Curves\nby Tom Weston (UM
ass Amherst) as part of Harvard number theory seminar\n\nLecture held in S
cience 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 ar
e $l$-Diophantine stable in the sense of Mazur and Rubin. This is joint w
ork with Anwesh Ray.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jun Yang (Harvard University)
DTSTART;VALUE=DATE-TIME:20231101T190000Z
DTEND;VALUE=DATE-TIME:20231101T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/90
DESCRIPTION:Title: The limit multiplicities and von Neumann dimensions\nby Jun
Yang (Harvard University) as part of Harvard number theory seminar\n\nLect
ure held in Science Center Room 507.\n\nAbstract\nGiven an arithmetic subg
roup $\\Gamma$ in a semi-simple Lie group $G$\, the multiplicity of an irr
educible representation of $G$ in $L^2(\\Gamma\\backslash G)$ is unknown i
n general.\nWe observe the multiplicity of any discrete series representat
ion $\\pi$ of $\\rm{SL}(2\,\\mathbb{R})$ in $L^2(\\Gamma(n)\\backslash \\r
m{SL}(2\,\\mathbb{R}))$ is close to the von Neumann dimension of $\\pi$ ov
er the group algebra of $\\Gamma(n)$.\nWe extend this result to other Lie
groups and bounded families of irreducible representations of them.\nBy ap
plying the trace formulas\, we show the multiplicities are exactly the von
Neumann dimensions if we take certain towers of descending lattices in so
me Lie groups.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Padmavathi Srinivasan (ICERM)
DTSTART;VALUE=DATE-TIME:20230913T190000Z
DTEND;VALUE=DATE-TIME:20230913T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/91
DESCRIPTION:Title: Towards a unified theory of canonical heights on abelian varieti
es\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 rationa
l points on a curve. In this talk\, we will outline a new construction of
canonical $p$-adic heights on abelian varieties from $p$-adic adelic metri
cs\, using $p$-adic Arakelov theory developed by Besser. This construction
closely mirrors Zhang's construction of canonical real valued heights fro
m real-valued adelic metrics. We will use this new construction to give di
rect explanations (avoiding $p$-adic Hodge theory) of the key properties o
f height pairings needed for the quadratic Chabauty method for rational po
ints. This is joint work with Amnon Besser and Steffen Mueller.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Lemke Oliver (Tufts University)
DTSTART;VALUE=DATE-TIME:20231129T200000Z
DTEND;VALUE=DATE-TIME:20231129T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/92
DESCRIPTION:Title: Faithful induction theorems and the Chebotarev density theorem\nby Robert Lemke Oliver (Tufts University) as part of Harvard number th
eory seminar\n\nLecture held in Science Center Room 507.\n\nAbstract\nThe
Chebotarev density theorem is a powerful and ubiquitous tool in number the
ory used to guarantee the existence of infinitely many primes satisfying s
plitting conditions in a Galois extension of number fields. In many appli
cations\, however\, it is necessary to know not just that there are many s
uch primes in the limit\, but to know that there are many such primes up t
o a given finite point. This is the domain of so-called effective Chebota
rev density theorems. In forthcoming joint work with Alex Smith that exte
nds previous joint work of the author with Thorner and Zaman and earlier w
ork of Pierce\, Turnage-Butterbaugh\, and Wood\, we prove that in any fami
ly of irreducible complex Artin representations\, almost all are subject t
o a very strong effective prime number theorem. This implies that almost
all number fields with a fixed Galois group are subject to a similarly str
ong effective form of the Chebotarev density theorem. Under the hood\, th
e key result is a new theorem in the character theory of finite groups tha
t is similar in spirit to classical work of Artin and Brauer on inductions
of one-dimensional characters.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robin Zhang (MIT)
DTSTART;VALUE=DATE-TIME:20230920T190000Z
DTEND;VALUE=DATE-TIME:20230920T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/93
DESCRIPTION:Title: Harris–Venkatesh plus Stark\nby Robin Zhang (MIT) as part
of Harvard number theory seminar\n\nLecture held in Science Center Room 50
7.\n\nAbstract\nThe class number formula describes the behavior of the Ded
ekind zeta function at $s=0$ and $s=1$. The Stark conjecture extends the c
lass number 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–V
enkatesh conjecture describes the residue of Stark units modulo $p$\, givi
ng a modular analogue to the Stark and Gross conjectures while also servin
g as the first verifiable part of the broader conjectures of Venkatesh\, P
rasanna\, and Galatius. In this talk\, I will draw an introductory picture
\, formulate a unified conjecture combining Harris–Venkatesh and Stark f
or weight one modular forms\, and describe the proof of this in the imagin
ary dihedral case.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Betts (Harvard)
DTSTART;VALUE=DATE-TIME:20231011T190000Z
DTEND;VALUE=DATE-TIME:20231011T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/94
DESCRIPTION:Title: A relative Oda's criterion\nby Alex Betts (Harvard) as part
of Harvard number theory seminar\n\nLecture held in Science Center Hall A.
