Mini-workshop on arithmetic and geometry of moduli spaces

Date:

July 6(Sat) - 7(Sun), 2024

Style:

Hybrid format (Lecture Room + Zoom)

Location (Lecture Room):

Lecture room 110, North campus, Graduate School of Science Bldg. No.3, Department of Mathematics, Kyoto University

North Campus Map

Official Program:

Official Program (PDF version, in Japanese)

Registration Form (for Online Participants):

Registration Form

Zoom ID:

Zoom ID: 942 6603 5932

Passcode: ????? (The largest known Fermat prime)

Zoom ID will be available on this website at least 30 minutes before the beginning of the workshop.

Speakers:

Kentaro Inoue (Kyoto University, Department of Mathematics)

Kai-Wen Lan (University of Minnesota/Kyoto University, Department of Mathematics)

Shohei Ma (Tokyo Institute of Technology)

Yuji Odaka (Kyoto University, Department of Mathematics)

Yasuhiro Oki (Hokkaido University)

Stefan Reppen (University of Tokyo)

Ken Sato (Tokyo Institute of Technology)

Teppei Takamatsu (Kyoto University, Department of Mathematics & Hakubi Center)

Yuta Takaya (University of Tokyo)

Organizers:

Yuji Odaka (Kyoto University, Department of Mathematics)

Teppei Takamatsu (Kyoto University, Department of Mathematics & Hakubi Center)

Tetsushi Ito (Kyoto University, Department of Mathematics)

Support:

This workshop is supported by KAKENHI (Grants-in-Aid for Scientific Research) 23K20786.

Program

July 6(Sat)

10:00-12:00  Discussion Session (Informal)

13:00-14:00  Takamatsu

14:15-15:15  Sato

15:30-16:30  Ma

16:45-17:45  Odaka

July 7(Sun)

10:00-11:00  Oki

11:10-12:10  Inoue

13:30-14:30  Reppen

14:45-15:45  Takaya

16:00-17:00  Lan

Titles & Abstracts

July 6(Sat)

13:00-14:00  Teppei Takamatsu (Kyoto University, Department of Mathematics & Hakubi Center)

Title: Arithmetic finiteness of Fano varieties of genus 7 

Abstract: The Shafarevich conjecture, which was proved by Faltings and Zarhin, states the finiteness of isomorphism classes of abelian varieties of a fixed dimension over a fixed number field admitting good reduction away from a fixed finite set of finite places. Recently, Javanpeykar-Loughran and Licht proved analogous results for some classes of Fano threefolds. In this talk, we discuss the Shafarevich conjecture forprime Fano 3-folds of genus 7, which is one of the missing cases in Javanpeykar-Loughran and Licht's works. This talk is based on a joint work with Tetsushi Ito, Akihiro Kanemitsu, and Yuuji Tanaka.

14:15-15:15  Ken Sato (Tokyo Institute of Technology)

Title: On higher Chow cycles on K3 surfaces

Abstract: The higher Chow groups introduced by Bloch are a generalization of the classical Chow groups. In this talk, I will explain an explicit construction of higher Chow cycles of type (2, 1) on several families of K3 surfaces by using cyclic covering structures. Combined with some specialization/deformation arguments on their (higher) normal functions and the explicit computation of their images under the Beilinson regulator map, I will show that for a very general member of the families, these cycles cannot be obtained by intersection products of cycles of lower codimension. Part of this talk is a joint work with Shohei Ma.

15:30-16:30  Shohei Ma (Tokyo Institute of Technology)

Title: Mixed Hodge structures of locally symmetric varieties

Abstract: I will talk about the mixed Hodge structures on the cohomology of locally symmetric varieties. In the middle degree, I relate the weight filtration to the Siegel operators for certain modular forms.  This has application to a classical problem on the Siegel operators. In the general degrees, I construct a spectral sequence which converges to the edge components in the Hodge triangle, and whose E1 page is expressed by some simple geometric invariants associated to the cusps. This already degenerates at E1 in a certain range. MHS approach also has application to the restriction map to the Borel-Serre boundary. 

