The 24th International Conference, Graduate School of Mathematics, Nagoya University

Moduli Spaces and Arithmetic
September 9 – 13, 2024
Rm. 509 in Mathematics Bldg., Nagoya University, Nagoya

Speakers
Margaret Bilu (Institut de Mathématiques de Bordeaux),
Huy Dang (National Center for Theoretical Sciences in Taipei),
Ratko Darda (Sabanci University),
Jordan Ellenberg* (University of Wisconsin-Madison),
Kenta Hashizume (Niigata University),
Reimi Irokawa (NTT),
Ayako Kubota (Saitama University),
Eric Larson* (Brown University),
Zhiyuan Li (Fudan University),
Shouhei Ma (Tokyo Institute of Technology),
Shusuke Otabe (Nagoya Institute of Technology),
Eric Riedl (University of Notre Dame),
Masa-Hiko Saito (Kobe Gakuin University),
Isabel Vogt (Brown University),
Yasuhiro Wakayabashi (Osaka University),
Yu Yasufuku (Waseda University).
* Zoom talks.
Organizing Committee
Sho Tanimoto (Nagoya University),
Takehiko Yasuda (Osaka University).
Sponsored byNagoya University,
Graduate School of Mathematics, Nagoya University,
JST FOREST Program Grant number JPMJFR212Z (Tanimoto),
JSPS KAKENHI Grant numbr 23K25764 (Tanimoto),
JSPS KAKENHI Grant number 20H00112 (Kondo).

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PROGRAM
Mon., September 9, 2024
10:00–11:00

Yu Yasufuku (Waseda University)
Towards rational representatives for the moduli spaces of dynamical systems on P1

11:30–12:30

Zhiyuan Li (Fudan University)
Supersingular Tate conjecture for Hyper-Kähler varieties

12:30–14:30— Lunch Break —14:30–15:30

Yasuhiro Wakayabashi (Osaka University)
Arithmetic liftings and 2d TQFT for differential equations in positive characteristic

16:00–17:00

Reimi Irokawa (NTT )Hybrid dynamics of hyperbolic automorphisms over K3 surfaces

Tue., September 10, 2024
10:00–11:00

Shouhei Ma (Tokyo Institute of Technology)
Mixed Hodge structures of locally symmetric varieties11:30–12:30

Isabel Vogt (Brown University)
Interpolation for Brill–Noether curves

12:30–14:30— Lunch Break —14:30–15:30

Kenta Hashizume (Niigata University)
On boundedness and moduli space of special klt-trivial fibrations

16:00–17:00

Ratko Darda (Sabanci University)
The Manin conjecture for toric stacks

Wed., September 11, 2024
10:00–11:00

Eric Larson (Brown University) zoom
Normal bundles of rational curves in Grassmannians

11:30–12:30

Margaret Bilu (Institut de Mathématiques de Bordeaux)
A motivic circle method

12:30–14:30— Lunch Break —14:30–17:00— Free Discussion —Thu., September 12, 2024
10:00–11:00

Masa-hiko Saito (Kobe Gakuin University)
On a family of K3 surfaces of Beukers and Peters and elliptic modular surafces

11:30–12:30

Eric Riedl (University of Notre Dame)
Non-free curves and Geometric Manin’s Conjecture

12:30–14:30— Lunch Break —14:30–15:30

Ayako Kubota (Saitama University)
Examples of the invariant Hilbert scheme of the Cox realization

16:00–17:00

Huy Dang (National Center for Theoretical Sciences in Taipei)
The moduli spaces of cyclic covers in characteristic 𝑝2

Fri., September 13, 2024
10:00–11:00

Shusuke Otabe (Nagoya Institute of Technology)
On rationality and inverse Galois problems for the Witt algebras

11:30–12:30

Jordan Ellenberg (University of Wisconsin-Madison) zoom
Homological stability and variation of Selmer groups in quadratic twist families overfunction fields