\n\nAbstract\nThe Neron--Ogg--Shafarevich criterion asserts that an abelia
n variety over $\\mathbb{Q}_p$ has good reduction if and only if the Galoi
s action on its $\\mathbb{Z}_\\ell$-linear Tate module is unramified (for
$\\ell$ different from $p$). In 1995\, Oda formulated and proved an analog
ue of the Neron--Ogg--Shafarevich criterion for smooth projective curves $
X$ of genus 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 this talk\, I will explain a relative version of Oda's criterion\, du
e to myself and Netan Dogra\, in which we answer the question of when the
Galois action on the pro-$\\ell$ torsor of paths between two points $x$ an
d $y$ is unramified in terms of the relative position of $x$ and $y$ on th
e reduction of $X$. On the way\, we will touch on topics from mapping clas
s groups and the theory of electrical circuits\, and\, time permitting\, w
ill outline some consequences for the Chabauty--Kim method.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Naomi Sweeting (Harvard)
DTSTART;VALUE=DATE-TIME:20231004T190000Z
DTEND;VALUE=DATE-TIME:20231004T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/95
DESCRIPTION:Title: Tate Classes and Endoscopy for $\\operatorname{GSp}_4$\nby N
aomi Sweeting (Harvard) as part of Harvard number theory seminar\n\nLectur
e held in Science Center Room 507.\n\nAbstract\nWeissauer proved using the
theory of endoscopy that the Galois representations associated to classic
al modular forms of weight two appear in the middle cohomology of both a m
odular curve and a Siegel modular threefold. Correspondingly\, there are
large families of Tate classes on the product of these two Shimura varieti
es\, and it is natural to ask whether one can construct algebraic cycles g
iving rise to these Tate classes. It turns out that a natural algebraic cy
cle generates some\, but not all\, of the Tate classes: to be precise\, it
generates exactly the Tate classes which are associated to generic member
s of the endoscopic $L$-packets on $\\operatorname{GSp}_4$. In the non-gen
eric case\, one can at least show that all the Tate classes arise from Hod
ge cycles. For this talk\, I'll focus on the behavior of the algebraic cyc
le class. NB: This talk is independent of the one in last week's number th
eorists' seminar.\n
LOCATION:
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:20241013T133215Z
UID:HarvardNT/96
DESCRIPTION:Title: Variation of Selmer groups in quadratic twist families of abelia
n varieties over function fields\nby Jordan Ellenberg (University of W
isconsin-Madison) as part of Harvard number theory seminar\n\nLecture held
in Science Center Room 507.\n\nAbstract\nA basic question in arithmetic s
tatistics is: what does the Selmer group of a random abelian variety look
like? This question is governed by the Poonen-Rains heuristics\, later g
eneralized by Bhargava-Kane-Lenstra-Poonen-Rains\, which predict\, for ins
tance\, that the mod p Selmer group of an elliptic curve has size p+1 on a
verage. Results towards these heuristics have been very partial but have
nonetheless enabled major progress in studying the distribution of ranks o
f abelian varieties.\n\n \n\nWe will describe new work\, joint with Aaron
Landesman\, 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 prob
ability that a random quadratic twist of an abelian variety A over a globa
l function field has rank at least 2. The method is very much in the spir
it of earlier work with Venkatesh and Westerland which proved a version of
the Cohen-Lenstra heuristics over function fields by means of homological
stabilization for Hurwitz spaces\; in other words\, the main argument is
topological in nature. I will try to embed the talk in a general discussi
on of how one gets from topological results to consequences in arithmetic
statistics\, and what the prospects for further developments in this area
look like.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART;VALUE=DATE-TIME:20231122T200000Z
DTEND;VALUE=DATE-TIME:20231122T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
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:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jennifer Balakrishnan (Boston University)
DTSTART;VALUE=DATE-TIME:20240424T190000Z
DTEND;VALUE=DATE-TIME:20240424T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/98
DESCRIPTION:Title: Shadow line distributions\nby Jennifer Balakrishnan (Boston
