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Takehiko Yasuda

Publications and source records attributed to Takehiko Yasuda.

At least 37 records · Page 2Linked to original sources
arXiv2022

Torsors for finite group schemes of bounded height

Let $F$ be a global field. Let $G$ be a non trivial finite étale tame $F$-group scheme. We define height functions on the set of $G$-torsors over $F,$ which generalize the usual heights such as discriminant. As an analogue of the Malle conjecture for group schemes, we formulate a conjecture on the asymptotic behavior of the number of $G$-torsors over $F$ of bounded height. This is a special case of our more general Stacky Batyrev-Manin conjecture from arXiv:2207.03645. The conjectured asymptotic is proven for the case $G$ is commutative. When $F$ is a number field, the leading constant is expressed as a product of certain arithmetic invariants of $G$ and a volume of a space attached to $G$. Moreover, an equidistribution property of $G$-torsors in the space is established.

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arXiv2021

The isomorphism problem of projective schemes and related algorithmic problems

We discuss the isomorphism problem of projective schemes; given two projective schemes, can we algorithmically decide whether they are isomorphic? We give affirmative answers in the case of one-dimensional projective schemes, the case of smooth irreducible varieties with a big canonical sheaf or a big anti-canonical sheaf, and the case of K3 surfaces with a finite automorphism group. As related algorithmic problems, we also discuss decidability of positivity properties of invertible sheaves, and approximation of the nef cone and the pseudo-effective cone.

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arXiv2021

Open problems in the wild McKay correspondence and related fields

The wild McKay correspondence is a form of McKay correspondence in terms of stringy invariants that is generalized to arbitrary characteristics. It gives rise to an interesting connection between the geometry of wild quotient varieties and arithmetic on extensions of local fields. The principal purpose of this article is to collect open problems on the wild McKay correspondence, as well as those in related fields that the author believes are interesting or important. It also serves as a survey on the present state of these fields.

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arXiv2021

Moduli of formal torsors II

Applying the authors' preceding work, we construct a version of the moduli space of $G$-torsors over the formal punctured disk for a finite group $G$. To do so, we introduce two Grothendieck topologies, the sur (surjective) and luin (locally universally injective) topologies, and define P-schemes using them as variants of schemes. Our moduli space is defined as a P-scheme approximating the relevant moduli functor. We then prove that Fröhlich's module resolvent gives a locally constructible function on this moduli space, which implies that motivic integrals appearing the wild McKay correspondence are well-defined.

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arXiv2021

The Wild McKay Correspondence for Cyclic Groups of Prime Power Order

The $\boldsymbol{v}$-function is a key ingredient in the wild McKay correspondence. In this paper, we give a formula to compute it in terms of valuations of Witt vectors, when the given group is a cyclic group of prime power order. We apply it to study singularities of a quotient variety by a cyclic group of prime square order. We give a criterion whether the stringy motive of the quotient variety converges or not. Furthermore, if the given representation is indecomposable, then we also give a simple criterion for the quotient variety being terminal, canonical, log canonical, and not log canonical.

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arXiv2020

Motivic integration over wild Deligne-Mumford stacks

We develop the motivic integration theory over formal Deligne-Mumford stacks over a power series ring of arbitrary characteristic. This is a generalization of the corresponding theory for tame and smooth Deligne-Mumford stacks constructed in earlier papers of the author. As an application, we obtain the wild motivic McKay correspondence for linear actions of arbitrary finite groups, which has been known only for cyclic groups of prime order. In particular, this implies the motivic version of Bhargava's mass formula as a special case. In fact, we prove a more general result, the invariance of stringy motives of (stacky) log pairs under crepant morphisms.

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arXiv2020

Moduli of formal torsors

We construct the moduli stack of torsors over the formal punctured disk in characteristic p > 0 for a finite group isomorphic to the semidirect product of a p-group and a tame cyclic group. We prove that the stack is a limit of separated Deligne-Mumford stacks with finite and universally injective transition maps.

