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BEGIN:VEVENT
SUMMARY:Andrew Obus
DTSTART;VALUE=DATE-TIME:20211102T130000Z
DTEND;VALUE=DATE-TIME:20211102T143000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070729Z
UID:viasmag/1
DESCRIPTION:Title:
Mac Lane valuations and an application to resolution of quotient singulari
ties\nby Andrew Obus as part of VIASM Arithmetic Geometry Online Semin
ar\n\n\nAbstract\nMac Lane's technique of "inductive valuations" is over 8
0 years old\, but has only recently been used to attack problems about ari
thmetic surfaces. We will give an explicit\, hands-on introduction to the
theory\, requiring little background beyond the definition of a non-archim
edean valuation. \n\nWe will then outline how this theory is helpful for r
esolving "weak wild" quotient singularities of arithmetic surfaces\, a cla
ss of singularity studied by Lorenzini that shows up naturally when comput
ing models of curves with potentially good reduction.\n\nSlides.\n
LOCATION:https://researchseminars.org/talk/viasmag/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amadou Bah
DTSTART;VALUE=DATE-TIME:20211216T130000Z
DTEND;VALUE=DATE-TIME:20211216T140000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070729Z
UID:viasmag/2
DESCRIPTION:Title:
Variation of the Swan conductor of an $\\mathbb{F}_{\\ell}$-sheaf on a rig
id annulus\nby Amadou Bah as part of VIASM Arithmetic Geometry Online
Seminar\n\n\nAbstract\nLet $C$ be a closed annulus of radii $r < r' \\in \
\mathbb{Q}_{\\geq 0}$ over a complete discrete valuation field with algebr
aically closed residue field of characteristic $p>0$. To an étale sheaf o
f $\\mathbb{F}_{\\ell}$-modules $\\mathcal{F}$ on $C$\, ramified at most a
t a finite set of rigid points of $C$\, one associates an Abbes-Saito Swan
conductor function ${\\rm sw}_{\\mathcal{F}}: [r\, r']\\cap \\mathbb{Q}_{
\\geq 0} \\to \\mathbb{Q}$ which\, for a radius $t$\, measures the ramific
ation of $\\mathcal{F}_{\\lvert C^{[t]}}$ — the restriction of $\\mathca
l{F}$ to the sub-annulus $C^{[t]}$ of $C$ of radius $t$ with $0$-thickness
— along the special fiber of the normalized integral model of $C^{[t]}$
. This function has the following remarkable properties: it is continuous\
, convex and piecewise linear outside the radii of the ramification points
of $\\mathcal{F}$\, with finitely many integer slopes whose variation bet
ween radii $t$ and $t'$ can be expressed as the difference of the orders o
f the characteristic cycles of $\\mathcal{F}$ at $t$ and $t'$. In this tal
k\, I will explain the construction of ${\\rm sw}_{\\mathcal{F}}$ and the
key nearby cycles formula in establishing the aforementioned properties of
${\\rm sw}_{\\mathcal{F}}$.\n
LOCATION:https://researchseminars.org/talk/viasmag/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vaidehee Thatte
DTSTART;VALUE=DATE-TIME:20211124T100000Z
DTEND;VALUE=DATE-TIME:20211124T110000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070729Z
UID:viasmag/3
DESCRIPTION:Title:
Arbitrary Valuation Rings and Wild Ramification\nby Vaidehee Thatte as
part of VIASM Arithmetic Geometry Online Seminar\n\nLecture held in C101\
, VIASM.\n\nAbstract\nClassical ramification theory deals with complete di
screte valuation fields $k((X))$ with perfect residue fields $k$. Invarian
ts such as the Swan conductor capture important information about extensio
ns of these fields. Many fascinating complications arise when we allow non
-discrete valuations and imperfect residue fields $k$. Particularly in pos
itive residue characteristic\, we encounter the mysterious phenomenon of t
he defect (or ramification deficiency). The occurrence of a non-trivial de
fect is one of the main obstacles to long-standing problems\, such as obta
ining resolution of singularities in positive characteristic.\n\n\nDegree
$p$ extensions of valuation fields are building blocks of the general case
. In this talk\, we will present a generalization of ramification invarian
ts for such extensions and discuss how this leads to a better understandin
g of the defect. If time permits\, we will briefly discuss their connectio
n with some recent work (joint with K. Kato) on upper ramification groups.