16:45-17:45  Yuji Odaka (Kyoto University, Department of Mathematics)

Title: On compactification of Shimura varieties from moduli perspectives

Abstract: This will be a informal survey talk on how the compactification theories of connected Shimura varieties over \C i.e., locally Hermitian symmetric domains are related to some topics in complex algebraic geometry, differential geometry, reflecting our recent works in this 15 years, especially in the context of moduli: the topics include some subset of moduli of varieties, their compactifications (e.g. AMRT/Looijenga, Satake), K-stability, Kahler-Einstein metrics, their asymptotic behaviours and tropical/Berkovich geometry (with Oshima), MMP-based birational geometry, Arakelov geometry and so on.

July 7(Sun)

10:00-11:00  Yasuhiro Oki (Hokkaido University)

Title: On the Hasse norm principle for extensions of degree $p\ell$ with $p\neq \ell$

Abstract: The Hasse norm principle is one of the most classic problems in algebraic number theory. It is known that the Hasse norm principle may or may not hold in general. Hence it is natural to investigate an equivalence condition on the validity of the Hasse norm principle for finite extensions of degree a given positive integer $d$. This problem has been settled if $d$ is a prime number or $d\geq 16$, however it is still open in general. In this talk, we discuss the above question where $d$ is a product of two distinct prime numbers under the assumption that the considered extensions have non-trivial intermediate fields.

10:10-12:10  Kentaro Inoue (Kyoto University, Department of Mathematics)

Title: Log prismatic Dieudonné theory for log p-divisible groups

Abstract: Bhatt and Scholze introduced a new integral p-adic cohomology theory called prismatic cohomology, and Anschutz-Le Bras established a prismatic version of Dieudonné theory. In this talk, we talk about generalization to log p-divisible groups in the framework of log prismatic cohomology established by Koshikawa and Yao. If time permits, we will explain the application to Shimura varieties.

13:30-14:30  Stefan Reppen (Stockholm University/University of Tokyo)

Title: On a principle of Ogus: the Hasse invariant's order of vanishing and ``Frobenius and the Hodge filtration''

Abstract: In joint work with W. Goldring, we construct Hasse invariants on the stack of $G$-zips for any triple $(G,\mu,r)$ consisting of a connected reductive $\mathbb{F}_p$-group $G$, a cocharacter $\mu$ and a representation $r$ of $G$. We define a notion of conjugate line position which generalises the notion of $a$-number for abelian varieties to such triples. For large classes of groups (all classical types) we compute the vanishing order of the Hasse invariant and show that it agrees with the conjugate line position. We deduce the corresponding result for the special fibre of Hilbert modular varieties, Siegel varieties and orthogonal varieties, all of arbitrary dimension, as well as certain unitary Shimura varieties. These results are analogous to Ogus' on families of Calabi-Yau varieties, and we also recover his result for K3-surfaces. In the talk I will present this work.

14:45-15:45  Yuta Takaya (University of Tokyo)

Title: Moduli spaces of level structures on mixed characteristic local shtukas

Abstract: Shimura varieties are of central interest in arithmetic geometry. Recently, Pappas and Rapoport introduced axiomatic characterization of canonical integral models of Shimura varieties. In this characterization, local shtukas play an important role as counterparts of p-divisible groups uniformly applicable to any Shimura datum. In this talk, I will explain the relation of canonical integral models at different parahoric levels in terms of local shtukas, restricting to those under hyperspecial levels. I will give a representability criterion of v-sheaves, which is similar in spirit to Artin's theorem on dilatations, and use it to show that canonical integral models under hyperspecial levels represent the flat moduli spaces of level structures on the universal local shtukas over those at hyperspecial levels. If time permits, I will explain the local counterparts of the above results.

16:00-17:00  Kai-Wen Lan (University of Minnesota/Kyoto University, Department of Mathematics)

Title: De Rham comparison for boundary cohomology

Abstract: Let X be any variety smooth but not necessarily proper over the p-adic numbers.  In joint works with Hansheng Diao, Ruochuan Liu, and Xinwen Zhu, we constructed a Riemann-Hilbert functor sending any p-adic etale local system L over X to a regular integrable connection RH(L) over X, and we showed that, when L is de Rham, we have a comparison isomorphism between the etale cohomology of L and the de Rham cohomology of RH(L) after tensoring with Fontaine's period ring B_{dR}, together with compatible analogues for the compactly supported cohomology and the so-called interior cohomology.  In this talk, I will explain an analogue for the so-called boundary cohomology, which naturally complements the interior cohomology, and some further generalizations, based on joint work with David Sherman. I will start with a general review, and explain our strategy. If time permits, I will also explain some applications to Shimura varieties.