arXiv subjects

Tetsushi Ito

Publications and source records attributed to Tetsushi Ito.

34 records · Page 2Linked to original sources

arXiv2018

CM liftings of K3 surfaces over finite fields and their applications to the Tate conjecture

We give applications of integral canonical models of orthogonal Shimura varieties and the Kuga-Satake morphism to the arithmetic of K3 surfaces over finite fields. We prove every K3 surface of finite height over a finite field admits a characteristic 0 lifting whose generic fiber is a K3 surface with complex multiplication. Combined with the results of Mukai and Buskin, we prove the Tate conjecture for the square of a K3 surface over a finite field. To obtain these results, we construct an analogue of Kisin's algebraic group for a K3 surface of finite height, and construct characteristic 0 liftings of the K3 surface preserving the action of tori in the algebraic group. We obtain these results for K3 surfaces over finite fields of any characteristics, including those of characteristic 2 or 3.

math.NT↗

arXiv2018

Galois representations associated with a non-selfdual automorphic representation of GL(3)

In 1994, van Geemen and Top constructed a non-selfdual motive of rank three over $\mathbb{Q}$ conjecturally associated with a cuspidal non-selfdual automorphic representation of $\mathrm{GL}_3(\mathbb{A}_{\mathbb{Q}})$ of level $Γ_0(128)$. They experimentally confirmed the coincidence of the local $L$-factors at finitely many primes using computer. In this paper, we shall prove the coincidence of the local $L$-factors at every prime. To show this, we use the recent results of Harris-Lan-Taylor-Thorne and Scholze on the construction of Galois representations, and Grenié's results to compare three-dimensional $2$-adic Galois representations. We also prove the local-global compatibility at $p = 2$, including the case $p = \ell$.

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arXiv2017

On log motives

We define the categories of log motives and log mixed motives. The latter gives a new formulation for the category of mixed motives. We prove that the former is a semisimple abelian category if and only if the numerical equivalence and homological equivalence coincide, and that it is also equivalent to that the latter is a Tannakian category. We discuss various realizations, formulate Tate and Hodge conjectures, and verify them in curve case.

math.AG↗

arXiv2016

The local-global principle for symmetric determinantal representations of smooth plane curves in characteristic two

We give an application of Mumford's theory of canonical theta characteristics to a Diophantine problem in characteristic two. We prove that a smooth plane curve over a global field of characteristic two is defined by the determinant of a symmetric matrix with entries in linear forms in three variables if and only if such a symmetric determinantal representation exists everywhere locally. It is a special feature in characteristic two because analogous results are not true in other characteristics.

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arXiv2016

The local-global principle for symmetric determinantal representations of smooth plane curves

A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local-global principle for the existence of symmetric determinantal representations of smooth plane curves over a global field of characteristic different from two. When the degree of the plane curve is less than or equal to three, we relate the problem of finding symmetric determinantal representations to more familiar Diophantine problems on the Severi-Brauer varieties and mod 2 Galois representations, and prove that the local-global principle holds for conics and cubics. We also construct counterexamples to the local-global principle for quartics using the results of Mumford, Harris, and Shioda on theta characteristics.

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arXiv2015

On the symmetric determinantal representations of the Fermat curves of prime degree

We prove that the defining equations of the Fermat curves of prime degree cannot be written as the determinant of symmetric matrices with entries in linear forms in three variables with rational coefficients. In the proof, we use a relation between symmetric matrices with entries in linear forms and non-effective theta characteristics on smooth plane curves. We also use some results of Gross-Rohrlich on the rational torsion points on the Jacobian varieties of the Fermat curves of prime degree.

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arXiv2015

Sheaf theoretic classifications of pairs of square matrices over arbitrary fields

We give classifications of linear orbits of pairs of square matrices with non-vanishing discriminant polynomials over a field in terms of certain coherent sheaves with additional data on closed subschemes of the projective line. Our results are valid in a uniform manner over arbitrary fields including those of characteristic two. This work is based on the previous work of the first author on theta characteristics on hypersurfaces. As an application, we give parametrizations of orbits of pairs of symmetric matrices under special linear groups with fixed discriminant polynomials generalizing some results of Wood and Bhargava-Gross-Wang.

math.AG↗

arXiv2011

On the cohomology of the Lubin-Tate curve of level 2 and the Lusztig theory over finite rings

An etale cohomology group $W$ of some irreducible components, which is the smooth compactification of an affine curve $(X^{q^2}-X)^{q-1}=(Y^{q(q+1)}-Y^{q+1})^{q-1},$ in the stable reduction the Lubin-Tate curve of level two is related to the Lusztig theory over finite rings. In this paper, we investigate a relationship between the cohomology group $W$ and the Lusztig theory.

