BEGIN:VCALENDAR
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PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Stefan Patrikis (Ohio State Univ.)
DTSTART;VALUE=DATE-TIME:20200922T170000Z
DTEND;VALUE=DATE-TIME:20200922T180000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/1
DESCRIPTION:Title: L
ifting Galois representations\nby Stefan Patrikis (Ohio State Univ.) a
s part of University of Arizona Algebra and Number Theory Seminar\n\n\nAbs
tract\nI will survey joint work with Najmuddin Fakhruddin and Chandrashekh
ar Khare in which we prove in many cases existence of geometric p-adic lif
ts of "odd" mod p Galois representations\, valued in general reductive gro
ups. Then I will discuss applications to modularity of reducible mod p Gal
ois representations.\n
LOCATION:https://researchseminars.org/talk/UAANTS/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wanlin Li (MIT)
DTSTART;VALUE=DATE-TIME:20200929T210000Z
DTEND;VALUE=DATE-TIME:20200929T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/2
DESCRIPTION:Title: T
he Ceresa class: tropical\, topological\, and algebraic\nby Wanlin Li
(MIT) as part of University of Arizona Algebra and Number Theory Seminar\n
\n\nAbstract\nThe Ceresa cycle is an algebraic cycle attached to a smooth
algebraic curve\, which is trivial in the Chow ring when the curve is hype
relliptic. Its image under a cycle class map provides a class in étale co
homology called the Ceresa class. There are few examples where the Ceresa
class is known for non-hyperelliptic curves. We explain how to define a Ce
resa class for a tropical algebraic curve\, and also for a Riemann surface
endowed with a multiset of commuting Dehn twists (where it is related to
the Morita cocycle on the mapping class group). Finally\, we explain how t
hese are related to the Ceresa class of a smooth algebraic curve over C((t
))\, and show that in this setting the Ceresa class is torsion.\n
LOCATION:https://researchseminars.org/talk/UAANTS/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jessica Fintzen (Univ. of Cambridge)
DTSTART;VALUE=DATE-TIME:20201006T170000Z
DTEND;VALUE=DATE-TIME:20201006T180000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/3
DESCRIPTION:Title: R
epresentations of p-adic groups and applications\nby Jessica Fintzen (
Univ. of Cambridge) as part of University of Arizona Algebra and Number Th
eory Seminar\n\n\nAbstract\nThe Langlands program is a far-reaching collec
tion of conjectures that relate different areas of mathematics including n
umber theory and representation theory. A fundamental problem on the repre
sentation theory side of the Langlands program is the construction of all
(irreducible\, smooth\, complex) representations of p-adic groups. I will
provide an overview of our understanding of the representations of p-adic
groups\, with an emphasis on recent progress.\n\nI will also outline how n
ew results about the representation theory of p-adic groups can be used to
obtain congruences between arbitrary automorphic forms and automorphic fo
rms which are supercuspidal at p\, which is joint work with Sug Woo Shin.
This simplifies earlier constructions of attaching Galois representations
to automorphic representations\, i.e. the global Langlands correspondence\
, for general linear groups. Moreover\, our results apply to general p-adi
c groups and have therefore the potential to become widely applicable beyo
nd the case of the general linear group.\n
LOCATION:https://researchseminars.org/talk/UAANTS/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesc Castella (UCSB)
DTSTART;VALUE=DATE-TIME:20201027T210000Z
DTEND;VALUE=DATE-TIME:20201027T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/4
DESCRIPTION:Title: I
wasawa theory of elliptic curves at Eisenstein primes and applications
\nby Francesc Castella (UCSB) as part of University of Arizona Algebra and
Number Theory Seminar\n\n\nAbstract\nIn the study of Iwasawa theory of el
liptic curves $E/\\mathbb{Q}$\, it is often assumed that $p$ is a non-Eise
nstein prime\, meaning that $E[p]$ is irreducible as a $G_{\\mathbb{Q}}$-m
odule. Because of this\, most of the recent results on the $p$-converse to
the theorem of Gross–Zagier and Kolyvagin (following Skinner and Wei Zh
ang) and on the $p$-part of the Birch–Swinnerton-Dyer formula in analyti
c rank 1 (following Jetchev–Skinner–Wan) were only known for non-Eisen
stein primes $p$. In this talk\, I’ll explain some of the ingredients in
a joint work with Giada Grossi\, Jaehoon Lee\, and Christopher Skinner in
which we study the (anticyclotomic) Iwasawa theory of elliptic curves ove
r $\\mathbb{Q}$ at Eisenstein primes. As a consequence of our study\, we o
btain an extension of the aforementioned results to the Eisenstein case. I
n particular\, for $p=3$ this leads to an improvement on the best known re
sults towards Goldfeld’s conjecture in the case of elliptic curves over
$\\mathbb{Q}$ with a rational $3$-isogeny.\n
LOCATION:https://researchseminars.org/talk/UAANTS/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Gemuenden (ETH Zurich)
DTSTART;VALUE=DATE-TIME:20201103T170000Z
DTEND;VALUE=DATE-TIME:20201103T180000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/5
DESCRIPTION:Title: N
on-Abelian Orbifold Theory\nby Thomas Gemuenden (ETH Zurich) as part o
f University of Arizona Algebra and Number Theory Seminar\n\n\nAbstract\nI
n this talk\, we will discuss the theory of holomorphic extensions of non-
abelian orbifold vertex operator algebras. We will give a brief overview o
f the concepts and motivations of vertex operator algebras and their modul
es. Then we will construct the module category of non-abelian orbifold ver
tex operator algebras and classify their holomorphic extensions. If time p
ermits we will prove that there exist holomorphic vertex operator algebras
at central charge 72 that cannot be constructed as a holomorphic extensio
n of a cyclic orbifold of a lattice vertex operator algebra.\n
LOCATION:https://researchseminars.org/talk/UAANTS/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Melissa Emory (Univ. of Toronto)
DTSTART;VALUE=DATE-TIME:20201110T210000Z
DTEND;VALUE=DATE-TIME:20201110T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/6
DESCRIPTION:Title: A
multiplicity one theorem for general spin groups\nby Melissa Emory (U
niv. of Toronto) as part of University of Arizona Algebra and Number Theor
y Seminar\n\n\nAbstract\nA classical problem in representation theory is h
ow a\nrepresentation of a group decomposes when restricted to a subgroup.
In the\n1990s\, Gross-Prasad formulated a conjecture regarding the\nrestri
ction of representations\, also known as branching laws\, of special\north
ogonal groups. Gan\, Gross and Prasad extended this conjecture\, now\nkno
wn as the local Gan-Gross-Prasad (GGP) conjecture\, to the remaining\nclas
sical groups. There are many ingredients needed to prove a local GGP\nconj
ecture. In this talk\, we will focus on the first ingredient: a\nmultipli
city at most one theorem.\nAizenbud\, Gourevitch\, Rallis and Schiffmann p
roved a multiplicity at\nmost one theorem for restrictions of irreducible
representations of\ncertain p-adic classical groups and Waldspurger proved
the same theorem\nfor the special orthogonal groups. We will discuss work
that establishes a\nmultiplicity at most one theorem for restrictions of
irreducible\nrepresentations for a non-classical group\, the general spin
group. This is\njoint work with Shuichiro Takeda.\n
LOCATION:https://researchseminars.org/talk/UAANTS/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrea Dotto (Univ. of Chicago)
DTSTART;VALUE=DATE-TIME:20201117T210000Z
DTEND;VALUE=DATE-TIME:20201117T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/7
DESCRIPTION:Title: M
od p Bernstein centres of p-adic groups\nby Andrea Dotto (Univ. of Chi
cago) as part of University of Arizona Algebra and Number Theory Seminar\n
\n\nAbstract\nThe centre of the category of smooth mod p representations o
f a p-adic reductive group does not distinguish the blocks of finite lengt
h representations\, in contrast with Bernstein's theory in characteristic
zero. Motivated by this observaton and the known connections between the B
ernstein centre and the local Langlands correspondence in families\, we co
nsider the case of GL_2(Q_p) and we prove that its category of representat
ions extends to a stack on the Zariski site of a simple geometric object:
a chain X of projective lines\, whose points are in bijection with Paskuna
s's blocks. Taking the centre over each open subset we obtain a sheaf of r
ings on X\, and we expect the resulting space to be closely related to the
Emerton--Gee stack for 2-dimensional representations of the absolute Galo
is group of Q_p. Joint work in progress with Matthew Emerton and Toby Gee.
\n
LOCATION:https://researchseminars.org/talk/UAANTS/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Pollack (UCSD)
DTSTART;VALUE=DATE-TIME:20201124T210000Z
DTEND;VALUE=DATE-TIME:20201124T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/8
DESCRIPTION:Title: S
ingular modular forms on quaternionic E_8\nby Aaron Pollack (UCSD) as
part of University of Arizona Algebra and Number Theory Seminar\n\n\nAbstr
act\nThe exceptional group $E_{7\,3}$ has a symmetric space with Hermitian
tube structure. On it\, Henry Kim wrote down low weight holomorphic modul
ar forms that are "singular" in the sense that their Fourier expansion has
many terms equal to zero. The symmetric space associated to the exception
al group $E_{8\,4}$ does not have a Hermitian structure\, but it has what
might be the next best thing: a quaternionic structure and associated "mod
ular forms". I will explain the construction of singular modular forms on
$E_{8\,4}$\, and the proof that these special modular forms have rational
Fourier expansions\, in a precise sense. This builds off of work of Wee Te
ck Gan and uses key input from Gordan Savin.\n
LOCATION:https://researchseminars.org/talk/UAANTS/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuan Liu (Univ. of Michigan)
DTSTART;VALUE=DATE-TIME:20201201T210000Z
DTEND;VALUE=DATE-TIME:20201201T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/9
DESCRIPTION:Title: P
resentations of Galois groups of maximal extensions with restricted ramifi
cations\nby Yuan Liu (Univ. of Michigan) as part of University of Ariz
ona Algebra and Number Theory Seminar\n\n\nAbstract\nIn this talk\, we are
going to discuss how to use Galois cohomology to study the presentation o
f Galois groups of maximal extensions with restricted ramifications. In pr
evious work with Melanie Matchett Wood and David Zureick-Brown\, we conjec
ture that an explicitly-defined random profinite group model can predict t
he distribution of the Galois groups of maximal unramified extension of gl
obal fields that range over $\\Gamma$-extensions of $\\mathbb{Q}$ or $\\ma
thbb{F}_q(t)$. In the function field case\, our conjecture is supported by
the moment computation\, but very little is known in the number field cas
e. Interestingly\, our conjecture suggests that the random group should si
mulate the maximal unramified Galois groups\, and hence suggests some part
icular requirements on the structure of these Galois groups. In this talk\
, we will prove that the maximal unramified Galois groups are always achie
vable by our random group model\, which verifies those interesting require
ments.\n
LOCATION:https://researchseminars.org/talk/UAANTS/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Wootton (Univ. of Arizona)
DTSTART;VALUE=DATE-TIME:20201208T210000Z
DTEND;VALUE=DATE-TIME:20201208T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/10
DESCRIPTION:Title:
Non-Abelian simple groups act with almost all signatures\nby Aaron Woo
tton (Univ. of Arizona) as part of University of Arizona Algebra and Numbe
r Theory Seminar\n\n\nAbstract\nThe topological data of a finite group $G$
acting conformally on a compact Riemann surface is often encoded using a
tuple of non-negative integers $(h\;m_1\,\\ldots \,m_s)$ called its signa
ture\, where the $m_i$ are orders of non-trivial elements in the group. Th
ere are two easily verifiable arithmetic conditions on a tuple which are n
ecessary for it to be a signature of some group action. We derive necessar
y and sufficient conditions on a group for the situation where all but fin
itely many tuples that satisfy these arithmetic conditions actually occur
as the signature for an action of $G$ on some Riemann surface. As a conseq
uence\, we show that all non-abelian finite simple groups exhibit this pro
perty.\n
LOCATION:https://researchseminars.org/talk/UAANTS/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robin Bartlett (Univ. of Münster)
DTSTART;VALUE=DATE-TIME:20210209T170000Z
DTEND;VALUE=DATE-TIME:20210209T180000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/11
DESCRIPTION:Title:
Some Breuil-Mezard identities in moduli spaces of Breuil-Kisin modules
\nby Robin Bartlett (Univ. of Münster) as part of University of Arizona A
lgebra and Number Theory Seminar\n\n\nAbstract\nThe Breuil--Mezard conject
ure predicts certain identities between cycles in moduli spaces of mod p G
alois representations in terms of the Fp-representation theory of GLn(Fq).