12:30–14:30— Lunch Break —14:30–17:00— Free Discussion —3

ABSTRACTS

Yu Yasufuku (Waseda University) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mon., September 9, 2024, 10:00–11:00
Towards rational representatives for the moduli spaces of dynamical systems on P1
Using GIT, we can construct the moduli space M 𝑑 of rational functions of degree 𝑑 up toconjugation. For 𝑑 = 2, it is well-known that M2  A2 , determined by the elementary symmetric−functions →𝜎 of the derivatives at the fixed points. Almost 15 years ago in a joint work with Manes,
we explicitly constructed the inverse of this isomorphism over number fields, i.e. from a 𝐾-vector→−
𝜎 we gave an explicit formula for the corresponding rational function defined over 𝐾. Generalizing
this to 𝑑 ≥ 3 faces several complications, including: (1) data at the fixed points may not be sufficient
to determine the conjugacy class (2) we might not be able to move all the fixed points to wherever
we like (3) the field of moduli might be different from the field of definition. In this talk, I will
discuss these complications as well as related works by West, Hutz–Tepper, Sugiyama, Gotou, etc.,
and then provide some partial results.
Zhiyuan Li (Fudan University) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mon., September 9, 2024, 11:30–12:30

Supersingular Tate conjecture for Hyper-Kähler varieties
The Tate conjecture asserts that all Tate classes on a smooth projective variety are algebraic. In
this lecture, I will talk about so called supersingular Tate conjecture for Hyper-Kähler varieties, which
predicts that all cohomology classes of a supersingular Hyper-Kähler variety are algebraic. This is
motivated from the work of Artin and Shioda on supersingular K3 surfaces. I will foucs on HyperKähler varieties of known types and confirm this conjecture over fields with large characteristic.
This is an on going work with Lie Fu and Xuanlin Huang.
Yasuhiro Wakayabashi (Osaka University) . . . . . . . . . . . . . . . . . . . . Mon., September 9, 2024, 14:30–15:30

Arithmetic liftings and 2d TQFT for differential equations in positive characteristic
Dormant opers are certain flat bundles on an algebraic curve in positive characteristic, related tolinear differential equations having a full set of solutions. The moduli space of such objects (in specialcases) has been discussed in the context of 𝑝-adic Teichmüller theory, and has many remarkableaspects, including the connections with the Gromov–Witten theory of Grassmann varieties and thecombinatorics of graphs and rational polytopes. One goal of my research is to solve the counting
problem of dormant opers while deepening our understanding of these connections. As an approachto that problem in the case of prime-power characteristic, I have recently been studying a kind ofarithmetic lifting of dormant opers, which I call “canonical diagonal lifting”. I would like to talkabout that topic, starting with some basics on flat bundles.

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Reimi Irokawa (NTT ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mon., September 9, 2024, 16:00–17:00

Hybrid dynamics of hyperbolic automorphisms over K3 surfaces
We study degenerating families of hyperbolic dynamics over complex K3 surfaces by meansof the theory of hybrid spaces by Boucksom, Favre, and Jonsson. For an analytic family ofhyperbolic automorphisms { 𝑓𝑡 : 𝑋𝑡 – 𝑋𝑡 }𝑡 ∈D∗ over K3 surfaces 𝑋𝑡 that is possibly meromorphicallydegenerating at the origin, we consider the family of invariant measures {𝜂𝑡 } on 𝑋𝑡 constructed byananan– 𝑋C((𝑡over the inducedCantat. The family 𝑓𝑡 induces a hyperbolic automorphism 𝑓C((𝑡: 𝑋C((𝑡))))))
non-archimedean K3 surface, where we also have a measure 𝜂0 by Filip. Our main theorem states
the weak convergence of {𝜂𝑡 } to 𝜂0 as 𝑡 – 0 over the induced so-called hybrid space.
Shouhei Ma (Tokyo Institute of Technology) . . . . . . . . . . . . . . . . . . . . Tue., September 10, 2024, 10:00–11:00

Mixed Hodge structures of locally symmetric varieties
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 constructa spectral sequence which converges to the edge components in the Hodge triangle, and whose E1
page is expressed by simpler group cohomology 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–Serreboundary.
Isabel Vogt (Brown University) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tue., September 10, 2024, 11:30–12:30

