BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Andrew Obus DTSTART:20211102T130000Z DTEND:20211102T143000Z DTSTAMP:20260422T225929Z 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:20211216T130000Z DTEND:20211216T140000Z DTSTAMP:20260422T225929Z 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:20211124T100000Z DTEND:20211124T110000Z DTSTAMP:20260422T225929Z 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:20220104T070000Z DTEND:20220104T083000Z DTSTAMP:20260422T225929Z 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:20220222T023000Z DTEND:20220222T040000Z DTSTAMP:20260422T225929Z 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:20220215T023000Z DTEND:20220215T040000Z DTSTAMP:20260422T225929Z 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/ END:VEVENT END:VCALENDAR