\n\nIn this talk I will discuss work in progress which considers the situt
ation arising from (the reduction modulo p of) two dimensional crystalline
Galois representation with suitably small* Hodge--Tate weights. We will d
iscuss how the predected identities can also seen in ``resolutions`` of th
ese spaces of Galois representations described in terms of semilinar algeb
ra.\n\n*small will be precisely the bound which ensures that the Fp-repres
entation theory of GL2(Fq) appearing behaves precisely as it would with ch
ar 0 coefficients.\n
LOCATION:https://researchseminars.org/talk/UAANTS/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carl Wang Erickson (Univ. of Pittsburgh)
DTSTART;VALUE=DATE-TIME:20210223T210000Z
DTEND;VALUE=DATE-TIME:20210223T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/12
DESCRIPTION:Title:
Small non-Gorenstein residually Eisenstein Hecke algebras\nby Carl Wan
g Erickson (Univ. of Pittsburgh) as part of University of Arizona Algebra
and Number Theory Seminar\n\n\nAbstract\nIn Mazur's work proving the torsi
on theorem for rational elliptic curves\, he studied congruences between c
usp forms and Eisenstein series in weight two and prime level. One of his
innovations was to measure such congruences using a residually Eisenstein
Hecke algebra. He asked for generalizations of his theory to squarefree le
vels. The speaker made progress toward such generalizations in joint work
with Preston Wake\; however\, a crucial condition in their work was that t
he Hecke algebra be Gorenstein\, which is often but by no means always tru
e. We present joint work with Catherine Hsu and Preston Wake in which we s
tudy the smallest possible non-Gorenstein case and leverage this smallness
to draw an explicit link between its size and an invariant from algebraic
number theory.\n
LOCATION:https://researchseminars.org/talk/UAANTS/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chao Li (Columbia Univ.)
DTSTART;VALUE=DATE-TIME:20210126T210000Z
DTEND;VALUE=DATE-TIME:20210126T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/13
DESCRIPTION:Title:
Beilinson-Bloch conjecture for unitary Shimura varieties\nby Chao Li (
Columbia Univ.) as part of University of Arizona Algebra and Number Theory
Seminar\n\n\nAbstract\nFor certain automorphic representations $\\pi$ on
unitary groups\, we show that if $L(s\, \\pi)$ vanishes to order one at th
e center $s=1/2$\, then the associated $\\pi$-localized Chow group of a un
itary Shimura variety is nontrivial. This proves part of the Beilinson-Blo
ch conjecture for unitary Shimura varieties\, which generalizes the BSD co
njecture. Assuming the modularity of Kudla's generating series of special
cycles\, we further prove a precise height formula for $L'(1/2\, \\pi)$. T
his proves the conjectural arithmetic inner product formula\, which genera
lizes the Gross-Zagier formula to Shimura varieties of higher dimension. W
e will motivate these conjectures and discuss some aspects of the proof. T
his is joint work with Yifeng Liu.\n
LOCATION:https://researchseminars.org/talk/UAANTS/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yong-Suk Moon (Univ. of Arizona)
DTSTART;VALUE=DATE-TIME:20210216T210000Z
DTEND;VALUE=DATE-TIME:20210216T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/14
DESCRIPTION:Title:
Relative Fontaine-Messing theory over power series rings\nby Yong-Suk
Moon (Univ. of Arizona) as part of University of Arizona Algebra and Numbe
r Theory Seminar\n\n\nAbstract\nLet k be a perfect field of char p > 2. Fo
r a smooth proper scheme over W(k)\, Fontaine-Messing theory gives a nice
way to compare its torsion crystalline cohomology H^i_cris and torsion eta
le cohomology H^i_et when i < p-1. We will explain how one can generalize
Fontaine-Messing theory in the relative setting over power series rings\,
and discuss some applications. This is joint work with Tong Liu and Deepam
Patel.\n
LOCATION:https://researchseminars.org/talk/UAANTS/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andreas Mihatsch (Univ. of Bonn)
DTSTART;VALUE=DATE-TIME:20210323T170000Z
DTEND;VALUE=DATE-TIME:20210323T180000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/15
DESCRIPTION:Title:
AFL over F\nby Andreas Mihatsch (Univ. of Bonn) as part of University
of Arizona Algebra and Number Theory Seminar\n\n\nAbstract\nI will report
on the recent proof of the AFL over a general p-adic local field (p > n).
The previous proof of the AFL (due to W. Zhang) was restricted to Q_p sinc
e it relied on the modularity of Kudla divisor generating series on integr
al models of unitary Shimura varieties\, which is only known over Q. The n
ew proof merely requires modularity for the generic fiber generating serie
s\, allowing us to work with an arbitrary totally real field. This is join
t work with W. Zhang.\n
LOCATION:https://researchseminars.org/talk/UAANTS/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Congling Qiu (Yale Univ.)
DTSTART;VALUE=DATE-TIME:20210406T210000Z
DTEND;VALUE=DATE-TIME:20210406T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/16
DESCRIPTION:Title:
Modularity and Heights of CM cycles on Kuga-Sato varieties\nby Conglin
g Qiu (Yale Univ.) as part of University of Arizona Algebra and Number The
ory Seminar\n\n\nAbstract\nWe study CM cycles on Kuga-Sato varieties over
X(N). Our first result is the modularity of the unramified Hecke module
generated by CM cycles. This result enable us to decompose the space of C
M cycles according to the unramified Hecke action. Our second result is t
he full modularity of all CM cycles in the components of representations w
ith vanishing central (base change) L-values. Finally\, we prove a higher
weight analog of the general Gross-Zagier formula of Yuan\, S. Zhang and W
. Zhang.\n
LOCATION:https://researchseminars.org/talk/UAANTS/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jason Quinones (Gallaudet Univ.)
DTSTART;VALUE=DATE-TIME:20210413T210000Z
DTEND;VALUE=DATE-TIME:20210413T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/17
DESCRIPTION:Title:
Slow-Growing Weak Jacobi Forms\nby Jason Quinones (Gallaudet Univ.) as
part of University of Arizona Algebra and Number Theory Seminar\n\n\nAbst
ract\nWeak Jacobi forms of weight 0 can be exponentially lifted to meromor
phic Siegel paramodular forms. It was recently observed that the Fourier c
oefficients of such lifts are then either fast growing or slow growing. Th
ose weak Jacobi forms with slow growing behavior could describe the ellipt
ic genus of a CFT whose symmetric orbifold exhibits a slow supergravity-li
ke growth. In this talk\, we investigate the space of weak Jacobi forms th
at lead to slow growth. We provide analytic and numerical evidence for the
conjecture that there are such slow growing forms for any index.\n
LOCATION:https://researchseminars.org/talk/UAANTS/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yihang Zhu (Univ. of Maryland)
DTSTART;VALUE=DATE-TIME:20210420T210000Z
DTEND;VALUE=DATE-TIME:20210420T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/18
DESCRIPTION:Title:
Irreducible components of affine Deligne-Lusztig varieties\nby Yihang
Zhu (Univ. of Maryland) as part of University of Arizona Algebra and Numbe
r Theory Seminar\n\n\nAbstract\nAffine Deligne-Lusztig varieties naturally
arise from the study of Shimura varieties. We prove a formula for the num
ber of their irreducible components\, which was a conjecture of Miaofen Ch
en and Xinwen Zhu. Our method is to count the number of F_q points\, and t
o relate it to certain twisted orbital integrals. We then study the growt
h rate of these integrals using the Base Change Fundamental Lemma of Cloze
l and Labesse. In an ongoing work we also give the number of irreducible c
omponents in the basic Newton stratum of a Shimura variety. This is joint
work with Rong Zhou and Xuhua He.\n
LOCATION:https://researchseminars.org/talk/UAANTS/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiuya Wang (Duke Univ.)
DTSTART;VALUE=DATE-TIME:20210504T210000Z
DTEND;VALUE=DATE-TIME:20210504T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/19
DESCRIPTION:Title:
Average of $3$-torsion in class groups of $2$-extensions\nby Jiuya Wan
g (Duke Univ.) as part of University of Arizona Algebra and Number Theory
Seminar\n\n\nAbstract\nIn 1971\, Davenport and Heilbronn prove the celebra
ted theorem\ndetermining the average of $3$-torsion in class groups of qua
dratic\nextensions. In this talk\, we will study the average of $3$-torsio
n in\nclass groups of $2$-extensions\, which are towers of relative quadra
tic\nextensions. As an example\, we determine the average of $3$-torsion i
n\nclass groups of $D_4$ quartic extension. This is a joint work with\nRob
ert J. Lemke Oliver and Melanie Matchett Wood.\n
LOCATION:https://researchseminars.org/talk/UAANTS/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yanshuai Qin (UC Berkeley)
DTSTART;VALUE=DATE-TIME:20210302T210000Z
DTEND;VALUE=DATE-TIME:20210302T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/20
DESCRIPTION:Title:
A relation between Brauer groups and Tate-Shafarevich groups for high dime
nsional fibrations\nby Yanshuai Qin (UC Berkeley) as part of Universit
y of Arizona Algebra and Number Theory Seminar\n\n\nAbstract\nLet $\\mathc
al{X} \\rightarrow C$ be a dominant morphism between smooth geometrically
connected varieties over a finitely generated field such that the generic
fiber $X/K$ is smooth\, projective and geometrically connected. We prove
a relation between the Tate-Shafarevich group of $Pic^0_{X/K}$ and the g
eometric Brauer groups of $ \\mathcal{X}$\, $X$ and $C$\, generalizing a t
heorem of Artin and Grothendieck for fibered surfaces to arbitrary relativ
e dimension.\n
LOCATION:https://researchseminars.org/talk/UAANTS/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Schwein (Univ. of Cambridge)
DTSTART;VALUE=DATE-TIME:20210831T210000Z
DTEND;VALUE=DATE-TIME:20210831T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/21
DESCRIPTION:Title:
Recent progress on the formal degree conjecture\nby David Schwein (Uni
v. of Cambridge) as part of University of Arizona Algebra and Number Theor
y Seminar\n\n\nAbstract\nThe local Langlands correspondence is more than a
\ncorrespondence: it promises an extensive dictionary between the\nreprese
ntation theory of reductive p-adic groups and the arithmetic of\ntheir L-p
arameters. One entry in this dictionary is a conjectural\nformula of Hirag
a\, Ichino\, and Ikeda for the size of a discrete series\nrepresentation
– its “formal degree” – in terms of a gamma factor of\nits L-param
eter. In the first part of the talk\, I’ll explain why the\nconjecture i
s true for almost all supercuspidal representations. In\nthe second part\,
I’ll compute the sign of the gamma factor\, verifying\na conjecture of
Gross and Reeder.\n
LOCATION:https://researchseminars.org/talk/UAANTS/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Huajie Li (Aix-Marseille Univ.)
DTSTART;VALUE=DATE-TIME:20210914T170000Z
DTEND;VALUE=DATE-TIME:20210914T180000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/22
DESCRIPTION:Title:
On the comparison of an infinitesimal variant of Guo-Jacquet trace formula
e\nby Huajie Li (Aix-Marseille Univ.) as part of University of Arizona
Algebra and Number Theory Seminar\n\n\nAbstract\nGuo-Jacquet have propose
d a comparison of two relative trace formulae in order to gerneralise Wald
spurger’s well-known theorem relating toric periods to central values of
automorphic L-functions for $GL(2)$. In our previous works\, we have esta
blished an infinitesimal variant of Guo-Jacquet trace formulae and the wei
ghted fundamental lemma in this case. In this talk\, we shall explain seve
ral local results for the comparison of regular semi-simple terms. In part
icular\, we shall talk about certain identities between Fourier transforms
of local weighted orbital integrals\, which are proved by Waldspurger’s
global method on the endoscopic transfer. During the proof\, we shall als
o need some results in local harmonic analysis such as local trace formula
e for certain $p$-adic infinitesimal symmetric spaces.\n
LOCATION:https://researchseminars.org/talk/UAANTS/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anwesh Ray (University of British Columbia)
DTSTART;VALUE=DATE-TIME:20210921T210000Z
DTEND;VALUE=DATE-TIME:20210921T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/23
DESCRIPTION:Title:
Rank jumps and growth of Tate-Shafarevich groups of elliptic curves\nb
y Anwesh Ray (University of British Columbia) as part of University of Ari
zona Algebra and Number Theory Seminar\n\n\nAbstract\nWe use techniques fr
om Iwasawa theory to study the growth of the Mordell Weil group and Tate-S
hafarevich groups of elliptic curves in cyclic extensions of prime degree.