University) as part of Harvard number theory seminar\n\nLecture held in Sc
ience Center Room 507.\n\nAbstract\nLet $E/\\mathbb{Q}$ be an elliptic cur
ve of analytic rank $2$\, and let $p$\nbe an odd prime of good\, ordinary
reduction such that the $p$-torsion of\n$E(\\mathbb{Q})$ is trivial. Let $
K$ be an 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\nthes
e assumptions\, there is a $1$-dimensional $\\mathbb{Q}_p$-vector space at
tached\nto the triple $(E\, p\, K)$\, known as the shadow line\, and it ca
n be\ncomputed using anticyclotomic $p$-adic heights. We describe the\ncom
putation of these heights and shadow lines. Furthermore\, fixing\npairs $
(E\, p)$ and varying $K$\, we present some data on the distribution\nof th
ese shadow lines. This is joint work with Mirela Çiperiani\,\nBarry Mazu
r\, and Karl Rubin.\n
LOCATION:
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:20241013T133215Z
UID:HarvardNT/99
DESCRIPTION:Title: On the distribution of class groups — beyond Cohen-Lenstra and
Gerth\nby Yuan Liu (University of Illinois Urbana-Champaign) as part
of Harvard number theory seminar\n\nLecture held in Science Center Room 50
7.\n\nAbstract\nThe Cohen-Lenstra heuristic studies the distribution of th
e p-part 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
group of quadratic fields. The main difference between these conjectures
is that while the (odd) $p$-part of the class group behaves completely “
randomly”\, the $2$-part of the class group does not since the $2$-torsi
on 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 con
jectures. The techniques involve Galois cohomology and the embedding probl
em of global fields.\n
LOCATION:
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:20241013T133215Z
UID:HarvardNT/101
DESCRIPTION:Title: Compatibility of the canonical $l$-adic local systems on except
ional Shimura varieties\nby Stefan Patrikis (The Ohio State University
) as part of Harvard number theory seminar\n\nLecture held in Science Cent
er Room 507.\n\nAbstract\nLet $(G\, X)$ be a Shimura datum\, and let $K$ b
e a compact open subgroup of $G(\\mathbb{A}_f)$. One hopes that under mild
assumptions on $G$ and $K$\, the points of the Shimura variety $Sh_K(G\,
X)$ parametrize a family of motives\; in abelian type this is well-underst
ood\, but in non-abelian type it is completely mysterious. I will discuss
joint work with Christian Klevdal showing that for exceptional Shimura var
ieties the points (over number fields\, say) at least yield compatible sys
tems of l-adic representations\, which should be the l-adic realizations o
f the conjectural motives.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven Achenjang (MIT)
DTSTART;VALUE=DATE-TIME:20240221T200000Z
DTEND;VALUE=DATE-TIME:20240221T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/102
DESCRIPTION:Title: The Average Size of 2-Selmer Groups of Elliptic Curves over Fun
ction Fields\nby Niven Achenjang (MIT) as part of Harvard number theor
y seminar\n\nLecture held in Science Center Room 507.\n\nAbstract\nGiven a
n elliptic curve $E$ over a global field $K$\, the abelian group $E(K)$ is
finitely generated\, and so much effort has been put into trying to under
stand the behavior of $\\operatorname{rank}E(K)$\, as $E$ varies. Of note\
, it is a folklore conjecture that\, when all elliptic curves $E/K$ are or
dered by a suitably defined height\, the average value of their ranks is e
xactly $1/2$. One fruitful avenue for understanding the distribution of $\
\operatorname{rank}E(K)$ has been to first understand the distribution of
the sizes of Selmer groups of elliptic curves. In this direction\, various
authors (including Bhargava-Shankar\, Poonen-Rains\, and Bhargava-Kane-Le
nstra-Poonen-Rains) have made conjectures which predict\, for example\, th
at the average size of the $n$-Selmer group of $E/K$ is equal to the sum o
f the divisors of $n$. In this talk\, I will report on some recent work ve
rifying this average size prediction\, "up to small error term\," whenever
$n=2$ and $K$ is any global *function* field. Results along these lines w
ere previously known whenever $K$ was a number field or function field of
characteristic $\\ge 5$\, so the novelty of my work is that it applies eve
n in "bad" characteristic.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Mundy (Princeton University)
DTSTART;VALUE=DATE-TIME:20240410T190000Z