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arXiv2018

Discrepancies of $p$-cyclic quotient varieties

We consider the quotient variety associated to a linear representation of the cyclic group of order p in characteristic p>0. We estimate the minimal discrepancy of exceptional divisors over the singular locus. In particular, we give criteria for the quotient variety being terminal, canonical and log canonical. As an application, we obtain new examples of non-Cohen-Macaulay terminal singularities, adding to examples recently announced by Totaro.

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arXiv2018

Notes on the motivic McKay correspondence for the group scheme $α_{p}$

We formulate a conjecture on the motivic McKay correspondence for the group scheme $ α_{p}$ in characteristic $p>0$ and give a few evidences. The conjecture especially claims that there would be a close relation between quotient varieties by $α_{p}$ and ones by the cyclic group of order $p$.

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arXiv2017

Motivic Serre invariants modulo the square of $\mathbb{L}-1$

Motivic Serre invariants defined by Loeser and Sebag are elements of the Grothendieck ring of varities modulo $\mathbb{L}-1$. In this paper, we show that we can lift these invariants to modulo the square of $\mathbb{L}-1$ after tensoring the Grothendieck ring with $\mathbb{Q}$, under certain assumptions.

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arXiv2016

Vojta's conjecture for singular varieties

We formulate a generalization of Vojta's conjecture in terms of log pairs and variants of multiplier ideals. In this generalization, a variety is allowed to have singularities. It turns out that the generalized conjecture for a log pair is equivalent to the original conjecture applied to a log resolution of the pair. A special case of the generalized conjecture can be interpreted as representing a general phenomenon that there tend to exist more rational points near singular points than near smooth points. The same phenomenon is also observed in relation between greatest common divisors of integer pairs satisfying an algebraic equation and plane curve singularities, which is discussed in Appendix. As an application of the generalization of Vojta's conjecture, we also derive a generalization of a geometric conjecture of Lang concering varieties of general type to singular varieties and log pairs.

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arXiv2015

Toward motivic integration over wild Deligne-Mumford stacks

We discuss how the motivic integration will be generalized to wild Deligne-Mumford stacks, that is, stabilizers may have order divisible by the characteristic of the base or residue field. We pose several conjectures on this topic. We also present some possible applications concerning stringy invariants, resolution of singularities, and weighted counts of extensions of local fields.

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arXiv2015

Mass formulas for local Galois representations and quotient singularities II: dualities and resolution of singularities

A total mass is the weighted count of continuous homomorphisms from the absolute Galois group of a local field to a finite group. In the preceding paper, the authors observed that in a particular example, two total masses coming from two different weightings are dual to each other. We discuss the problem how general such a duality holds and relate it to the existence of simultaneous resolution of singularities, using the wild McKay correspondence and the Poincaré duality for stringy invariants. We also exhibit several examples.

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arXiv2015

Manin's conjecture vs. Malle's conjecture

By a heuristic argument, we relate two conjectures. One is a version of Manin's conjecture about the distribution of rational points on a Fano variety. We concern specific singular Fano varieties, namely quotients of projective spaces by finite group actions, and their singularities play a key role. The other conjecture is a generalization of Malle's conjecture about the distribution of extensions of a number field. Main tools are several Dirichlet series and previously obtained techniques, especially the untwisting, for the counterpart over a local field.

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arXiv2014

Densities of rational points and number fields

We relate the problem of counting number fields, in particular, Malle's conjecture with the problem of counting rational points on singular Fano varieties, in particular, Batyrev and Tschinkel's generalization of Manin's conjecture.

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arXiv2014

Wilder McKay correspondences

A conjectural generalization of the McKay correspondence in terms of stringy invariants to arbitrary characteristic, including the wild case, was recently formulated by the author in the case where the given finite group linearly acts on an affine space. In cases of very special groups and representations, the conjecture has been verified and related stringy invariants have been explicitly computed. In this paper, we try to generalize the conjecture and computations to more complicated situations such as non-linear actions on possibly singular spaces and non-permutation representations of non-abelian groups.

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