This is joint work with Lea Beneish and Debanjana Kundu.\n
LOCATION:https://researchseminars.org/talk/UAANTS/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Feng (MIT)
DTSTART;VALUE=DATE-TIME:20211005T210000Z
DTEND;VALUE=DATE-TIME:20211005T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/24
DESCRIPTION:Title:
Higher arithmetic theta series\nby Tony Feng (MIT) as part of Universi
ty of Arizona Algebra and Number Theory Seminar\n\n\nAbstract\nArithmetic
theta series are incarnations of theta functions in arithmetic algebraic g
eometry. The first examples were constructed by Kudla as generating series
of special cycles on Shimura varieties. Their conjectural key features ar
e (1) modularity of the generating series\, and (2) the arithmetic Siegel-
Weil formula\, relating their enumerative geometry to the first derivative
of Eisenstein series at special values. In joint work with Zhiwei Yun and
Wei Zhang\, we construct "higher" arithmetic theta series on moduli space
s of shtukas\, which we conjecture to also enjoy (1) modularity and (2) a
higher arithmetic Siegel-Weil formula relating their enumerative geometry
to all derivatives of Eisenstein series at special values. We prove severa
l results towards these conjectures\, drawing upon ideas from Ngo's proof
of the Fundamental Lemma in addition to new ingredients from Springer theo
ry and derived algebraic geometry.\n
LOCATION:https://researchseminars.org/talk/UAANTS/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yousheng Shi (Univ. of Wisconsin)
DTSTART;VALUE=DATE-TIME:20211026T210000Z
DTEND;VALUE=DATE-TIME:20211026T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/25
DESCRIPTION:Title:
Kudla-Rapoport conjecture at a ramified prime\nby Yousheng Shi (Univ.
of Wisconsin) as part of University of Arizona Algebra and Number Theory S
eminar\n\n\nAbstract\nKudla-Rapoport conjecture predicts that there is an
identity between the intersection number of special cycles on unitary Rapo
port-Zink space and the derivative of local density of certain Hermitian f
orm. However\, the original conjecture was only formulated for RZ space wi
th hyperspecial level structure over unramified primes . In this talk\, I
will motivate the original conjecture and discuss how to modify it at a ra
mified prime. Finally\, I will sketch how to verify the modified conjectur
e for n=3. This is a joint work with Qiao He and Tonghai Yang.\n
LOCATION:https://researchseminars.org/talk/UAANTS/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shizhang Li (Univ. of Michigan)
DTSTART;VALUE=DATE-TIME:20211109T210000Z
DTEND;VALUE=DATE-TIME:20211109T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/26
DESCRIPTION:Title:
On u-power torsions in prismatic cohomology\nby Shizhang Li (Univ. of
Michigan) as part of University of Arizona Algebra and Number Theory Semin
ar\n\n\nAbstract\nI will report a joint work with Tong\, concerning the st
ructure of a certain submodule inside prismatic cohomology of a smooth pro
per scheme over a p-adic ring of integers. I will explain how this part of
prismatic cohomology causes various pathologies\, then say a few correspo
nding consequences of our structural result. If time permits\, I shall als
o mention an interesting example.\n
LOCATION:https://researchseminars.org/talk/UAANTS/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Felix Baril Boudreau (Western Univ.)
DTSTART;VALUE=DATE-TIME:20211102T210000Z
DTEND;VALUE=DATE-TIME:20211102T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/27
DESCRIPTION:Title:
Computing an L-function modulo a prime\nby Felix Baril Boudreau (Weste
rn Univ.) as part of University of Arizona Algebra and Number Theory Semin
ar\n\n\nAbstract\nLet $E$ be an elliptic curve with non-constant $j$-invar
iant over a function field $K$ with constant field of size an odd prime po
wer $q$. Its $L$-function $L(T\,E/K)$ belongs to $1 + T\\mathbb{Z}[T]$. In
spired by the algorithms of Schoof and Pila for computing zeta functions o
f curves over finite fields\, we propose an approach to compute $L(T\,E/K)
$. The idea is to compute\, for sufficiently many primes $\\ell$ coprime w
ith $q$\, the reduction $L(T\,E/K) \\bmod{\\ell}$. The $L$-function is the
n recovered via the Chinese remainder theorem. When $E(K)$ has a subgroup
of order $N \\geq 2$ coprime with $q$\, Chris Hall showed how to explicitl
y calculate $L(T\,E/K) \\bmod{N}$. We present novel theorems going beyond
Hall's.\n
LOCATION:https://researchseminars.org/talk/UAANTS/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Garcia (UCL)
DTSTART;VALUE=DATE-TIME:20211123T210000Z
DTEND;VALUE=DATE-TIME:20211123T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/28
DESCRIPTION:Title:
Eisenstein cocycles and values of L-functions\nby Luis Garcia (UCL) as
part of University of Arizona Algebra and Number Theory Seminar\n\n\nAbst
ract\nThere are several recent constructions by many authors of Eisenstein
cocycles of arithmetic groups. I will discuss a point of view on these co
nstructions using equivariant cohomology and equivariant differential form
s. The resulting objects behave like theta kernels relating the homology o
f arithmetic groups to algebraic objects. As an application\, I will expla
in the proof of some conjectures of Sczech and Colmez on critical values o
f Hecke L-functions. The talk is based on joint work with Nicolas Bergeron
and Pierre Charollois.\n
LOCATION:https://researchseminars.org/talk/UAANTS/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aparna Upadhyay (Univ. of Arizona)
DTSTART;VALUE=DATE-TIME:20211019T210000Z
DTEND;VALUE=DATE-TIME:20211019T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/29
DESCRIPTION:Title:
The non-projective part of modular representations of finite groups\nb
y Aparna Upadhyay (Univ. of Arizona) as part of University of Arizona Alge
bra and Number Theory Seminar\n\n\nAbstract\nIn a recent paper\, Dave Bens
on and Peter Symonds introduced a new invariant for modular representation
s of a finite group. This invariant is a result of studying the asymptotic
s of the direct sum decomposition of the non-projective part of tensor pow
ers of a finite dimensional representation of a finite group in prime char
acteristic. In this talk\, we will see some interesting properties of this
invariant. We will obtain a closed formula for computing the invariant fo
r a family of modules of the symmetric group and for trivial source module
s of a finite group. Benson and Symonds conjectured that the growth of the
non-projective part of tensor powers of a module is linear recursive. We
will also see some results towards this conjecture.\n
LOCATION:https://researchseminars.org/talk/UAANTS/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Serin Hong (Univ. of Michigan)
DTSTART;VALUE=DATE-TIME:20211207T210000Z
DTEND;VALUE=DATE-TIME:20211207T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/30
DESCRIPTION:Title:
Classification theorems for vector bundles on the Fargues-Fontaine curve\nby Serin Hong (Univ. of Michigan) as part of University of Arizona Alg
ebra and Number Theory Seminar\n\n\nAbstract\nThe Fargues-Fontaine curve h
as played a pivotal role in the recent development of arithmetic geometry.
Most notably\, the work of Fargues-Scholze constructs the local Langlands
correspondence in a form of the geometric Langlands correspondence for th
e Fargues-Fontaine curve. In addition\, Fargues shows that the Fargues-Fon
taine curve provides a geometric interpretation for Galois cohomology of l
ocal fields and much of the classical p-adic Hodge theory. \n\nIn this tal
k\, we discuss several classification theorems for vector bundles on the F
argues-Fontaine curve. In particular\, we give a complete classification o
f all subsheaves\, quotients\, and minuscule modifications of a given vect
or bundle on the Fargues-Fontaine curve. We also discuss some applications
of these theorems in the context of the local Langlands correspondence.\n
LOCATION:https://researchseminars.org/talk/UAANTS/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Darlayne Addabbo (Univ. of Arizona)
DTSTART;VALUE=DATE-TIME:20211116T210000Z
DTEND;VALUE=DATE-TIME:20211116T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/31
DESCRIPTION:Title:
Zhu algebras and their applications\nby Darlayne Addabbo (Univ. of Ari
zona) as part of University of Arizona Algebra and Number Theory Seminar\n
\n\nAbstract\nZhu algebras and their generalizations\, higher level Zhu al
gebras\, are associative algebras that are important in the study of verte
x operator algebras. In this talk\, I will define Zhu algebras and higher
level Zhu algebras and discuss motivation for their study. This talk will
be expository and prior knowledge of vertex operator algebras will not be
assumed. (Based on joint work with Barron\, Batistelli\, Orosz-Hunziker\,
Pedi\\'{c}\, and Yamskulna.)\n
LOCATION:https://researchseminars.org/talk/UAANTS/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ayan Maiti (Oklahoma State Univ.)
DTSTART;VALUE=DATE-TIME:20211130T210000Z
DTEND;VALUE=DATE-TIME:20211130T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/32
DESCRIPTION:Title:
WEYL’S LAW FOR CUSP FORMS OF ARBITRARY $K_{\\infty}$-TYPE\nby Ayan M
aiti (Oklahoma State Univ.) as part of University of Arizona Algebra and N
umber Theory Seminar\n\n\nAbstract\nLet $M$ be a compact Riemannian manifo
ld. It was proved by Weyl that number of\nLaplacian eigenvalues less than
$T$\, is asymptotic to $C(M)T^{dim(M)/2}$\, where $C(M)$ is the\nproduct o
f the volume of $M$\, volume of the unit ball and $(2π)^{−dim(M)}$. Let
$\\Gamma$ be an\narithmetic subgroup of $SL_2(\\mathbb{Z})$ and \\mathbb{
H}^2 be an upper-half plane. When $M = \\Gamma \\backslash \\mathbb{H}^2$\
, Weyl’sasymptotic holds true for the discrete spectrum of Laplacian. It
was proved by Selberg\, who used his celebrated trace formula.\nLet $G$ b
e a semisimple algebraic group of Adjoint and split type over $\\mathbb{Q}
$. Let $G(\\mathbb{R})$ be\nthe set of $\\mathbb{R}$-points of $G$. For si
mplicity of this exposition let us assume that $\\Gamma \\subset G(\\mathb
b{R})$ be an torsion free arithmetic subgroup. Let $K_{\\infty}$ be the ma
ximal compact subgroup.\nLet $L^2(\\Gamma \\backslash G(\\mathbb{R})$ be s
pace of square integrable $\\Gamma$ invariant functions on $G(\\mathbb{R})
$. Let $L^2_{cusp}(\\Gamma \\backslash G(\\mathbb{R})$ be the cuspidal sub
space. Let $M = \\Gamma \\backslash G(\\mathbb{R})/K_{\\infty}$ be a local
ly symmetric space. Suppose $d = dim(\\Gamma \\backslash G(\\mathbb{R})/K_
{\\infty})$. Then it was proved by Lindenstrauss and Venkatesh\,\nthat num
ber of spherical\, i.e. bi-$K_{\\infty}$ invariant cuspidal Laplacian eige
nfunctions\, whose\neigenvalues are less than T is asymptotic to $C(M)T^{d
im(M)/2}$\, where $C(M)$ is the same\nconstant as above.\nWe are going to
prove the same Weyl’s asymptotic estimates for $K_{\\infty}$-finite cusp
forms for\nthe above space.\n
LOCATION:https://researchseminars.org/talk/UAANTS/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pavel Coupek (Purdue University)
DTSTART;VALUE=DATE-TIME:20220201T210000Z
DTEND;VALUE=DATE-TIME:20220201T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/33
DESCRIPTION:Title:
Crystalline condition for Ainf-cohomology and ramification bounds\nby
Pavel Coupek (Purdue University) as part of University of Arizona Algebra
and Number Theory Seminar\n\n\nAbstract\nLet $p>2$ be a prime and let $X$
be a proper smooth formal scheme over $\\mathcal{O}_K$ where $K/\\mathbb{Q
}_p$ is a local number field. In this talk\, we describe a series of condi
tions $(\\mathrm{Cr}_s)$ that provide control on the Galois action on the
Breuil--Kisin cohomology $\\mathrm{R}\\Gamma_{\\Delta}(X/\\mathfrak{S})$ i
nside the $A_{\\inf}$--cohomology $\\mathrm{R}\\Gamma_{\\Delta}(X_{\\mathb
b{C}_K}/A_{\\inf})$. When $s=0$\, the resulting condition is essentially t
he crystallinity criterion of Gee and Liu for Breuil--Kisin--Fargues $G_K$
--modules\, and it leads to an alternative proof of crystallinity of the $
p$--adic \\'{e}tale cohomology $H^i_{\\mathrm{et}}(X_{\\mathbb{C}_K}\, \\m
athbb{Q}_p)$. Adapting a strategy of Caruso and Liu\, the conditions $(\\m
athrm{Cr}_s)$ for higher $s$ then lead to upper bounds on ramification of
the mod $p$ \\'{e}tale cohomology $H^i_{\\mathrm{et}}(X_{\\mathbb{C}_K}\,
\\mathbb{Z}/p\\mathbb{Z})$\, expressed in terms of $i\, p$ and $e=e(K/\\ma
thbb{Q}_p)$ that work without any restrictions on the size of $i$ and $e$.