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arXiv2011

On the cuspidal representations of ${\rm GL}_2(F)$ of level 1 or 1/2 in the cohomology of the Lubin-Tate space $\mathcal{X}(π^2)$

In this paper, we compute irreducible components which appear in the stable reduction of the Lubin-Tate curve of level two, in the mixed characteristic case. We also compute the action of the central division algebra of invariant 1/2, the action of ${\rm GL}_2$, and the inertia action explicitly. As a result, in a sense, we observe that, in the cohomology group of the stable reduction of the Lubin-Tate curve for ${\rm GL}_2$, the local Langlands correspondence and the local Jacquet-Langlands correspondence for ${\rm GL}_2$ are realized for the cuspidal representations of level 1 or 1/2.

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arXiv2010

Cuspidal representations in the l-adic cohomology of the Rapoport-Zink space for GSp(4)

In this paper, we study the l-adic cohomology of the Rapoport-Zink tower for GSp(4). We prove that the smooth representation of GSp_4(Q_p) obtained as the i-th compactly supported l-adic cohomology of the Rapoport-Zink tower has no quasi-cuspidal subquotient unless i=2,3,4. Our proof is purely local and does not require global automorphic methods.

math.RT↗

arXiv2005

Weight-monodromy conjecture over equal characteristic local fields

The aim of this paper is to study certain properties of the weight spectral sequences of Rapoport-Zink by a specialization argument. By reducing to the case over finite fields previously treated by Deligne, we prove that the weight filtration and the monodromy filtration defined on the $l$-adic étale cohomology coincide, up to shift, for proper smooth varieties over equal characteristic local fields. We also prove that the weight spectral sequences degenerate at $E_2$ in any characteristic without using log geometry. Moreover, as an application, we give a modulo $p>0$ reduction proof of a Hodge analogue previously considered by Steenbrink.

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arXiv2004

On the Manin-Mumford conjecture for abelian varieties with a prime of supersingular reduction

We give a short proof of the "prime-to-$p$ version" of the Manin-Mumford conjecture for an abelian variety over a number field, when it has supersingular reduction at a prime dividing $p$, by combining the methods of Bogomolov, Hrushovski, and Pink-Roessler. Our proof here is quite simple and short, and neither $p$-adic Hodge theory nor model theory is used. The observation is that a power of a lift of the Frobenius element at a supersingular prime acts on the prime-to-$p$ torsion points via nontrivial homothety.

math.NT↗

arXiv2004

Weight-monodromy conjecture for p-adically uniformized varieties

The aim of this paper is to prove the weight-monodromy conjecture (Deligne's conjecture on the purity of monodromy filtration) for varieties p-adically uniformized by the Drinfeld upper half spaces of any dimension. The ingredients of the proof are to prove a special case of the Hodge standard conjecture, and apply a positivity argument of Steenbrink, M. Saito to the weight spectral sequence of Rapoport-Zink. As an application, by combining our results with the results of Schneider-Stuhler, we compute the local zeta functions of p-adically uniformized varieties in terms of representation theoretic invariants. We also consider a p-adic analogue by using the weight spectral sequence of Mokrane.

math.NT↗

arXiv2003

Stringy Hodge numbers and p-adic Hodge theory

The aim of this paper is to give an application of p-adic Hodge theory to stringy Hodge numbers introduced by V. Batyrev for a mathematical formulation of mirror symmetry. Since the stringy Hodge numbers of an algebraic variety are defined by choosing a resolution of singularities, the well-definedness is not clear from the definition. We give a proof of the well-definedness based on arithmetic results such as p-adic integration and p-adic Hodge theory. Note that another proof of the well-definedness was already obtained by V. Batyrev himself by motivic integration. This is a generalization of the author's earlier work in math.NT/0209269, where he treats only the smooth case.

math.NT↗

arXiv2003

Weight-monodromy conjecture for certain threefolds in mixed characteristic

The weight-monodromy conjecture claims the coincidence of the weight filtration and the monodromy filtration, up to shift, on the $l$-adic étale cohomology of a proper smooth variety over a complete discrete valuation field. Although it has been proved in some cases, the case of dimension $\geq 3$ in mixed characteristic is still open so far. The aim of this paper is to give a proof of the weight-monodromy conjecture for a threefold which has a projective strictly semistable model such that, for each irreducible component of the special fiber, the Picard number is equal to the second $l$-adic Betti number. Our proof is based on a careful analysis of the weight spectral sequence of Rapoport-Zink by the Hodge index theorem for surfaces. We also prove a $p$-adic analogue by using the weight spectral sequence of Mokrane.

math.NT↗

arXiv2002

Birational smooth minimal models have equal Hodge numbers in all dimensions

This is a resume of the author's talk at the Worhshop on Arithmetic, Geometry and Physics around Calabi-Yau Varieties and Mirror Symmetry (July 23-29, 2001), the Fields Institute. The aim of this note is to prove that birational smooth minimal models over C have equal Hodge numbers in all dimensions by an arithmetic method. Our method is a refinement of the method of V. Batyrev and C.-L. Wang on Betti numbers who used p-adic integration and the Weil conjecture. Our ingredient is to use further arithmetic results such as the Chebotarev density theorem and p-adic Hodge theory.

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