DTEND;VALUE=DATE-TIME:20240410T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
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\n
Lecture held in Science Center Room 507.\n\nAbstract\nLet $\\pi$ be a cusp
idal automorphic representation of $\\mathrm{Sp}_{2n}$ over $\\mathbb{Q}$
which is holomorphic discrete series at infinity\, and $\\chi$ a Dirichlet
character. Then one can attach to $\\pi$ an orthogonal $p$-adic Galois re
presentation $\\rho$ of dimension $2n+1$. Assume $\\rho$ is irreducible\,
that $\\pi$ is ordinary at $p$\, and that $p$ does not divide the conducto
r of $\\chi$. I will describe work in progress which aims to prove that th
e Bloch--Kato Selmer group attached to the twist of $\\rho$ by $\\chi$ van
ishes\, under some mild ramification assumptions on $\\pi$\; this is what
is predicted by the Bloch--Kato conjectures.\n\n\nThe proof uses "ramified
Eisenstein congruences" by constructing $p$-adic families of Siegel cusp
forms degenerating to Klingen Eisenstein series of nonclassical weight\, a
nd using these families to construct ramified Galois cohomology classes fo
r the Tate dual of the twist of $\\rho$ by $\\chi$.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shiva Chidambaram (MIT)
DTSTART;VALUE=DATE-TIME:20240228T200000Z
DTEND;VALUE=DATE-TIME:20240228T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/104
DESCRIPTION:Title: Computing Galois images of Picard curves\nby Shiva Chidamba
ram (MIT) as part of Harvard number theory seminar\n\nLecture held in Scie
nce Center Room 507.\n\nAbstract\nLet $C$ be a genus $3$ curve whose Jacob
ian is geometrically simple and has geometric endomorphism algebra equal t
o an imaginary quadratic field. In particular\, consider Picard curves $y^
3 = f_4(x)$ where the geometric endomorphism algebra is $\\mathbb{Q}(\\zet
a_3)$. We study the associated mod-$\\ell$ Galois representations and thei
r images. I will discuss an algorithm\, developed in ongoing joint work wi
th Pip Goodman\, to compute the set of primes $\\ell$ for which the images
are not maximal. By running it on several datasets of Picard curves\, the
largest non-maximal prime we obtain is $13$. This may be compared with ge
nus 1\, where Serre's uniformity question asks if the mod-$\\ell$ Galois i
mage of non-CM elliptic curves over $\\Q$ is maximal for all primes $\\ell
> 37$.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Salim Tayou (Harvard University)
DTSTART;VALUE=DATE-TIME:20240501T190000Z
DTEND;VALUE=DATE-TIME:20240501T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/106
DESCRIPTION:Title: Modularity of special cycles in orthogonal and unitary Shimura
varieties\nby Salim Tayou (Harvard University) as part of Harvard numb
er theory 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 quadratic lattices produce modular forms. In a vast\ngeneralization\,
Kudla and Millson have proved that the generating series\nof special cycle
s in orthogonal and unitary Shimura varieties are\nmodular forms. In this
talk\, I will explain an extension of these\nresults to toroidal compactif
ications where we prove that the generating\nseries of divisors is a mixed
mock modular form. This recovers and\nrefines earlier results of Bruinier
and Zemel. The results of this talk\nare joint work with Philip Engel and
François Greer.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frank Calegari (University of Chicago)
DTSTART;VALUE=DATE-TIME:20240327T190000Z
DTEND;VALUE=DATE-TIME:20240327T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/107
DESCRIPTION:Title: “everywhere unramified” objects in number theory and the co
homology of $\\mathrm{GL}_n(\\mathbb{Z})$\nby Frank Calegari (Universi
ty of Chicago) as part of Harvard number theory seminar\n\nLecture held in
Science Center Room 507.\n\nAbstract\nOne theme in number theory is to st
udy objects via their ramification: the discriminant of a number field\, t
he conductor of an elliptic curve\, the level of a modular form\, and so o
n.\nThere is\, however\, some particular interest in understanding objects
which are “everywhere unramified” — and also understanding when suc
h objects don’t exist. Such non-existence results\nare often the startin
g point for inductive arguments. For example\, Minkowski’s theorem that
there are no unramified extensions of $\\mathbb{Q}$ can be used to prove t
he Kronecker-Weber theorem\, and the vanishing\nof a certain space of modu