\n
LOCATION:https://researchseminars.org/talk/UAANTS/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tarun Dalal (IIT)
DTSTART;VALUE=DATE-TIME:20220208T210000Z
DTEND;VALUE=DATE-TIME:20220208T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/34
DESCRIPTION:Title:
The structure of Drinfeld modular forms of level $\\Gamma_0(T)$ and applic
ations\nby Tarun Dalal (IIT) as part of University of Arizona Algebra
and Number Theory Seminar\n\n\nAbstract\nLet $q$ be a power of an odd prim
e $p$. Let $A:=\\mathbb{F}_q[T]$ and $C$ denote the completion of an algeb
raic closure of $\\mathbb{F}_q((\\frac{1}{T}))$. For any ring $R$ with $A
\\subseteq R \\subseteq C$\, we let $M(\\Gamma_0(\\mathfrak{n}))_R$ denote
the ring of Drinfeld modular forms of level $\\Gamma_0(\\mathfrak{n})$ wi
th coefficients in $R$.\nIn 1988\, Gekeler showed that the $C$-algebra $M(
\\mathrm{GL}_2(A))_C$ is isomorphic to $C[X\,Y]$. As a result\, the proper
ties of the weight filtration for Drinfeld modular forms for $\\mathrm{GL}
_2(A)$ are studied by Gekeler in 1988 and by Vincent in 2010.\n\nIn this t
alk\, we discuss about the structure of the $R$-algebra $M(\\Gamma_0(T))_R
$ and study the properties of the weight filtration for Drinfeld modular f
orms of level $\\Gamma_0(T)$. As an application\, we prove a result on mod
-$\\mathfrak{p}$ congruences for Drinfeld modular forms of level $\\Gamma_
0(\\mathfrak{p} T)$ for $\\mathfrak{p} \\neq (T)$. This is a joint work wi
th Narasimha Kumar.\n
LOCATION:https://researchseminars.org/talk/UAANTS/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhixiang Wu (Paris-Saclay)
DTSTART;VALUE=DATE-TIME:20220215T210000Z
DTEND;VALUE=DATE-TIME:20220215T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/35
DESCRIPTION:Title:
Companion forms with non-regular weights\nby Zhixiang Wu (Paris-Saclay
) as part of University of Arizona Algebra and Number Theory Seminar\n\n\n
Abstract\nIn general\, for an overconvergent p-adic automorphic form of a
definite unitary group\, there exist other p-adic automorphic forms with p
ossibly different weights which are associated with the same Galois repres
entation (the companion forms). Under the Taylor-Wiles hypothesis\, we det
ermine all the companion forms whose associated Galois representations are
generic crystalline over p and with Hodge-Tate weights possibly non-regul
ar. This generalizes the result of Breuil-Hellmann-Schraen in regular case
s.\n
LOCATION:https://researchseminars.org/talk/UAANTS/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zahi Hazan (Tel Aviv Univ.)
DTSTART;VALUE=DATE-TIME:20220125T210000Z
DTEND;VALUE=DATE-TIME:20220125T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/36
DESCRIPTION:Title:
An Identity Relating Eisenstein Series on General Linear Groups\nby Za
hi Hazan (Tel Aviv Univ.) as part of University of Arizona Algebra and Num
ber Theory Seminar\n\n\nAbstract\nEisenstein series are key objects in the
theory of automorphic forms. They play an important role in the study of
automorphic $L$-functions\, and they figure out in the spectral decomposit
ion of the $L^2$-space of automorphic forms. In recent years\, new constru
ctions of global integrals generating identities relating Eisenstein serie
s were discovered. In 2018 Ginzburg and Soudry introduced two general iden
tities relating Eisenstein series on split classical groups (generalizing
Mœglin 1997\, Ginzburg-Piatetski-Shapiro-Rallis 1997\, and Cai-Friedberg-
Ginzburg-Kaplan 2016)\, as well as double covers of symplectic groups (gen
eralizing Ikeda 1994\, and Ginzburg-Rallis-Soudry 2011).\n\nWe consider th
e Kronecker product embedding of two general linear groups\, $\\mathrm{GL}
{m}(\\mathbb{A})$ and $\\mathrm{GL}{n}(\\mathbb{A})$\, in $\\mathrm{GL}{mn
}(\\mathbb{A})$. Now\, similarly to Ginzburg and Soudry's construction\, w
e use a degenerate Eisenstein series of $\\mathrm{GL}{mn}(\\mathbb{A})$ as
a kernel function on $\\mathrm{GL}{m}(\\mathbb{A}) \\otimes \\mathrm{GL}{
n}(\\mathbb{A})$. Integrating it against a cusp form on $\\mathrm{GL}{n}(\
\mathbb{A})$\, we obtain a 'semi-degenerate' Eisenstein series on $\\mathr
m{GL}{m}(\\mathbb{A})$. Locally\, we find an interesting relation to the l
ocal Godement-Jacquet integral.\n\nThis construction demonstrates the rise
of interesting $L$-functions from integrals of doubling type\, as suggest
ed by the philosophy of Ginzburg and Soudry.\n
LOCATION:https://researchseminars.org/talk/UAANTS/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tristan Phillips (University of Arizona)
DTSTART;VALUE=DATE-TIME:20220222T210000Z
DTEND;VALUE=DATE-TIME:20220222T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/37
DESCRIPTION:Title:
Counting Elliptic Curves over Number Fields\nby Tristan Phillips (Univ
ersity of Arizona) as part of University of Arizona Algebra and Number The
ory Seminar\n\n\nAbstract\nIn this talk I will discuss some results on cou
nting elliptic curves over number fields. In particular\, I will give asym
ptotics for the number of isomorphism classes of elliptic curves over arbi
trary number fields with certain prescribed level structures and prescribe
d local conditions. This is done by counting the number of points of bound
ed height on genus zero modular curves which are isomorphic to a weighted
projective space. This includes the cases of X(N) for N\\in\\{1\,2\,3\,4\
,5\\}\, X_1(N) for N\\in\\{1\,2\,\\dots\,10\,12\\}\, and X_0(N) for N\\in\
\{1\,2\,4\,6\,8\,9\,12\,16\,18\\}. Using these results for counting ellipt
ic curves over number fields with a prescribed local condition\, one can s
how that the average analytic rank of elliptic curves over any number fiel
d K is bounded above by 3\\text{deg}(K)+1/2\, under the assumptions that a
ll elliptic curves over K are modular and have L-functions which satisfy t
he Generalized Riemann Hypothesis\n
LOCATION:https://researchseminars.org/talk/UAANTS/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shichen Tang (UC Irvine)
DTSTART;VALUE=DATE-TIME:20220412T210000Z
DTEND;VALUE=DATE-TIME:20220412T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/38
DESCRIPTION:Title:
Slope stability for higher rank Artin--Schreier--Witt towers\nby Shich
en Tang (UC Irvine) as part of University of Arizona Algebra and Number Th
eory Seminar\n\n\nAbstract\nFor a curve in characteristic p\, consider the
p-adic valuations of the reciprocal roots of its zeta function. These are
rational numbers between 0 and 1\, and they are also the slopes of the p-
adic Newton polygon of the numerator polynomial of the zeta function. In g
eneral\, these numbers depend on the curve\, and all we have is an upper b
ound and a lower bound for the Newton polygon. But for curves in an Artin-
-Schreier--Witt tower satisfying certain conditions\, the slopes behave in
a stable way. It can be shown that the data of the slopes of the Newton p
olygon for all the curves in the tower is determined by the data for finit
ely many curves\, and for each curve\, the slopes can be explicitly writte
n as a union of finitely many arithmetic progressions.\n\nLet d be the ran
k of the Galois group of this tower as a free Z_p-module. In rank d=1 case
\, this was proved by Kosters--Zhu in 2017. In this talk\, we will explain
the proof for the higher rank case.\n
LOCATION:https://researchseminars.org/talk/UAANTS/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Keerthi Madapusi (Boston College)
DTSTART;VALUE=DATE-TIME:20220315T210000Z
DTEND;VALUE=DATE-TIME:20220315T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/39
DESCRIPTION:Title:
Derived homomorphisms of abelian varieties and special cycles on Shimura v
arieties\nby Keerthi Madapusi (Boston College) as part of University o
f Arizona Algebra and Number Theory Seminar\n\n\nAbstract\nI will present
a 'derived' version of homomorphisms between abelian varieties that in a s
ense explains their deformation theory\, and will give some indication of
how this can be applied to give a uniform construction of special cycle cl
asses on Shimura varieties of Hodge type using methods from derived algebr
aic geometry.\n
LOCATION:https://researchseminars.org/talk/UAANTS/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhiyu Zhang (MIT)
DTSTART;VALUE=DATE-TIME:20220405T210000Z
DTEND;VALUE=DATE-TIME:20220405T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/40
DESCRIPTION:Title:
Modularity and Bruhat--Tits stratification\nby Zhiyu Zhang (MIT) as pa
rt of University of Arizona Algebra and Number Theory Seminar\n\n\nAbstrac
t\nIt is well-known that the theta series of a positive definite quadratic
lattice is a modular form. In this talk\, I will present new modularity r
esults for arithmetic theta series of some “parahoric" hermitian lattice
s of sign (n-1\,1)\, which live on unitary Shimura varieties. One key step
is to understand a local analog of the story\, which happens on unitary R
apoport-Zink spaces. I will explain the use of Bruhat-Tits stratification
studied by S. Cho in the computation.\n
LOCATION:https://researchseminars.org/talk/UAANTS/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Qiao He (University of Wisconsin-Madison)
DTSTART;VALUE=DATE-TIME:20220426T210000Z
DTEND;VALUE=DATE-TIME:20220426T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/41
DESCRIPTION:Title:
Kudla-Rapoport conjecture at a ramified prime.\nby Qiao He (University
of Wisconsin-Madison) as part of University of Arizona Algebra and Number
Theory Seminar\n\n\nAbstract\nKudla-Rapoport conjecture predicts that the
re is an identity between the intersection number of special cycles on uni
tary Rapoport-Zink space and the derivative of local density of certain He
rmitian form. However\, the original conjecture was only formulated for RZ
space with hyperspecial level structure over unramified primes. In this t
alk\, I will motivate the original conjecture and discuss how to modify it
at a ramified prime. Finally\, I will sketch a surprisingly simple proof
of the modified conjecture by taking partial Fourier transform. This is a
joint work with Chao Li\, Yousheng Shi and Tonghai Yang.\n
LOCATION:https://researchseminars.org/talk/UAANTS/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maria Fox (University of Oregon)
DTSTART;VALUE=DATE-TIME:20220503T210000Z
DTEND;VALUE=DATE-TIME:20220503T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/42
DESCRIPTION:Title:
Supersingular Loci of Unitary (2\,m-2) Shimura Varieties\nby Maria Fox
(University of Oregon) as part of University of Arizona Algebra and Numbe
r Theory Seminar\n\n\nAbstract\nThe supersingular locus of a Unitary (2\,m
-2) Shimura variety parametrizes supersingular abelian varieties of dimens
ion m\, with an action of a quadratic imaginary field meeting the "signatu
re (2\,m-2)" condition.\nIn some cases\, for example when m=3 or m=4\, eve
ry irreducible component of the supersingular locus is isomorphic to a Del
igne-Lusztig variety\, and the intersection combinatorics are governed by
a Bruhat-Tits building. We'll consider these cases for motivation\, and th
en see how the structure of the supersingular locus becomes very different