Interpolation for Brill–Noether curves
The interpolation problem is one of the oldest in mathematics. In its most broad form it asks:
when can a curve of a given type be passed through a given number of general points? I’ll discuss
my joint work with Eric Larson that completely solves this problem for curves of general moduli.
Kenta Hashizume (Niigata University) . . . . . . . . . . . . . . . . . . . . . . . . Tue., September 10, 2024, 14:30–15:30

On boundedness and moduli space of special klt-trivial fibrations
A klt-trivial fibration is a kind of fibration which often appears in birational geometry. In thistalk, I will introduce the boundedness and the coarse moduli space of special klt-trivial fibrations
over curves. I will mainly explain the boundedness of special klt-trivial fibrations over curves withsome fixed invariants. This talk is based on a part of a joint work with Masafumi Hattori.
Ratko Darda (Sabanci University) . . . . . . . . . . . . . . . . . . . . . . . . . . . Tue., September 10, 2024, 16:00–17:00

The Manin conjecture for toric stacks
The Manin conjecture predicts the number of rational points of bounded height on varieties,
the anticanonical line bundle of which is big. Recently, the Manin conjecture has been generalized
to Deligne–Mumford stacks. Recall that the Deligne–Mumford stacks are algebro-geometric objects
which arise as solutions to the questions of parametrizing objects having finite (not necessarilytrivial) automorphism groups. In this talk, we study the Manin conjecture for toric stacks, which
are Deligne–Mumford stacks, whose geometry can be described using combinatorial data similar tothat describing toric varieties. The talk is based on a joint work with T. Yasuda.
5Eric Larson (Brown University) zoom . . . . . . . . . . . . . . . . . . . . . . . Wed., September 11, 2024, 10:00–11:00

Normal bundles of rational curves in Grassmannians
Let 𝐶 be a general rational curve of degree 𝑑 in a Grassmannian 𝐺 (𝑘, 𝑛). The natural expectation
Éis that its normal bundle is balanced, i.e., isomorphic toO (𝑒 𝑖 ) with all |𝑒 𝑖 − 𝑒 𝑗 | ≤ 1. In this talk,I will describe several counterexamples to this expectation, propose a suitably revised conjecture,and describe recent progress towards this conjecture.
Margaret Bilu (Institut de Mathématiques de Bordeaux) . . . . . . . . . Wed., September 11, 2024, 11:30–12:30

A motivic circle method
The circle method is a technique developed by Ramanujan, Hardy and Littlewood and which
has allowed to solve many difficult counting problems in number theory throughout the last century.
More recently, a version of the method over function fields, combined with spreading out techniques,
has led to new results about the geometry of moduli spaces of rational curves on hypersurfaces oflow degree, thus expanding the scope of the circle method to the realm of algebraic geometry. In
this talk I will show how to implement a circle method with an even more geometric flavour, where
the computations take place in a suitable Grothendieck ring of varieties, and explain how this leads
to a more precise description of the geometry of the above moduli spaces. This is joint work withTim Browning.
Masa-Hiko Saito (Kobe Gakuin University) . . . . . . . . . . . . . . . . . . . Thu., September 12, 2024, 10:00–11:00

On a family of K3 surfaces of Beukers and Peters and elliptic modular surafces
A family of K3 surfaces studied by Beukers and Peters are related to Apéry’s numbers and𝜁 (3). This family is also related to a elliptic modular surface of Γ1 (6). Following Peters, we will
see that how the family of K3 surfaces are related to elliptic modular surfaces. Moreover we would
like to explain some relations between rigid Calabi–Yau threefolds and modular forms by recallinga result due to Saito and Yui.
Eric Riedl (University of Notre Dame) . . . . . . . . . . . . . . . . . . . . . . . . Thu., September 12, 2024, 11:30–12:30