lar forms is the starting point for Wiles’ proof of Fermat’s Last Theo
rem. In this talk\, I will begin by describing many such vanishing results
both in arithmetic and in the\ntheory of automorphic forms\, and how they
are related by the Langlands program (sometimes only conjecturally). Then
I will descibe the construction of a new example of an automorphic form o
f level one\nand “weight zero”. This construction also gives the firs
t non-zero classes in the cohomology of $\\mathrm{GL}_n(\\mathbb{Z})$ (for
some $n$) that come from “cuspidal” modular forms (for $n > 0$).\n\nT
his is joint work with George Boxer and Toby Gee.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Weinstein (Boston University)
DTSTART;VALUE=DATE-TIME:20240306T200000Z
DTEND;VALUE=DATE-TIME:20240306T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/109
DESCRIPTION:Title: Integral Ax-Sen-Tate theory\nby Jared Weinstein (Boston Uni
versity) as part of Harvard number theory seminar\n\nLecture held in Scien
ce Center Room 507.\n\nAbstract\nLet $K$ be a local field of mixed charact
eristic\, let $G$ be the absolute Galois group of $K$\, and let $C$ be the
completion of an algebraic closure of $K$. The Ax-Sen-Tate theorem state
s 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 continuo
us cocycles): $H^1(G\,C)=K$ 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 sub
group of $H^i(G\,O_C)$ is killed by a constant which only depends on the r
esidue characteristic $p$ (in fact $p^6$ suffices). This is a part of a p
roject with coauthors Tobias Barthel\, Tomer Schlank\, and Nathaniel Stapl
eton.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sasha Petrov (MIT)
DTSTART;VALUE=DATE-TIME:20240911T190000Z
DTEND;VALUE=DATE-TIME:20240911T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/110
DESCRIPTION:Title: Characteristic classes of p-adic local systems\nby Sasha Pe
trov (MIT) as part of Harvard number theory seminar\n\nLecture held in Sci
ence Center Room 507.\n\nAbstract\nGiven an étale Z_p-local system of ran
k 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 var
iety with coefficients in Q_p. These characteristic classes can be thought
of as p-adic analogs of Chern-Simons characteristic classes of vector bun
dles with a flat connection.\n\nOn a smooth projective variety over comple
x numbers\, Chern-Simons classes of all flat bundles are torsion in degree
s >1 by a theorem of Reznikov. But for varieties over non-closed fields th
e characteristic classes of p-adic local systems turn out to often be non-
zero even rationally. When X is defined over a p-adic field\, characterist
ic classes of a p-adic local system on it can be partially expressed in te
rms of Hodge-theoretic invariants of the local system. This relation is es
tablished through considering an analog of Chern classes for vector bundle
s on the pro-étale site of X.\n\nThis is joint work with Lue Pan.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jit Wu Yap (Harvard University)
DTSTART;VALUE=DATE-TIME:20240918T190000Z
DTEND;VALUE=DATE-TIME:20240918T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/111
DESCRIPTION:Title: Quantitative Equidistribution of Small Points for Canonical Hei
ghts\nby Jit Wu Yap (Harvard University) as part of Harvard number the
ory seminar\n\nLecture held in Science Center Room 507.\n\nAbstract\nLet K
be a number 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 h
eight tending to 0\, Szpiro-Ullmo-Zhang proved that the Galois orbits of x
_n converges weakly to the Haar measure of A. Yuan then generalized Szpiro
-Ullmo-Zhang's result to the setting of polarized endomorphisms on a proje
ctive variety X defined over K. In this talk\, I will explain how to prove
a quantitative version of Yuan's result when X is assumed to be smooth. T
his was previously only known when dim X = 1.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hélène Esnault (Freie Universität Berlin)
DTSTART;VALUE=DATE-TIME:20240925T190000Z
DTEND;VALUE=DATE-TIME:20240925T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/112
DESCRIPTION:Title: Diophantine Properties of the Betti Moduli Space\nby Hélè
ne Esnault (Freie Universität Berlin) as part of Harvard number theory se
minar\n\nLecture held in Science Center Room 507.\n\nAbstract\nWe prove in
particular that when the Betti moduli space of a smooth quasi-projective
variety\nover the complex number with some quasi-unipotent monodromies at