for m>4. (The new result in this talk is joint with Naoki Imai.)\n
LOCATION:https://researchseminars.org/talk/UAANTS/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elad Zelingher (Yale University)
DTSTART;VALUE=DATE-TIME:20220510T210000Z
DTEND;VALUE=DATE-TIME:20220510T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/43
DESCRIPTION:Title:
On regularization of integrals of matrix coefficients associated to spheri
cal Bessel models\nby Elad Zelingher (Yale University) as part of Univ
ersity of Arizona Algebra and Number Theory Seminar\n\n\nAbstract\nThe Gan
-Gross-Prasad conjecture relates a special value of an L-function of two c
uspidal automorphic representations to the non-vanishing of a certain peri
od. The Ichino-Ikeda conjecture is a refinement of this conjecture. It rou
ghly states that the absolute value of the square of the period in questio
n can be expressed as a product of the special value of the L-function and
a product of normalized local periods. However\, in order to formulate th
is conjecture\, one needs to assume that the representations in question a
re tempered everywhere\, or else the convergence of the local periods is n
ot guaranteed. The generalized Ramanujan conjecture speculates that the re
presentations in question (cuspidal automorphic representations lying in g
eneric packets) are already tempered everywhere. However\, the generalized
Ramanujan conjecture is far from being known. In this talk\, I will expla
in how to drop the assumption that the representations are tempered almost
everywhere. I will explain how to extend the definition of the normalized
local periods for places where the local components are given by principa
l series representations.\n
LOCATION:https://researchseminars.org/talk/UAANTS/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ziquan Yang (UW-Madison)
DTSTART;VALUE=DATE-TIME:20220913T210000Z
DTEND;VALUE=DATE-TIME:20220913T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/44
DESCRIPTION:Title:
The Tate Conjecture over Finite Fields for Varieties with $h^{2\,0}=1$.\nby Ziquan Yang (UW-Madison) as part of University of Arizona Algebra an
d Number Theory Seminar\n\n\nAbstract\nThe past decade has witnessed a gre
at advancement on the Tate conjecture for varieties with Hodge number $h^{
2\,0}=1$. Charles\, Madapusi-Pera and Maulik completely settled the conjec
ture for K3 surfaces over finite fields\, and Moonen proved the Mumford-Ta
te (and hence also Tate) conjecture for more or less\narbitrary $h^{2\,0}=
1$ varieties in characteristic $0$.\nIn this talk\, I will explain that th
e Tate conjecture is true for mod $p$ reductions of complex projective $h^
{2\,0}=1$ varieties when $p >> 0$\, under a mild assumption on moduli. By
refining this general result\, we prove that in characteristic $p \\geq 5$
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 a
ll singular fibers in its minimal compactification are irreducible. We als
o 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 connection between the Tate conjecture over finite fields and the Lef
schetz (1\, 1)-theorem over C is very robust for $h^{2\,0}=1$ varieties\,
and works well beyond the hyperkähler world.\nThis is based on joint work
with Paul Hamacher and Xiaolei Zhao.\n
LOCATION:https://researchseminars.org/talk/UAANTS/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pan Yan (U of Arizona)
DTSTART;VALUE=DATE-TIME:20220830T210000Z
DTEND;VALUE=DATE-TIME:20220830T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/45
DESCRIPTION:Title:
L-function for Sp(4)xGL(2) via a non-unique model\nby Pan Yan (U of Ar
izona) as part of University of Arizona Algebra and Number Theory Seminar\
n\nLecture held in ENR2 S395.\n\nAbstract\nThe theory of L-functions of au
tomorphic forms or automorphic representations is a central topic in moder
n number theory. A fruitful way to study L-functions is through an integra
l formula\, commonly referred to as an integral representation. The most c
ommon examples of Eulerian integrals are the ones which unfold to a unique
model such as the Whittaker model. Integrals which unfold to non-unique m
odels fall outside of this paradigm\, and there are only a few such exampl
es which are known to represent L-functions. In this talk\, we prove a con
jecture of Ginzburg and Soudry [IMRN\, 2020] on an integral representation
for the tensor product partial L-function for Sp(4)×GL(2) which is deriv
ed from the twisted doubling method of Cai\, Friedberg\, Ginzburg\, and Ka
plan. We show that the integral unfolds to a non-unique model and analyze
it using the New Way method of Piatetski-Shapiro and Rallis.\n
LOCATION:https://researchseminars.org/talk/UAANTS/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Hazeltine (Purdue University)
DTSTART;VALUE=DATE-TIME:20220920T210000Z
DTEND;VALUE=DATE-TIME:20220920T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/46
DESCRIPTION:Title:
Intersections of local Arthur packets for classical groups\nby Alexand
er Hazeltine (Purdue University) as part of University of Arizona Algebra
and Number Theory Seminar\n\n\nAbstract\nRecently\, Atobe gave a refinemen
t of Moeglin's construction of local Arthur packets for symplectic and spl
it odd special orthogonal groups. In joint work with Baiying Liu and Chi-H
eng Lo\, using Atobe's refinement\, we define certain operators which dete
rmine when two local Arthur packets intersect. From these operators\, we c
an define an ordering on the set of all local Arthur packets containing a
fixed representation for which there is a unique maximal and minimal eleme
nt. In this talk\, we will discuss Atobe's construction\, introduce the op
erators\, define the unique maximal and minimal element\, and discuss how
these elements behave with respect to other orderings.\n
LOCATION:https://researchseminars.org/talk/UAANTS/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arvind Suresh (U of Arizona)
DTSTART;VALUE=DATE-TIME:20220906T210000Z
DTEND;VALUE=DATE-TIME:20220906T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/47
DESCRIPTION:Title:
Curves with large rank via the PTE problem\nby Arvind Suresh (U of Ari
zona) as part of University of Arizona Algebra and Number Theory Seminar\n
\nLecture held in ENR2 S395.\n\nAbstract\nIt is an open question whether t
he rank of a curve X/Q (i.e. the Mordell--Weil rank of the group of ration
al points of the Jacobian J/Q) is bounded in terms of the genus g of X. Sh
ioda extended a construction of Mestre to produce infinite families of g>1
curves over Q with rank at least 4g+7. \nIn this talk\, I will present a
refinement of the Mestre--Shioda construction which leads to some interest
ing families of curves over Q (and over cyclotomic fields) with rank large
r than 4g+7. These families are parametrized by certain highly symmetric r
ational varieties associated to the Prouhet--Tarry--Escott (PTE) problem\,
a classical problem in number theory.\n
LOCATION:https://researchseminars.org/talk/UAANTS/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jialiang Zou (University of Michigan)
DTSTART;VALUE=DATE-TIME:20221101T210000Z
DTEND;VALUE=DATE-TIME:20221101T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/48
DESCRIPTION:Title:
On some Hecke algebra modules arising from theta correspondence and its de
formation\nby Jialiang Zou (University of Michigan) as part of Univers
ity of Arizona Algebra and Number Theory Seminar\n\n\nAbstract\nThis talk
is based on the joint work with Jiajun Ma and Congling Qiu on theta corres
pondence of type I dual pairs over a finite field $F_q$. We study the Hec
ke algebra modules arising from theta correspondence between certain Haris
h-Chandra series for these dual pairs. We first show that the normalizatio
n of the corresponding Hecke algebra is related to the first occurrence in
dex\, which leads to proof of the conservation relation. We then study the
deformation of this Hecke algebra module at $q=1$ and generalize the resu
lts of Aubert-Michel-Rouquier and Pan on theta correspondence between unip
otent representations along this way.\n
LOCATION:https://researchseminars.org/talk/UAANTS/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicholas Ovenhouse (Yale)
DTSTART;VALUE=DATE-TIME:20221108T210000Z
DTEND;VALUE=DATE-TIME:20221108T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/49
DESCRIPTION:Title:
Super Ptolemy Relation and Double Dimer Covers\nby Nicholas Ovenhouse
(Yale) as part of University of Arizona Algebra and Number Theory Seminar\
n\nLecture held in ENR2-S395.\n\nAbstract\nGiven a quadrilateral inscribed
in a circle\, Ptolemy's Theorem relates the lengths of the diagonals and
sides. In general\, for an inscribed polygon\, Ptolemy's relation allows o
ne to write the length of any diagonal as a Laurent polynomial in terms of
the lengths of the diagonals coming from some fixed triangulation. Schiff
ler and Musiker showed that these Laurent polynomials can be written in te
rms of perfect matchings (or "dimer covers") of some planar graph. Recentl
y\, Penner and Zeitlin defined a super-symmetric version of Ptolemy's rela
tion\, involving anti-commuting variables. In recent work with Musiker and
Zhang\, we showed that iterated applications of the super Ptolemy relatio
n gives a sum over double dimer covers of the same planar graph.\n
LOCATION:https://researchseminars.org/talk/UAANTS/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Soumya Sankar (Ohio State University)
DTSTART;VALUE=DATE-TIME:20221115T210000Z
DTEND;VALUE=DATE-TIME:20221115T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/50
DESCRIPTION:Title:
Counting elliptic curves with a rational N-isogeny\nby Soumya Sankar (
Ohio State University) as part of University of Arizona Algebra and Number
Theory Seminar\n\n\nAbstract\nThe classical problem of counting elliptic
curves with a rational N-isogeny can be phrased in terms of counting ratio
nal points on certain moduli stacks of elliptic curves. Counting points on
stacks poses various challenges\, and I will discuss these along with a f
ew ways to overcome them. I will also talk about height functions on certa
in stacks\, focusing on the theory of heights on stacks developed in recen
t work of Ellenberg\, Satriano and Zureick-Brown. I will then use their fr
amework to count elliptic curves with an N-isogeny for certain N. The talk
assumes no prior knowledge of stacks and is based on joint work with Bran
don Boggess.\n
LOCATION:https://researchseminars.org/talk/UAANTS/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Landesman (MIT)
DTSTART;VALUE=DATE-TIME:20221014T200000Z
DTEND;VALUE=DATE-TIME:20221014T210000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/51
DESCRIPTION:Title:
Arithmetic representations on generic curves\nby Aaron Landesman (MIT)
as part of University of Arizona Algebra and Number Theory Seminar\n\n\nA
bstract\nOver the last century\, the Hodge and Tate conjectures have inspi
red much activity in algebraic and arithmetic geometry. These conjectures
give predictions for when certain topological objects come from geometry.
