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Wataru Kai

Publications and source records attributed to Wataru Kai.

13 recordsLinked to original sources
arXiv2024

Equi-dimensionalization via subdivision of simplices

We give an alternative proof of Suslin's equi-dimensionalization moving lemma using a different geometric construction. The new construction provides better control of the degrees of the polynomials describing the geometric procedure. The new degree bound can be used to improve an earlier result of Hiroyasu Miyazaki and the present author on algebraic cycles with modulus: the isomorphism in question is now valid for any fixed divisor, without the need to take the limit over thickenings.

math.AG↗
arXiv2023

Linear patterns of prime elements in number fields

We prove a number field analogue of the Green--Tao--Ziegler theorem on simultaneous prime values of degree 1 polynomials whose linear parts are pairwise linearly independent. Applications of our results include a Hasse principle of rational points for certain fibrations $X\to \mathbb{P}^1$ over number fields $K$ which had only been available over $\mathbb Q $ by Harpaz--Skorobogatov--Wittenberg, and construction of elliptic curves having some specified ranks due to Koymans--Pagano and Zywina. This latter family of results led to a negative answer to a generalized Hilbert Tenth Problem.

math.NT↗
arXiv2022

Notes on Mitsui's Prime Number Theorem with Siegel zeros

In these notes, we refine Mitsui's Prime Number Theorem from 1957, which for a number field $K$ predicts the number of prime elements in a bounded convex set in $K \otimes _{\mathbf Q} \mathbf R $ and in a specified congruence class modulo an ideal $\mathfrak q $, by incorporating potential Siegel zeros of Hecke $L$-functions. This allows the norm of the modulus $\mathbf N (\mathfrak q )$ to grow at a pseudopolynomial rate $e^{\sqrt{\log X}/O(1)}$ with respect to the size $X$ of the convex set as opposed to powers of $\log X$. The extra flexibility and precision are essential in our sequel work on linear patterns of prime elements. We also hope that our updated exposition will make Mitsui's work accessible to a wider mathematical audience.

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arXiv2022

The Green-Tao theorem for affine curves over F_q

Green and Tao famously proved in a 2008 paper that there are arithmetic progressions of prime numbers of arbitrary lengths. Soon after, analogous statements were proved by Tao for the ring of Gaussian integers and by Lê for the polynomial rings over finite fields. In 2020 this was extented to orders of arbitrary number fields by Kai-Mimura-Munemasa-Seki-Yoshino. We settle the case of the coordinate rings of affine curves over finite fields. The main contribution of this paper is subtle choice of a polynomial subring of the given ring which plays the role of $\mathbb Z$ in the number field case. This choice and the proof of its pleasant properties eventually depend on the Riemann-Roch formula.

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arXiv2022

Constellations in prime elements of number fields

Given any number field, we prove that there exist arbitrarily shaped constellations consisting of pairwise non-associate prime elements of the ring of integers. This result extends the celebrated Green-Tao theorem on arithmetic progressions of rational primes and Tao's theorem on constellations of Gaussian primes. Furthermore, we prove a constellation theorem on prime representations of binary quadratic forms with integer coefficients. More precisely, for a non-degenerate primitive binary quadratic form $F$ which is not negative definite, there exist arbitrarily shaped constellations consisting of pairs of integers $(x,y)$ for which $F(x,y)$ is a rational prime. The latter theorem is obtained by extending the framework from the ring of integers to the pair of an order and its invertible fractional ideal.

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arXiv2021

Unramified logarithmic Hodge-Witt cohomology and $\mathbb{P}^1$-invariance

Let $X$ be a smooth proper variety over a field $k$ and suppose that the degree map $\mathrm{CH}_0(X \otimes_k K) \to \mathbb{Z}$ is isomorphic for any field extension $K/k$. We show that $G(\mathrm{Spec} k) \to G(X)$ is an isomorphism for any $\mathbb{P}^1$-invariant Nisnevich sheaf with transfers $G$. This generalize a result of Binda-Rülling-Saito that proves the same conclusion for reciprocity sheaves. We also give a direct proof of the fact that the unramified logarithmic Hodge-Witt cohomology is a $\mathbb{P}^1$-invariant Nisnevich sheaf with transfers.

math.AG↗
arXiv2019

Chern classes with modulus

In this paper, we construct Chern classes from the relative $K$-theory of modulus pairs to the relative motivic cohomology defined by Binda-Saito. An application to relative motivic cohomology of henselian dvr is given.

math.KT↗
arXiv2018

A moving lemma for algebraic cycles with modulus and contravariance

We prove a moving lemma which implies the contravariance of Bloch-Esnault's additive higher Chow group in smooth affine varieties and Binda-Saito's higher Chow group (taken in the Nisnevich topology) in smooth varieties equipped with effective Cartier divisors. The new ingredients in the moving method are parallel translation {\em with modulus} in the affine space that involves a new integer parameter, and Noether's normalization lemma over a Dedekind base.

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arXiv2017

Notes on a p-adic exponential map for the Picard group

Part of these notes was written as the author's 2013 master thesis. For proper flat schemes over a complete discrete valuation ring of mixed characteristic, we construct an isomorphism of certain subgroups of the Picard group and the first cohomology group of the structure sheaf. When the Picard scheme is available and smooth, it recovers the isomorphism coming from its formal completion. A reinterpretation of an old theorem of Mattuck is given.

math.AG↗
arXiv2016

Torsion and divisibility for reciprocity sheaves and 0-cycles with modulus

The notion of modulus is a striking feature of Rosenlicht-Serre's theory of generalized Jacobian varieties of curves. It was carried over to algebraic cycles on general varieties by Bloch-Esnault, Park, Rülling, Krishna-Levine. Recently, Kerz-Saito introduced a notion of Chow group of $0$-cycles with modulus in connection with geometric class field theory with wild ramification for varieties over finite fields. We study the non-homotopy invariant part of the Chow group of $0$-cycles with modulus and show their torsion and divisibility properties. Modulus is being brought to sheaf theory by Kahn-Saito-Yamazaki in their attempt to construct a generalization of Voevodsky-Suslin-Friedlander's theory of homotopy invariant presheaves with transfers. We prove parallel results about torsion and divisibility properties for them.

math.AG↗
arXiv2016

A higher dimensional generalization of Lichtenbaum duality in terms of the Albanese map

We present a conjectural formula describing the cokernel of the Albanese map of zero-cycles of smooth projective varieties $X$ over $p$-adic fields in terms of the Néron-Severi group and provide a proof under additional assumptions on an integral model of $X$. The proof depends on a non-degeneracy result of Brauer-Manin pairing due to Saito-Sato and on Gabber-de Jong's comparison result of cohomological- and Azumaya-Brauer groups. We will also mention the local-global problem of the Albanese-cokernel; the abelian group on the "local side" turns out to be a finite group.

math.NT↗
arXiv2016

Suslin's moving lemma with modulus

The moving lemma of Suslin states that a cycle on $X\times \mathbb{A} ^n$ meeting all faces properly can be moved so that it becomes equidimensional over $\mathbb{A}^n$. This leads to an isomorphism of motivic Borel-Moore homology and higher Chow groups. In this short paper we formulate and prove a variant of this. It leads to an isomorphism of Suslin homology with modulus and higher Chow groups with modulus, in an appropriate pro setting.

math.AG↗