infinity. finite determinant\nis irreducible over the integers and over th
e complex 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-
projective variety. \n\n(Joint with J. de Jong\, based in part on joint wo
rk with M. Groechenig).\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sanath Devalapurkar (Harvard University)
DTSTART;VALUE=DATE-TIME:20241002T190000Z
DTEND;VALUE=DATE-TIME:20241002T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/113
DESCRIPTION:Title: The image of J and p-adic geometry\nby Sanath Devalapurkar
(Harvard University) as part of Harvard number theory seminar\n\nLecture h
eld 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 "unive
rsal" cohomology theory for p-adic formal schemes. I will describe joint w
ork with Hahn\, Raksit\, and Yuan (building on work of Hahn-Raksit-Wilson)
\, in which we give a new construction of prismatization using the methods
of homotopy theory (in particular\, the theory of topological Hochschild
homology\, aka THH). The case when R is Z_{p} turns out to be particularly
interesting\, and I will discuss joint work with Raksit which describes a
construction 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 Adam
s. This plays the same role for prismatic cohomology as the usual commutat
ive ring Z_{p} plays for crystalline cohomology. It gives an alternative p
erspective on results of Bhatt and Lurie\, and is also related to Lurie’
s "prismatization of F_{1}".\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sameera Vemulapalli (Harvard University)
DTSTART;VALUE=DATE-TIME:20241009T190000Z
DTEND;VALUE=DATE-TIME:20241009T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/114
DESCRIPTION:Title: Steinitz classes of number fields and Tschirnhausen bundles of
covers of the projective line\nby Sameera Vemulapalli (Harvard Univers
ity) as part of Harvard number theory seminar\n\nLecture held in Science C
enter Room 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 of the complex projective line by a smooth curve yields a vector bun
dle on the projective line by pushforward of the structure sheaf\; which b
undles arise this way? In this talk\, I'll describe recent work with Vakil
in which we use tools in arithmetic statistics (in particular\, binary fo
rms) to completely answer the first question and make progress towards the
second.\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Artane Siad (Princeton University)
DTSTART;VALUE=DATE-TIME:20241016T190000Z
DTEND;VALUE=DATE-TIME:20241016T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/115
DESCRIPTION:by Artane Siad (Princeton University) as part of Harvard numbe
r theory seminar\n\nLecture held in Science Center Room 507.\nAbstract: TB
A\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linus Hamann (Harvard University)
DTSTART;VALUE=DATE-TIME:20241023T190000Z
DTEND;VALUE=DATE-TIME:20241023T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
UID:HarvardNT/116
DESCRIPTION:by Linus Hamann (Harvard University) as part of Harvard number
theory seminar\n\nLecture held in Science Center Room 507.\nAbstract: TBA
\n
LOCATION:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wei Zhang (MIT)
DTSTART;VALUE=DATE-TIME:20241030T190000Z
DTEND;VALUE=DATE-TIME:20241030T200000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
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:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sachi Hashimoto (Brown University)
DTSTART;VALUE=DATE-TIME:20241106T200000Z
DTEND;VALUE=DATE-TIME:20241106T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
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:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Bertoloni Meli (Boston University)
DTSTART;VALUE=DATE-TIME:20241113T200000Z
DTEND;VALUE=DATE-TIME:20241113T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
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:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nina Zubrilina (Harvard University)
DTSTART;VALUE=DATE-TIME:20241120T200000Z
DTEND;VALUE=DATE-TIME:20241120T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
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:
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Smith (UCLA)
DTSTART;VALUE=DATE-TIME:20241204T200000Z
DTEND;VALUE=DATE-TIME:20241204T210000Z
DTSTAMP;VALUE=DATE-TIME:20241013T133215Z
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:
END:VEVENT
END:VCALENDAR