Simpson and Fontaine-Mazur introduced non-abelian analogs of these conject
ures. In joint work with Daniel Litt\, we prove these analogs for low rank
local systems on generic curves\, resolving conjectures of Esnault-Kerz a
nd Budur-Wang as well as answering questions of Kisin and Whang.\n
LOCATION:https://researchseminars.org/talk/UAANTS/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Heidi Goodson (Brooklyn College (CUNY))
DTSTART;VALUE=DATE-TIME:20221129T210000Z
DTEND;VALUE=DATE-TIME:20221129T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/52
DESCRIPTION:Title:
Degeneracy and Sato-Tate Groups in Dimension Greater than 3\nby Heidi
Goodson (Brooklyn College (CUNY)) as part of University of Arizona Algebra
and Number Theory Seminar\n\n\nAbstract\nThe term degenerate is used to d
escribe abelian varieties whose Hodge rings contain exceptional cycles --
Hodge cycles that are not generated by divisor classes. We can see the eff
ect of the exceptional cycles on the structure of an abelian variety throu
gh its Mumford-Tate group\, Hodge group\, and Sato-Tate group. In this tal
k I will discuss degeneracy through these different but related lenses\, s
pecializing to Jacobians of hyperelliptic curves of the form $y^2=x^m−1$
. Together\, we will explore the various forms of degeneracy for several e
xamples\, each illustrating different phenomena that can occur.\n
LOCATION:https://researchseminars.org/talk/UAANTS/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Disegni (Ben-Gurion Univ. of the Negev)
DTSTART;VALUE=DATE-TIME:20221018T170000Z
DTEND;VALUE=DATE-TIME:20221018T180000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/53
DESCRIPTION:Title:
Theta cycles\nby Daniel Disegni (Ben-Gurion Univ. of the Negev) as par
t of University of Arizona Algebra and Number Theory Seminar\n\n\nAbstract
\nI will introduce ‘canonical’ algebraic cycles for motives M enjoying
a certain symmetry - for instance\, some symmetric powers of elliptic cu
rves. The construction is based on works of Kudla and Liu on some (conject
urally modular) theta series valued in Chow groups of Shimura varieties. T
he cycles have Heegner-point-like features that allow\, under some assumpt
ions\, to establish an analogue of the BSD conjecture for M at an ordinary
prime p. Namely\, if the p-adic L-function of M vanishes at 0 to order ex
actly 1\, then the Selmer group of M has rank 1 and it is generated by cla
sses of algebraic cycles. Partly joint work with Yifeng Liu.\n
LOCATION:https://researchseminars.org/talk/UAANTS/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chen Wan (Rutgers University – Newark)
DTSTART;VALUE=DATE-TIME:20230112T210000Z
DTEND;VALUE=DATE-TIME:20230112T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/54
DESCRIPTION:Title:
Period integrals and multiplicities for some strongly tempered spherical v
arieties\nby Chen Wan (Rutgers University – Newark) as part of Unive
rsity of Arizona Algebra and Number Theory Seminar\n\n\nAbstract\nIn this
talk I will discuss the local and global conjectures for some strongly tem
pered spherical varieties. Both conjectures are very similar to the Gan-Gr
oss-Prasad models. More specifically\, globally the square of the period i
ntegrals should be related to the central value of some L-functions of sym
plectic type. Locally each tempered L-packet should contain a unique disti
nguished element with multiplicity one and the unique distinguished elemen
t should be determined by certain epsilon factors (i.e. epsilon dichotomy)
. I will also discuss the proof of the local conjecture in many cases. Thi
s is a joint work with Lei Zhang.\n
LOCATION:https://researchseminars.org/talk/UAANTS/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Baiying Liu (Purdue)
DTSTART;VALUE=DATE-TIME:20230221T210000Z
DTEND;VALUE=DATE-TIME:20230221T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/55
DESCRIPTION:Title:
Recent progress on certain problems related to local Arthur packets of cla
ssical groups\nby Baiying Liu (Purdue) as part of University of Arizon
a Algebra and Number Theory Seminar\n\n\nAbstract\nIn this talk\, I will i
ntroduce recent progress made on certain problems related to local Arthur
packets of classical groups. First\, I will introduce my joint work with F
reydoon Shahidi towards Jiang's conjecture on the wave front sets of repre
sentations in local Arthur packets of classical groups\, which is a natura
l generalization of Shahidi's conjecture\, confirming the relation between
the structure of wave front sets and the local Arthur parameters. Then\,
I will introduce my joint work with my students Alexander Hazeltine and Ch
i-Heng Lo on the intersection problem of local Arthur packets for symplect
ic and split odd special orthogonal groups\, with applications to the Enha
nced Shahidi's conjecture and the closure relation conjecture.\n
LOCATION:https://researchseminars.org/talk/UAANTS/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xu Gao (UC Santa Cruz)
DTSTART;VALUE=DATE-TIME:20230131T210000Z
DTEND;VALUE=DATE-TIME:20230131T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/56
DESCRIPTION:Title:
p-adic representations and simplicial balls in Bruhat-Tits buildings\n
by Xu Gao (UC Santa Cruz) as part of University of Arizona Algebra and Num
ber Theory Seminar\n\n\nAbstract\np-adic representations are important obj
ects in number theory\, and stable lattices serve as a connection between
the study of ordinary and modular representations. These stable lattices c
an be understood as stable vertices in Bruhat-Tits buildings. From this vi
ewpoint\, the study of fixed point sets in these buildings can aid researc
h on p-adic representations. The simplicial balls\, in particular\, hold a
n important role as they possess the most symmetry and fastest growth\, an
d are closely related to the Moy-Prasad filtrations. In this talk\, I'll e
xplain those new findings\, provide a characterization of such simplicial
balls\, and compute their simplicial volume under certain conditions.\n
LOCATION:https://researchseminars.org/talk/UAANTS/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Naomi Sweeting (Harvard)
DTSTART;VALUE=DATE-TIME:20230228T210000Z
DTEND;VALUE=DATE-TIME:20230228T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/57
DESCRIPTION:Title:
Tate Classes and Endoscopy for GSp4\nby Naomi Sweeting (Harvard) as pa
rt of University of Arizona Algebra and Number Theory Seminar\n\nLecture h
eld in ENR2 S395.\n\nAbstract\nWeissauer proved using the theory of endosc
opy that the Galois representations associated to classical modular forms
of weight two appear in the middle cohomology of both a modular curve and
a Siegel modular threefold. Correspondingly\, there are large families of
Tate classes on the product of these two Shimura varieties\, and it is nat
ural to ask whether one can construct algebraic cycles giving rise to thes
e Tate classes. It turns out that a natural algebraic cycle generates some
\, but not all\, of the Tate classes: to be precise\, it generates exactly
the Tate classes which are associated to generic members of the endoscopi
c L-packets on GSp4. In the non-generic case\, one can at least show that
all the Tate classes arise from Hodge cycles. I'll explain these results a
nd sketch their proofs\, which rely on the theta correspondence.\n
LOCATION:https://researchseminars.org/talk/UAANTS/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cheng Chen (University of Minnesota)
DTSTART;VALUE=DATE-TIME:20230214T210000Z
DTEND;VALUE=DATE-TIME:20230214T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/58
DESCRIPTION:Title:
On the local Gross-Prasad conjecture over archimedean local fields\nby
Cheng Chen (University of Minnesota) as part of University of Arizona Alg
ebra and Number Theory Seminar\n\n\nAbstract\nThe local Gross-Prasad conje
cture was introduced by Gross and Prasad in the 1990s. The conjecture for
tempered parameters over non-archimedean local fields was proved by Waldsp
urger using the trace formula and twisted formula\, and the conjecture for
generic parameters over non-archimedean local fields was later proved by
Mœglin and Waldspurger. I will present my proof for the conjecture for ge
neric parameters over archimedean local fields\, together with a multiplic
ity formula for future applications\, and part of the work (tempered cases
) is joint with Z. Luo.\n
LOCATION:https://researchseminars.org/talk/UAANTS/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Danielle Wang (MIT)
DTSTART;VALUE=DATE-TIME:20230328T210000Z
DTEND;VALUE=DATE-TIME:20230328T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/59
DESCRIPTION:Title:
Global twisted GGP conjecture for unramified quadratic extensions\nby
Danielle Wang (MIT) as part of University of Arizona Algebra and Number Th
eory Seminar\n\nLecture held in ENR2 S395.\n\nAbstract\nThe twisted Gan--G
ross--Prasad conjectures consider the restriction of representations from
GL_n to a unitary group over a quadratic extension E/F. In this talk\, I w
ill explain the adaptation of the relative trace formula comparison used i
n previous work on the global GGP conjecture for unitary groups\, to this
twisted version. In particular\, I will discuss the fundamental lemma that
arises\, which can be used to obtain the global twisted GGP conjecture (u
nder some local assumptions) in the case that everything is unramified\, a
nd how it can be reduced to the Jacquet--Rallis fundamental lemma.\n
LOCATION:https://researchseminars.org/talk/UAANTS/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steven Groen (University of Warwick)
DTSTART;VALUE=DATE-TIME:20230117T210000Z
DTEND;VALUE=DATE-TIME:20230117T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/60
DESCRIPTION:Title:
p-Torsion of Abelian varieties in characteristic p\nby Steven Groen (U
niversity of Warwick) as part of University of Arizona Algebra and Number
Theory Seminar\n\n\nAbstract\nLet A be an Abelian variety of dimension g o
ver an algebraically closed field k. We are interested in the group scheme
A[p]\, consisting of the elements of A whose order divides p. If the char
acteristic of k is not p\, then there is only one possibility for A[p]: as
a group it consists of 2g copies of Z/pZ. On the other hand\, if k has ch
aracteristic p\, then there are several distinct possibilities for A[p]\,
called Ekedahl-Oort strata. In particular\, the group will consist of at m
ost g copies of Z/pZ. An example of an Ekedahl-Oort stratification is the
distinction between ordinary and supersingular elliptic curves. If the dim
ension g is higher\, it is natural to ask which Ekedahl-Oort strata arise
from the Jacobian of a curve. In this talk\, we treat both previously know
n results and new results in this area. In many cases\, we add the restric
tion that the curves in question are Artin-Schreier covers.\n
LOCATION:https://researchseminars.org/talk/UAANTS/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yu Yang (RIMS\, Kyoto University)
DTSTART;VALUE=DATE-TIME:20230322T010000Z
DTEND;VALUE=DATE-TIME:20230322T020000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/61
DESCRIPTION:Title:
Moduli spaces of fundamental groups in positive characteristic\nby Yu
Yang (RIMS\, Kyoto University) as part of University of Arizona Algebra an
d Number Theory Seminar\n\n\nAbstract\nIn the 1980s\, A. Grothendieck sugg
ested a theory of arithmetic geometry called "anabelian geometry". This th
eory focuses on the following fundamental question: How much information a
bout algebraic varieties can be carried by their algebraic fundamental gro
ups? The conjectures based on this question are called Grothendieck's anab
elian conjectures which have been studied deeply when the base fields are
arithmetic (e.g. number fields\, p-adic fields\, finite fields\, etc.) sin
ce the 1990s\, and the non-trivial Galois representations play vital roles
. \n\nOn the other hand\, in 1996\, A. Tamagawa discovered surprisingly
that anabelian phenomena also exist for curves over algebraically closed f
ields of characteristic p>0 (i.e.\, no Galois actions). In this talk\, I w
ill explain these kinds of anabelian phenomena from the point of view of "
moduli spaces of fundamental groups" introduced by the speaker\, which giv
es a general framework for describing the anabelian phenomena for curves o
ver algebraically closed fields of characteristic p.\n
LOCATION:https://researchseminars.org/talk/UAANTS/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yasuhiro Wakabayashi (Osaka Univ.)
DTSTART;VALUE=DATE-TIME:20230315T010000Z
DTEND;VALUE=DATE-TIME:20230315T020000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/62
DESCRIPTION:Title:
Dormant opers and canonical diagonal liftings\nby Yasuhiro Wakabayashi
(Osaka Univ.) as part of University of Arizona Algebra and Number Theory
Seminar\n\n\nAbstract\nIn this talk\, we will discuss dormant opers\, whic
h are certain flat bundles on an algebraic curve in positive characteristi
c related to linear differential equations having a full set of solutions.
