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BEGIN:VEVENT
SUMMARY:Andrew Obus
DTSTART:20211102T130000Z
DTEND:20211102T143000Z
DTSTAMP:20260404T094804Z
UID:viasmag/1
DESCRIPTION:Title: Mac Lane valuations and an application to resolution of quotient si
ngularities\nby Andrew Obus as part of VIASM Arithmetic Geometry Onlin
e Seminar\n\n\nAbstract\nMac Lane's technique of "inductive valuations" is
over 80 years old\, but has only recently been used to attack problems ab
out arithmetic surfaces. We will give an explicit\, hands-on introduction
to the theory\, requiring little background beyond the definition of a non
-archimedean valuation. \n\nWe will then outline how this theory is helpfu
l for resolving "weak wild" quotient singularities of arithmetic surfaces\
, a class of singularity studied by Lorenzini that shows up naturally when
computing models of curves with potentially good reduction.\n\nSlides.\n
LOCATION:https://stable.researchseminars.org/talk/viasmag/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amadou Bah
DTSTART:20211216T130000Z
DTEND:20211216T140000Z
DTSTAMP:20260404T094804Z
UID:viasmag/2
DESCRIPTION:Title: Variation of the Swan conductor of an $\\mathbb{F}_{\\ell}$-sheaf o
n a rigid 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
algebraically closed residue field of characteristic $p>0$. To an étale
sheaf of $\\mathbb{F}_{\\ell}$-modules $\\mathcal{F}$ on $C$\, ramified at
most at a finite set of rigid points of $C$\, one associates an Abbes-Sai
to Swan conductor function ${\\rm sw}_{\\mathcal{F}}: [r\, r']\\cap \\math
bb{Q}_{\\geq 0} \\to \\mathbb{Q}$ which\, for a radius $t$\, measures the
ramification of $\\mathcal{F}_{\\lvert C^{[t]}}$ — the restriction of $\
\mathcal{F}$ to the sub-annulus $C^{[t]}$ of $C$ of radius $t$ with $0$-th
ickness — along the special fiber of the normalized integral model of $C
^{[t]}$. This function has the following remarkable properties: it is cont
inuous\, convex and piecewise linear outside the radii of the ramification
points of $\\mathcal{F}$\, with finitely many integer slopes whose variat
ion between radii $t$ and $t'$ can be expressed as the difference of the o
rders of the characteristic cycles of $\\mathcal{F}$ at $t$ and $t'$. In t
his talk\, I will explain the construction of ${\\rm sw}_{\\mathcal{F}}$ a
nd the key nearby cycles formula in establishing the aforementioned proper
ties of ${\\rm sw}_{\\mathcal{F}}$.\n
LOCATION:https://stable.researchseminars.org/talk/viasmag/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vaidehee Thatte
DTSTART:20211124T100000Z
DTEND:20211124T110000Z
DTSTAMP:20260404T094804Z
UID:viasmag/3
DESCRIPTION:Title: Arbitrary Valuation Rings and Wild Ramification\nby Vaidehee Th
atte as part of VIASM Arithmetic Geometry Online Seminar\n\nLecture held i
n C101\, VIASM.\n\nAbstract\nClassical ramification theory deals with comp
lete discrete valuation fields $k((X))$ with perfect residue fields $k$. I
nvariants such as the Swan conductor capture important information about e
xtensions of these fields. Many fascinating complications arise when we al
low non-discrete valuations and imperfect residue fields $k$. Particularly
in positive residue characteristic\, we encounter the mysterious phenomen
on of the defect (or ramification deficiency). The occurrence of a non-tri
vial defect is one of the main obstacles to long-standing problems\, such
as obtaining resolution of singularities in positive characteristic.\n\n\n
Degree $p$ extensions of valuation fields are building blocks of the gener
al case. In this talk\, we will present a generalization of ramification i
nvariants for such extensions and discuss how this leads to a better under
standing of the defect. If time permits\, we will briefly discuss their co
nnection with some recent work (joint with K. Kato) on upper ramification
groups.\n
LOCATION:https://stable.researchseminars.org/talk/viasmag/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liang Xiao (Peking International Center for Mathematical Research)
DTSTART:20220104T070000Z
DTEND:20220104T083000Z
DTSTAMP:20260404T094804Z
UID:viasmag/4
DESCRIPTION:Title: Beilinson--Bloch--Kato conjecture for some Rankin-Selberg motives.<
/a>\nby Liang Xiao (Peking International Center for Mathematical Research)
as part of VIASM Arithmetic Geometry Online Seminar\n\nLecture held in C1
01\, VIASM.\n\nAbstract\nThe Birch and Swinnerton-Dyer conjecture is known
in the case of rank $0$ and $1$ thanks to the foundational work of Kolyva
gin and Gross-Zagier. In this talk\, I will report on a joint work with Yi
feng Liu\, Yichao Tian\, Wei Zhang\, and Xinwen Zhu. We study the analogue
and generalizations of Kolyvagin's result to the unitary Gan-Gross-Prasad
paradigm. More precisely\, our ultimate goal is to show that\, under some
technical conditions\, if the central value of the Rankin-Selberg $L$-fun
ction of an automorphic representation of $U(n) \\ast U(n+1)$ is nonzero\,
then the associated Selmer group is trivial\; Analogously\, if the Selmer
class of certain cycle for the $U(n) \\ast U(n+1)$-Shimura variety is non
trivial\, then the dimension of the corresponding Selmer group is one. ($\
\href{https://drive.google.com/file/d/1crDh5JwaFv8HW7wZ3NZtDugaFQHOZrTG/vi
ew?usp=sharing}{{\\rm notes}}$).\n
LOCATION:https://stable.researchseminars.org/talk/viasmag/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joe Kramer-Miller (Lehigh University)
DTSTART:20220222T023000Z
DTEND:20220222T040000Z
DTSTAMP:20260404T094804Z
UID:viasmag/5
DESCRIPTION:Title: Ramification of geometric $p$-adic representations in positive char
acteristic\nby Joe Kramer-Miller (Lehigh University) as part of VIASM
Arithmetic Geometry Online Seminar\n\nLecture held in C101\, VIASM.\n\nAbs
tract\nA classical theorem of Sen describes a close relationship between t
he ramification filtration and the $p$-adic Lie filtration for $p$-adic re
presentations in mixed characteristic. Unfortunately\, Sen's theorem fails
miserably in positive characteristic. The extensions are just too wild! T
here 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 consider 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 tha
t the Abbes-Saito conductors along $D$ exhibit a remarkable regular growth
with respect to the $p$-adic Lie filtration.\n
LOCATION:https://stable.researchseminars.org/talk/viasmag/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeremy Booher (University of Canterbury)
DTSTART:20220215T023000Z
DTEND:20220215T040000Z
DTSTAMP:20260404T094804Z
UID:viasmag/6
DESCRIPTION:Title: Iwasawa Theory for $p$-torsion Class Group Schemes in Characteristi
c $p$\nby Jeremy Booher (University of Canterbury) as part of VIASM Ar
ithmetic Geometry Online Seminar\n\nLecture held in C101\, VIASM.\n\nAbstr
act\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 fi
eld 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 g
enus 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 characteristi
c $p$\, there are additional curve invariants like the $a$-number which ar
e poorly understood. They describe the group-scheme structure of the $p$-t
orsion of the Jacobian. I will discuss work with Bryden Cais studying thes
e invariants and suggesting that their growth is also "regular" in $\\math
bb{Z}_p$ towers. This is a new kind of Iwasawa theory for function fields.
\n
LOCATION:https://stable.researchseminars.org/talk/viasmag/6/
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