\n
LOCATION:https://researchseminars.org/talk/viasmag/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liang Xiao (Peking International Center for Mathematical Research)
DTSTART;VALUE=DATE-TIME:20220104T070000Z
DTEND;VALUE=DATE-TIME:20220104T083000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070729Z
UID:viasmag/4
DESCRIPTION:Title:
Beilinson--Bloch--Kato conjecture for some Rankin-Selberg motives.\nby
Liang Xiao (Peking International Center for Mathematical Research) as par
t of VIASM Arithmetic Geometry Online Seminar\n\nLecture held in C101\, VI
ASM.\n\nAbstract\nThe Birch and Swinnerton-Dyer conjecture is known in the
case of rank $0$ and $1$ thanks to the foundational work of Kolyvagin and
Gross-Zagier. In this talk\, I will report on a joint work with Yifeng Li
u\, Yichao Tian\, Wei Zhang\, and Xinwen Zhu. We study the analogue and ge
neralizations of Kolyvagin's result to the unitary Gan-Gross-Prasad paradi
gm. More precisely\, our ultimate goal is to show that\, under some techni
cal conditions\, if the central value of the Rankin-Selberg $L$-function o
f an automorphic representation of $U(n) \\ast U(n+1)$ is nonzero\, then t
he associated Selmer group is trivial\; Analogously\, if the Selmer class
of certain cycle for the $U(n) \\ast U(n+1)$-Shimura variety is nontrivial
\, then the dimension of the corresponding Selmer group is one. ($\\href{h
ttps://drive.google.com/file/d/1crDh5JwaFv8HW7wZ3NZtDugaFQHOZrTG/view?usp=
sharing}{{\\rm notes}}$).\n
LOCATION:https://researchseminars.org/talk/viasmag/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joe Kramer-Miller (Lehigh University)
DTSTART;VALUE=DATE-TIME:20220222T023000Z
DTEND;VALUE=DATE-TIME:20220222T040000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070729Z
UID:viasmag/5
DESCRIPTION:Title:
Ramification of geometric $p$-adic representations in positive characteris
tic\nby Joe Kramer-Miller (Lehigh University) as part of VIASM Arithme
tic Geometry Online Seminar\n\nLecture held in C101\, VIASM.\n\nAbstract\n
A classical theorem of Sen describes a close relationship between the rami
fication filtration and the $p$-adic Lie filtration for $p$-adic represent
ations in mixed characteristic. Unfortunately\, Sen's theorem fails misera
bly in positive characteristic. The extensions are just too wild! There is
some hope if we restrict to representations coming from geometry. Let $X$
be a smooth variety and let $D$ be a normal crossing divisor in $X$ and c
onsider a geometric $p$-adic lisse sheaf on $X \\setminus D$ (e.g. the $p$
-adic Tate module of a fibration of abelian varieties). We show that the A
bbes-Saito conductors along $D$ exhibit a remarkable regular growth with r
espect to the $p$-adic Lie filtration.\n
LOCATION:https://researchseminars.org/talk/viasmag/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeremy Booher (University of Canterbury)
DTSTART;VALUE=DATE-TIME:20220215T023000Z
DTEND;VALUE=DATE-TIME:20220215T040000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070729Z
UID:viasmag/6
DESCRIPTION:Title:
Iwasawa Theory for $p$-torsion Class Group Schemes in Characteristic $p$\nby Jeremy Booher (University of Canterbury) as part of VIASM Arithmeti
c Geometry Online Seminar\n\nLecture held in C101\, VIASM.\n\nAbstract\nA
$\\mathbb{Z}_p$ tower of curves in characteristic $p$ is a sequence $C_0\,
C_1\, C_2\, \\ldots$ of smooth projective curves over a perfect field of
characteristic $p$ such that $C_n$ is a branched cover of $C_{n-1}$ and $C
_n$ is a branched Galois $\\mathbb{Z}/(p^n)$-cover of $C_0$. The genus is
a well-understood invariant of algebraic curves\, and the genus of $C_n$
can be seen to depend on $n$ in a simple fashion. In characteristic $p$\,
there are additional curve invariants like the $a$-number which are poorl
y understood. They describe the group-scheme structure of the $p$-torsion
of the Jacobian. I will discuss work with Bryden Cais studying these invar
iants and suggesting that their growth is also "regular" in $\\mathbb{Z}_p
$ towers. This is a new kind of Iwasawa theory for function fields.\n
LOCATION:https://researchseminars.org/talk/viasmag/6/
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