The moduli theory of such objects (in special cases) has been studied in
the context of p-adic Teichmüller theory\, and has many different aspects
\, including the connections with the intersection theory of Quot schemes
and the combinatorics of colored graphs\, as well as rational polytopes. O
ne goal of my research is to solve the counting problem of dormant opers w
hile deepening our understanding of these connections. As an approach to t
hat problem in the case of prime-power characteristic\, I have recently be
en thinking about a kind of arithmetic lifting of dormant opers\, which I
call “canonical diagonal lifting”. I would like to talk about that top
ic\, starting with some basics on flat bundles.\n
LOCATION:https://researchseminars.org/talk/UAANTS/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Radu Laza (Stony Brook)
DTSTART;VALUE=DATE-TIME:20230502T210000Z
DTEND;VALUE=DATE-TIME:20230502T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/64
DESCRIPTION:Title:
Deformations of mildly singular Calabi-Yau varieties\nby Radu Laza (St
ony Brook) as part of University of Arizona Algebra and Number Theory Semi
nar\n\nLecture held in ENR2 S395.\n\nAbstract\nThe well-known Bogomolov-Ti
an-Todorov theorem says that the deformations of Calabi-Yau manifolds are
unobstructed. The unobstructedness of deformations continues to hold Calab
i-Yau varieties with ordinary nodal singularities (Kawamata\, Ran\, Tian)\
, but surprisingly the smoothability of such varieties is subject to topol
ogical constrains. These obstructions to the existence of smoothings are l
inear in dimension 3 (Friedman)\, and non-linear in higher dimensions (Rol
lenske-Thomas).\n\nIn this talk\, I will give vast generalizations to both
the unobstructedness of deformations for mildly singular Calabi-Yau varie
ties\, and to the constraints on the existence of smoothings for certain c
lasses of singular Calabi-Yau varieties. Additionally\, I will establish t
he proper context for these results: the Hodge theory of degenerations wit
h prescribed singularities (specifically higher rational/higher Du Bois an
d liminal singularities).\n\nThis is joint work with Robert Friedman.\n
LOCATION:https://researchseminars.org/talk/UAANTS/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arvind Suresh (U of Arizona)
DTSTART;VALUE=DATE-TIME:20230418T210000Z
DTEND;VALUE=DATE-TIME:20230418T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/65
DESCRIPTION:Title:
Realizing Galois representations in abelian varieties by specialization\nby Arvind Suresh (U of Arizona) as part of University of Arizona Algebr
a and Number Theory Seminar\n\nLecture held in ENR2 S395.\n\nAbstract\nWe
present a strategy for constructing abelian varieties J/K which realize a
given rational Galois representation $\\rho : G_K \\to GL_n(Q)$\, i.e. suc
h that $\\rho$ is a subrep of $J(\\Kbar)\\otimes \\Q$. When $\\rho$ is the
trivial rep.\, then $J/K$ realizes $\\rho$ if and only if $J(K)$ is of ra
nk at least $n$\, and such families are usually constructed by the special
ization method pioneered by Neron. Our strategy consists in taking an alre
ady existing construction of abelian varieties with large rank and\, provi
ded there is enough symmetry\, twisting the construction to obtain non-tri
vial Galois actions on the points. After twisting\, we use a simple genera
lization of the classical Neron specialization theorem (from trivial reps.
to non-trivial reps.) We apply this procedure to a construction of Mestre
and Shioda to prove the following: Given a representation $\\rho: G_K \\t
o GL_n(\\Q)$\, there exist infinitely many absolutely simple absolutely ab
elian varieties $J/K$ (which are in fact Jacobians of hyperelliptic curves
) such that $\\rho$ is a subrep. of the $G_K$ rep on $J(\\Kbar) \\otimes_{
\\Z} \\Q$.\n
LOCATION:https://researchseminars.org/talk/UAANTS/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Wear (University of Utah)
DTSTART;VALUE=DATE-TIME:20230411T210000Z
DTEND;VALUE=DATE-TIME:20230411T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/67
DESCRIPTION:Title:
The conjugate uniformization via 1-motives\nby Peter Wear (University
of Utah) as part of University of Arizona Algebra and Number Theory Semina
r\n\nLecture held in ENR2 S395.\n\nAbstract\nGiven an abelian variety $A$
over a finite extension $K$ of $\\mathbb{Q}_p$\, Fontaine constructed an i
ntegration map from the Tate module of A to its Lie algebra. This map give
s the splitting of the Hodge-Tate short exact sequence. Recent work of Iov
ita-Morrow-Zaharescu extends this integration map to the $\\overline{K}$ p
oints of the perfectoid universal cover of $A$. They used this result to g
ive a uniformization of the $\\mathcal{O}_{\\overline K}$ points of the un
derlying $p$-divisible group. In this talk\, we explain joint work with Se
an Howe and Jackson Morrow in which we give a different perspective on thi
s uniformization using 1-motives. We will first give some intuition from t
he complex uniformization of semi-abelian varieties and some background an
d motivation on $p$-divisible groups. Then we will explain how to construc
t the $p$-divisible group of a 1-motive and how this gives the desired uni
formization. Finally\, we will point out some interesting geometric featur
es of this map: it embeds the rigid analytic points of a $p$-divisible gro
up into an etale cover of a negative Banach-Colmez space.\n
LOCATION:https://researchseminars.org/talk/UAANTS/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liyang Yang (Princeton)
DTSTART;VALUE=DATE-TIME:20230829T210000Z
DTEND;VALUE=DATE-TIME:20230829T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/68
DESCRIPTION:Title:
Bessel Periods for $U(3)\\times U(2)$: Nonvanishing and Equidistribution\nby Liyang Yang (Princeton) as part of University of Arizona Algebra an
d Number Theory Seminar\n\nLecture held in ENR2 S395.\n\nAbstract\nWe will
introduce a simple relative trace formula to compute the second moment of
Bessel periods associated to $U(3)\\times U(2)$. Moreover\, we explore it
s arithmetic implications\, addressing the quantitative nonvanishing probl
em\, and the distribution in the weighted vertical Sato-Tate context. This
is joint work with Dinakar Ramakrishnan and Philippe Michel.\n
LOCATION:https://researchseminars.org/talk/UAANTS/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robin Zhang (MIT)
DTSTART;VALUE=DATE-TIME:20231121T210000Z
DTEND;VALUE=DATE-TIME:20231121T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/69
DESCRIPTION:Title:
Harris–Venkatesh plus Stark\nby Robin Zhang (MIT) as part of Univers
ity of Arizona Algebra and Number Theory Seminar\n\nLecture held in ENR2 S
395.\n\nAbstract\nThe class number formula describes the behavior of the D
edekind zeta function at $s = 0$ and $s = 1$. The Stark and Gross conjectu
res extend the class number formula\, describing the behavior of Artin $L$
-functions and $p$-adic $L$-functions at $s = 0$ and $s = 1$ in terms of u
nits. The Harris–Venkatesh conjecture describes the residue of Stark uni
ts modulo $p$\, giving a modular analogue to the Stark and Gross conjectur
es while also serving as the first verifiable part of the broader conjectu
res of Venkatesh\, Prasanna\, and Galatius. In this talk\, I will draw an
introductory picture\, formulate a unified conjecture combining Harris–V
enkatesh and Stark for weight one modular forms\, and describe the proof o
f this in the imaginary dihedral case.\n
LOCATION:https://researchseminars.org/talk/UAANTS/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Finn McGlade (UCSD)
DTSTART;VALUE=DATE-TIME:20231114T210000Z
DTEND;VALUE=DATE-TIME:20231114T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/70
DESCRIPTION:Title:
A Level 1 Maass Spezialschar for Modular Forms on $\\mathrm{SO}_8$\nby
Finn McGlade (UCSD) as part of University of Arizona Algebra and Number T
heory Seminar\n\nLecture held in ENR2 S395.\n\nAbstract\nThe classical Spe
zialschar is the subspace of the space of holomorphic modular forms on $\\
mathrm{Sp}_4(\\mathbb{Z})$ whose Fourier coefficients satisfy a particular
system of linear equations. An equivalent characterization of the Spezial
schar can be obtained by combining work of Maass\, Andrianov\, and Zagier\
, whose work identifies the Spezialschar in terms of a theta-lift from $\\
widetilde{\\mathrm{SL}_2}$. Inspired by work of Gan-Gross-Savin\, Weissman
and Pollack have developed a theory of modular forms on the split adjoint
group of type D_4. In this setting we describe an analogue of the classic
al Spezialschar\, in which Fourier coefficients are used to characterize t
hose modular forms which arise as theta lifts from holomorphic forms on $\
\mathrm{Sp}_4(\\mathbb{Z})$.\n
LOCATION:https://researchseminars.org/talk/UAANTS/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Serin Hong (University of Arizona)
DTSTART;VALUE=DATE-TIME:20230905T210000Z
DTEND;VALUE=DATE-TIME:20230905T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/71
DESCRIPTION:Title:
Newton stratification on the $B_{dR}^+$-Grassmannian\nby Serin Hong (U
niversity of Arizona) as part of University of Arizona Algebra and Number
Theory Seminar\n\nLecture held in ENR2 S395.\n\nAbstract\nThe $B_{dR}^+$-G
rassmannian is a p-adic (perfectoid) analogue of the classical affine Gras
smannian. It plays an important role in the geometrization of the local La
nglands program and the study of Shimura varieties. In this talk\, we disc
uss its geometry in terms of a natural stratification called the Newton st
ratification\, with a particular focus on the case where the underlying gr
oup is GLn.\n
LOCATION:https://researchseminars.org/talk/UAANTS/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Junehyuk Jung (Brown)
DTSTART;VALUE=DATE-TIME:20231107T210000Z
DTEND;VALUE=DATE-TIME:20231107T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/72
DESCRIPTION:Title:
Zelditch’s trace formula and effective Bowen’s theorem\nby Junehyu
k Jung (Brown) as part of University of Arizona Algebra and Number Theory
Seminar\n\nLecture held in ENR2 S395.\n\nAbstract\nIn 1989\, Zelditch cons
idered the trace of an invariant operator composed with a pseudo-different
ial operator. The resulting trace formula turned out to be extremely usefu
l in studying the distribution of closed geodesics on hyperbolic surfaces.