Non-free curves and Geometric Manin’s Conjecture
One important way to study a variety is to study the curves on it. Free curves, those whose
restricted tangent bundle has vanishing 𝐻 1 , are particularly nice. They are smooth points of the space
Hom(𝐵, 𝑋) of morphisms from 𝐵 to 𝑋, and they lie on a unique component of Hom(𝐵, 𝑋) of theexpected dimension. Non-free curves, on the other hand, are more mysterious, and the componentsof Hom(𝐵, 𝑋) consisting entirely of non-free curves are particularly challenging to study. In thistalk, we provide a geometric classification of the ways that a family of curves can be non-free. In
the more general setting of a Fano fibration, we show that these curves come from 𝑎-covers of 𝑋,and we show that the set of such 𝑎-covers up to equivalence form a bounded family. This proves
the first of Batyrev’s heuristics that make up Geometric Manin’s Conjecture for Fano fibrations overarbitrary base curves and provides strong evidence for Lehmann, Sengupta and Tanimoto’s proposedcharacterization of the thin set in Manin’s Conjecture. If time permits, we can discuss the restriction
theorems that allow us to prove these results, which may be of independent interest. This is joint
work with Brian Lehmann and Sho Tanimoto.

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Ayako Kubota (Saitama University) . . . . . . . . . . . . . . . . . . . . . . . . . . Thu., September 12, 2024, 14:30–15:30

Examples of the invariant Hilbert scheme of the Cox realization
The invariant Hilbert scheme is a moduli space of affine schemes that are stable under an actionof a reductive algebraic group. Under a suitable choice of the parameter, there exists a naturalmorphism, called the quotient-scheme map, from the invariant Hilbert scheme to an affine quotient
variety, and it makes the invariant Hilbert scheme a candidate for a resolution of singularities of
the quotient variety. As the quotient-scheme map depends on the quotient construction of the given
singularity, we may ask if it has a good presentation. In this talk, we focus on a quotient presentation
called the Cox realization and consider the associated invariant Hilbert scheme.
Huy Dang (National Center for Theoretical Sciences in Taipei). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thu., September 12, 2024, 16:00–17:00

The moduli spaces of cyclic covers in characteristic 𝑝
Galois covers in characteristic 𝑝 > 0 exhibit distinct properties compared to those in characteristic 0. One notable difference is the existence of families of covers with varying ramification data.
In this talk, we focus on the moduli space ASW (𝑑1 ,𝑑2 ,...,𝑑𝑛 ) , which parametrizes Z/𝑝 𝑛 -covers of
the projective line in characteristic 𝑝 > 0 with Z/𝑝 𝑖 -subcovers possessing conductor 𝑑𝑖 . In particular, we investigate the geometry of ASW (𝑑1 ,𝑑2 ,...,𝑑𝑛 ) by exploring the deformations of cycliccovers, particularly examining changes in the 𝑝-rank and the number of branch points. Our main
tool for analyzing these deformations is the refined Swan conductor, an important invariant from
Kato’s higher class field theory, which measures how far a Galois cover over a DVR is from having
separable reduction.
Shusuke Otabe (Nagoya Institute of Technology) . . . . . . . . . . . . . . . . Fri., September 13, 2024, 10:00–11:00

On rationality and inverse Galois problems for the Witt algebras
The Witt algebras in positive characteristic appear as a family of non-classical simple Liealgebras in the classification theory of Block–Wilson–Strade–Premet for modular Lie algebras. In
this talk, I will discuss a rationality problem and an inverse Galois problem for the finite simple
group schemes 𝐺 associated with the Witt algebras. First, I will explain about results for the inverse
Galois problem obtained through joint work with Fabio Tonini and Lei Zhang. Next, I will discuss
the stable birational types of the classifying spaces for 𝐺.
Jordan Ellenberg (University of Wisconsin-Madison) zoom . . . . . . . Fri., September 13, 2024, 11:30–12:30

Homological stability and variation of Selmer groups in quadratic twist families
over function fields
The Poonen–Rains heuristics, later generalized by Bhargava, Kane, and Lenstra as well as theprior two named authors, are a conjectural framework for answering the question: what does themod 𝑁 Selmer group of a random abelian variety over a global field look like? Of course, this may
depend on what one means by a random abelian variety. One natural family from which one mightdraw at random is the family of quadratic twists of bounded conductor of a fixed abelian variety. I
will talk about recent work with Aaron Landesman which establishes a weak form of the BKLPR
heuristics in this context when the global field is a function field over a finite field. The main tool is
an extension of the homological stability approach of Ellenberg–Venkatesh–Westerland to apply toa substantially wider class of sheaves on moduli spaces.
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