VIASM Arithmetic Geometry Online Seminar

Meeting Coordinates for the upcoming talk:

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Meeting ID: 795 146 5011

Password: The order of GL_2(F_3).

• Tuesday February 22th, 2022, 9:30-11:00 ICT (GMT +7)
  Speaker: Joe Kramer-Miller, Lehigh University
  Title: Ramification of geometric p-adic representations in positive characteristic
  Abstract: A classical theorem of Sen describes a close relationship between the ramification filtration and the p-adic Lie filtration for p-adic representations in mixed characteristic. Unfortunately, Sen's theorem fails miserably 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 consider a geometric p-adic lisse sheaf on X \ D (e.g. the p-adic Tate module of a fibration of abelian varieties). We show that the Abbes-Saito conductors along D exhibit a remarkable regular growth with respect to the p-adic Lie filtration.

• Tuesday February 15th, 2022, 9:30-11:00 ICT (GMT +7)
  Speaker: Jeremy Booher, University of Canterbury
  Title: Iwasawa Theory for p-torsion Class Group Schemes in Characteristic p
  Abstract: A Zp tower of curves in characteristic p is a sequence C_0, C_1, C_2, ... 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 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 poorly understood. They describe the group-scheme structure of the p-torsion of the Jacobian. I will discuss work with Bryden Cais studying these invariants and suggesting that their growth is also "regular" in Zp towers. This is a new kind of Iwasawa theory for function fields. (notes).

• Tuesday January 4th, 2022, 14:00-15:00 ICT (GMT +7)
  Speaker: Liang Xiao, Peking International Center for Mathematical Research
  Title: Beilinson-Bloch-Kato conjecture for some Rankin-Selberg motives.
  Abstract: The 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 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-function of an automorphic representation of U(n)*U(n+1) is nonzero, then the associated Selmer group is trivial; Analogously, if the Selmer class of certain cycle for the U(n)*U(n+1)-Shimura variety is nontrivial, then the dimension of the corresponding Selmer group is one. (notes).

• Thursday December 16th, 2021, 20:00-21:00 ICT (GMT +7)
  Speaker: Amadou Bah, Columbia University
  Title: Variation of the Swan conductor of an $\mathbb{F}_{\ell}$-sheaf on a rigid annulus (pdf)
  Abstract: Let $C$ 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-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 ramification of $\mathcal{F}_{\lvert C^{[t]}}$ — the restriction of $\mathcal{F}$ to the sub-annulus $C^{[t]}$ of $C$ of radius $t$ with $0$-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 between radii $t$ and $t'$ can be expressed as the difference of the orders of the characteristic cycles of $\mathcal{F}$ at $t$ and $t'$. In this talk, I will explain the construction of ${\rm sw}_{\mathcal{F}}$ and the key nearby cycles formula in establishing the aforementioned properties of ${\rm sw}_{\mathcal{F}}$. (notes).

• Wednesday November 24th, 2021, 17:00-18:00 ICT (GMT +7)
  Speaker: Vaidehee Thatte, King's College London
  Title: Arbitrary Valuation Rings and Wild Ramification
  Abstract: Classical ramification theory deals with complete discrete valuation fields k((X)) with perfect residue fields k. Invariants such as the Swan conductor capture important information about extensions of these fields. Many fascinating complications arise when we allow non-discrete valuations and imperfect residue fields k. Particularly in positive residue characteristic, we encounter the mysterious phenomenon of the defect (or ramification deficiency). The occurrence of a non-trivial defect is one of the main obstacles to long-standing problems, such as obtaining resolution of singularities in positive characteristic.
  Degree p extensions of valuation fields are building blocks of the general case. In this talk, we will present a generalization of ramification invariants for such extensions and discuss how this leads to a better understanding of the defect. If time permits, we will briefly discuss their connection with some recent work (joint with K. Kato) on upper ramification groups.

• Tuesday November 2nd, 2021, 20:00-21:00 ICT (GMT +7)
  Speaker: Andrew Obus, Baruch College/CUNY Graduate Center
  Title: Mac Lane valuations and an application to resolution of quotient singularities
  Abstract: Mac Lane's technique of "inductive valuations" is over 80 years old, but has only recently been used to attack problems about arithmetic surfaces. We will give an explicit, hands-on introduction to the theory, requiring little background beyond the definition of a non-archimedean valuation. We will then outline how this theory is helpful 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 (slides).