I will demonstrate the simplest case of the proof\, and discuss how thing
s can be generalized to higher dimensional hyperbolic manifolds. This is a
joint work with Insung Park.\n
LOCATION:https://researchseminars.org/talk/UAANTS/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART;VALUE=DATE-TIME:20230912T210000Z
DTEND;VALUE=DATE-TIME:20230912T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/73
DESCRIPTION:by TBA as part of University of Arizona Algebra and Number The
ory Seminar\n\nLecture held in ENR2 S395.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/UAANTS/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART;VALUE=DATE-TIME:20230919T210000Z
DTEND;VALUE=DATE-TIME:20230919T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/74
DESCRIPTION:by TBA as part of University of Arizona Algebra and Number The
ory Seminar\n\nLecture held in ENR2 S395.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/UAANTS/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kirti Joshi (University of Arizona)
DTSTART;VALUE=DATE-TIME:20231003T210000Z
DTEND;VALUE=DATE-TIME:20231003T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/76
DESCRIPTION:Title:
Construction of Arithmetic Teichmuller Spaces and applications to Mochizuk
i's Theory\nby Kirti Joshi (University of Arizona) as part of Universi
ty of Arizona Algebra and Number Theory Seminar\n\nLecture held in ENR2 S3
95.\n\nAbstract\nThis talk is intended as an semi-informal introduction to
my recent work on Arithmetic Teichmuller Spaces and its relationship to M
ochizuki's Inter Universal Teichmuller Theory which is at the center of hi
s work on the abc-conjecture. My work establishes a p-adic analog of class
ical Teichmuller Theory of Riemann surfaces. Notably in this theory\, the
etale fundamental group remains fixed (just as in the classical theory of
Riemann surfaces) while the holomorphic structure varies. Given the nature
of this work and its close relationship to Mochizuki's work\, I have inte
ntionally divided this talk into two talks on (10/3/2023) and (10/10/2023
) both to provide a gentle introduction to my ideas and also to allow for
plenty of questions. [There will be plenty of technical material presented
as well but the emphasis will be on providing an introduction to my ideas
.] I will also include a discussion of Mochizuki-Scholze-Stix issues in t
he context of Mochizuki's Theory (and answer any questions in this context
).\n
LOCATION:https://researchseminars.org/talk/UAANTS/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kirti Joshi (University of Arizona)
DTSTART;VALUE=DATE-TIME:20231010T210000Z
DTEND;VALUE=DATE-TIME:20231010T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/77
DESCRIPTION:Title:
Construction of Arithmetic Teichmuller Spaces and applications to Mochizuk
i's Theory\nby Kirti Joshi (University of Arizona) as part of Universi
ty of Arizona Algebra and Number Theory Seminar\n\nLecture held in ENR2 S3
95.\n\nAbstract\nThis talk is intended as an semi-informal introduction to
my recent work on Arithmetic Teichmuller Spaces and its relationship to M
ochizuki's Inter Universal Teichmuller Theory which is at the center of hi
s work on the abc-conjecture. My work establishes a p-adic analog of class
ical Teichmuller Theory of Riemann surfaces. Notably in this theory\, the
etale fundamental group remains fixed (just as in the classical theory of
Riemann surfaces) while the holomorphic structure varies. Given the nature
of this work and its close relationship to Mochizuki's work\, I have inte
ntionally divided this talk into two talks on (10/3/2023) and (10/10/2023
) both to provide a gentle introduction to my ideas and also to allow for
plenty of questions. [There will be plenty of technical material presented
as well but the emphasis will be on providing an introduction to my ideas
.] I will also include a discussion of Mochizuki-Scholze-Stix issues in t
he context of Mochizuki's Theory (and answer any questions in this context
).\n
LOCATION:https://researchseminars.org/talk/UAANTS/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yu Fu (California Institute of Technology)
DTSTART;VALUE=DATE-TIME:20231031T210000Z
DTEND;VALUE=DATE-TIME:20231031T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/78
DESCRIPTION:Title:
How do generic properties spread?\nby Yu Fu (California Institute of T
echnology) as part of University of Arizona Algebra and Number Theory Semi
nar\n\nLecture held in ENR2 S395.\n\nAbstract\nGiven a family of algebraic
varieties\, a natural question to ask is what type of properties of the g
eneric fiber\, and how those properties extend to other fibers. Let's expl
ore this topic from an arithmetic point of view by looking at the scenari
o: Suppose we have a 1-dimensional family of pairs of elliptic curves ove
r a number field $K$\, with the generic fiber of this family being a pai
r of non-isogenous elliptic curves. Furthermore\, suppose the (projective)
height of the parametrizer is less than or equal to $B$. One may ask how
does the property of "being isogenous" extends to the special fibers. Can
we give a quantitative estimation for the number of specializations of hei
ght at most $B$\, such that the two elliptic curves at the specializations
are isogenous?\n
LOCATION:https://researchseminars.org/talk/UAANTS/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ellen Eischen (University of Oregon)
DTSTART;VALUE=DATE-TIME:20240312T210000Z
DTEND;VALUE=DATE-TIME:20240312T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/79
DESCRIPTION:Title:
Algebraic and p-adic aspects of L-functions\, with a view toward Spin L-fu
nctions for GSp_6\nby Ellen Eischen (University of Oregon) as part of
University of Arizona Algebra and Number Theory Seminar\n\nLecture held in
ENR2 S395.\n\nAbstract\nI will discuss recent developments and ongoing wo
rk for algebraic and p-adic aspects of L-functions. Interest in p-adic pro
perties of values of L-functions originated with Kummer’s study of congr
uences between values of the Riemann zeta function at negative odd integer
s\, as part of his attempt to understand class numbers of cyclotomic exten
sions. After presenting an approach to studying analogous congruences for
more general classes of L-functions\, I will conclude by introducing ongoi
ng joint work of G. Rosso\, S. Shah\, and myself (concerning Spin L-functi
ons for GSp 6). I will explain how this work fits into the context of earl
ier developments\, while also indicating where new technical challenges ar
ise. All who are curious about this topic are welcome at this talk\, even
without prior experience with p-adic L-functions or Spin L-functions.\n
LOCATION:https://researchseminars.org/talk/UAANTS/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bhawesh Mishra (University of Memphis)
DTSTART;VALUE=DATE-TIME:20231205T210000Z
DTEND;VALUE=DATE-TIME:20231205T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/81
DESCRIPTION:Title:
A Generalization of the Grunwald-Wang Theorem for $n^{th}$ Powers\nby
Bhawesh Mishra (University of Memphis) as part of University of Arizona Al
gebra and Number Theory Seminar\n\n\nAbstract\nThe Grunwald-Wang theorem f
or $n^{th}$ powers states that a rational number $a$ is an $n^{th}$ power
in $\\mathbb{Q}_{p}$ for almost every prime $p$ if and only if either $a$
is a perfect $n^{th}$ power in rationals or $8 \\mid n$ and $a = 2^{\\frac
{n}{2}} \\cdot b^{n}$ for some rational $b$. We will discuss an appropriat
e generalization of this theorem from a single rational number to a subset
$A$ of rational numbers. Let $n$ be an odd number and $q$ be the smallest
prime dividing $n$. A finite subset $A$ of rationals with cardinality $\\
leq q$ contains a $n^{th}$ power in $\\mathbb{Q}_{p}$ for almost every pri
me $p$ if and only if $A$ contains a perfect $n^{th}$ power. For even $n$\
, the result is analogous - up to a short list of exceptions\, as evident
in the Grunwald-Wang theorem. \n\nIf time permits\, we will also show that
this generalization is optimal\, i.e.\, for every $n \\geq 2$\, there are
infinitely many subsets $A$ of rationals of cardinality $q+1$ that contai
n a $n^{th}$ power in $\\mathbb{Q}_{p}$ for almost every prime $p$ but nei
ther contain a perfect $n^{th}$ power nor fall under the finite list of ex
ceptions.\n
LOCATION:https://researchseminars.org/talk/UAANTS/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mandi Schaeffer Fry (University of Denver)
DTSTART;VALUE=DATE-TIME:20240402T210000Z
DTEND;VALUE=DATE-TIME:20240402T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/82
DESCRIPTION:by Mandi Schaeffer Fry (University of Denver) as part of Unive
rsity of Arizona Algebra and Number Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/UAANTS/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Miller (UC Santa Cruz)
DTSTART;VALUE=DATE-TIME:20240116T210000Z
DTEND;VALUE=DATE-TIME:20240116T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/83
DESCRIPTION:by Sam Miller (UC Santa Cruz) as part of University of Arizona
Algebra and Number Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/UAANTS/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bryden Cais (University of Arizona)
DTSTART;VALUE=DATE-TIME:20240123T210000Z
DTEND;VALUE=DATE-TIME:20240123T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/84
DESCRIPTION:Title:
Iwasawa theory for class group schemes in characteristic p\nby Bryden
Cais (University of Arizona) as part of University of Arizona Algebra and
Number Theory Seminar\n\nLecture held in ENR2 S395.\n\nAbstract\nIn a land
mark 1959 paper\, Iwasawa studied the growth of class groups in Z_p-towers
of number fields\, establishing a remarkable formula for the exact power
of p dividing the order of the class group of the n-th layer of the tower.
Iwasawa's work was inspired by a profound analogy between number fields a
nd function fields over finite fields. In this setting\, the direct analog
ue of Iwasawa theory is the study of class groups in Z_p-towers of global
function fields over finite fields k of characteristic p\, and an analogou
s formula for the order of p dividing the class group was established by M
azur and Wiles in 1983. An extraordinary feature of this function field se
tting is that the class group can be realized as the k-rational points of
an algebraic variety---the Jacobian. We will briefly survey some of this
history\, and introduce a novel analogue of Iwasawa theory for function fi
elds by studying not just the k-points of these Jacobians\, but their full
p-torsion group schemes\, which are much richer\, geometric objects havin
g no analogue in the number field setting.\n
LOCATION:https://researchseminars.org/talk/UAANTS/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Doug Ulmer (University of Arizona)
DTSTART;VALUE=DATE-TIME:20240130T210000Z
DTEND;VALUE=DATE-TIME:20240130T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/85
DESCRIPTION:Title:
p-torsion for p-covers\nby Doug Ulmer (University of Arizona) as part
of University of Arizona Algebra and Number Theory Seminar\n\nLecture held
in ENR2 S395.\n\nAbstract\nWe consider Z/pZ covers of curves in character
istic p and the p-torsion group schemes of their Jacobians. Such covers a
re the building blocks of the towers Bryden Cais spoke about last week. O
ur aim is to understand the full Dieudonne structure\, not just the kernel
of Frobenius.\n\nWe will give more details on groups schemes killed by p
and their Dieudonne modules\, introduce the Ekedahl-Oort stratification of
Ag\, the moduli space of abelian varieties\, and state basic questions on
how it interacts with the locus of Jacobians in Ag. (This gives an intro
duction to the AWS2024 courses of Karemaker and Pries.) We end by stating
a structural result on Dieudonne modules of p-covers and some of the rest
rictions it places on group schemes.\n
LOCATION:https://researchseminars.org/talk/UAANTS/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Dillery (University of Maryland)
DTSTART;VALUE=DATE-TIME:20240416T210000Z
DTEND;VALUE=DATE-TIME:20240416T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/86
DESCRIPTION:Title:
Comparing local Langlands correspondences\nby Peter Dillery (Universit
y of Maryland) as part of University of Arizona Algebra and Number Theory
Seminar\n\nLecture held in ENR2 S395.\n\nAbstract\nBroadly speaking\, for
G a connected reductive group over a local field F\, the Langlands program
is the endeavor of relating Galois representations (more precisely\, "L-p
arameters"---certain homomorphisms from the Weil-Deligne group of F to the
dual group of G) to admissible smooth representations of G(F). There is c
onjectured to be a finite-to-one map from irreducible smooth representatio
ns of G(F) to L-parameters\, and there are many different approaches to pa
rametrizing the fibers of such a map. \n\nThe goal of this talk is to expl
ain some of these approaches\; a special focus will be placed on the so-c
alled "isocrystal" and "rigid" local Langlands correspondences. The former
is best suited for building on the recent breakthroughs of Fargues-Scholz
e\, while the latter is the broadest and is well-suited to endoscopy (a ve
rsion of functoriality). We will discuss a proof of the equivalence of the
se two approaches\, initiated by Kaletha for p-adic fields and extended to
arbitrary nonarchimedean local fields in my recent work.\n
LOCATION:https://researchseminars.org/talk/UAANTS/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nathan Kaplan (University of California\, Irvine)
DTSTART;VALUE=DATE-TIME:20240328T170000Z
DTEND;VALUE=DATE-TIME:20240328T180000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/87
DESCRIPTION:Title:
Codes from varieties over finite fields\nby Nathan Kaplan (University
of California\, Irvine) as part of University of Arizona Algebra and Numbe
r Theory Seminar\n\nLecture held in ENR2 N595.\n\nAbstract\nThere are $q^{
20}$ homogeneous cubic polynomials in four variables with coefficients in
the finite field $F_q$. How many of them define a cubic surface with $q^2+
7q+1$ $F_q$-rational points? What about other numbers of rational points?
How many of the $q^{20}$ pairs of homogeneous cubic polynomials in three v
ariables define cubic curves that intersect in 9 $F_q$-rational points? Th
e goal of this talk is to explain how ideas from the theory of error-corre
cting codes can be used to study families of varieties over a fixed finite
field. We will not assume any previous familiarity with coding theory. We
will start from the basics and emphasize examples.\n
LOCATION:https://researchseminars.org/talk/UAANTS/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Garcia (University College London)
DTSTART;VALUE=DATE-TIME:20240409T210000Z
DTEND;VALUE=DATE-TIME:20240409T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/88
DESCRIPTION:Title:
Elliptic units for complex cubic fields\nby Luis Garcia (University Co
llege London) as part of University of Arizona Algebra and Number Theory S
eminar\n\n\nAbstract\nThe elliptic Gamma function — a generalization of
the q-Gamma function\, which is itself the q-analog of the ordinary Gamma
function — is a meromorphic special function in several variables that m
athematical physicists have shown to satisfy modular functional equations
under SL(3\,Z). In this talk I will present evidence (numerical and theore
tical) that products of values of this function are often algebraic number
s that satisfy explicit reciprocity laws and are related to derivatives of
Hecke L-functions of cubic fields at s=0. We will discuss the relation to
Stark's conjectures and will see that this function conjecturally allows
to extend the theory of complex multiplication to complex cubic fields as
envisioned by Hilbert's 12th problem. This is joint work with Nicolas Berg
eron and Pierre Charollois.\n
LOCATION:https://researchseminars.org/talk/UAANTS/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linli Shi (University of Connecticut)
DTSTART;VALUE=DATE-TIME:20241015T210000Z
DTEND;VALUE=DATE-TIME:20241015T220000Z
DTSTAMP;VALUE=DATE-TIME:20240910T013814Z
UID:UAANTS/89
DESCRIPTION:by Linli Shi (University of Connecticut) as part of University
of Arizona Algebra and Number Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/UAANTS/89/
END:VEVENT
END:VCALENDAR