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Yoshiyasu Ozeki

Publications and source records attributed to Yoshiyasu Ozeki.

At least 19 recordsLinked to original sources
arXiv2026

Fields of definition of $p$-torsion points of elliptic curves and their ramification

Let $E$ be an elliptic curve over $\mathbb{Q}_p$. We study the field $\mathbb{Q}_p(E[p])$ generated by the $p$-torsion points of $E$. When $E$ has good reduction, we determine not only $\mathbb{Q}_p(E[p])$ but also the field $\mathbb{Q}_p(P)$ for every point $P\in E[p]$. This classification yields criteria for the existence of $p$-torsion over unramified and Lubin--Tate extensions of $\mathbb{Q}_p$. As a global application, let $E/\mathbb{Q}$ be an elliptic curve and let $p$ be an odd prime of good reduction. Define $f_p(E)$ to be the multiplicative order of $a_p(E)$ modulo $p$ in the ordinary case, and set $f_p(E)=p^2-1$ in the supersingular case. We prove that $E(K)[p]=0$ for every number field $K/\mathbb{Q}$ with $[K:\mathbb{Q}]<f_p(E)$. When $E$ has bad reduction, we describe the ramified part of $\mathbb{Q}_p(E[p])/\mathbb{Q}_p$. For every reduction type, we determine the maximal upper ramification break of $\mathbb{Q}_p(E[p])/\mathbb{Q}_p$.

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arXiv2025

Kummer-faithfulness over $p$-adic fields

The notion of a Kummer-faithful field, defined by Mochizuki, is expected as one of suitable base fields for anabelian geometry. In this paper, we study Kummer-faithfulness for algebraic extension fields of $p$-adic fields. We show that Kummer-faithfulness for such fields are deeply related with various finiteness properties on torsion points of (semi-)abelian varieties. For example, a Galois extension $K$ of a $p$-adic field is Kummer-faithful with finite residue field if and only if, for any finite extension $L$ of $K$ and any abelian variety over $L$,its $L$-rational torsion subgroup is finite. In addition, we study Kummer-faithfulness for Lubin-Tate extension fields.

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arXiv2025

Some Kummer extensions over maximal cyclotomic fields, a finiteness theorem of Ribet and TKND-AVKF fields

It is a theorem of Ribet that an abelian variety defined over a number field $K$ has only finitely many torsion points with values in the maximal cyclotomic extension field $K^{\mathrm{cyc}}$ of $K$. Recently, Rössler and Szamuely generalized Ribet's theorem in terms of the étale cohomology with $\mathbb{Q}/\mathbb{Z}$-coefficients of a smooth proper variety. In this paper, we show that the same finiteness holds even after replacing $K^{\mathrm{cyc}}$ with the field obtained by adjoining to $K$ all roots of all elements of a certain subset of $K$. Furthermore, we give some new examples of TKND-AVKF fields; the notion of TKND-AVKF is introduced by Hoshi, Mochizuki and Tsujimura, and TKND-AVKF fields are expected as one of suitable base fields for anabelian geometry.

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arXiv2023

Explicit bounds on torsion of CM abelian varieties over $p$-adic fields with values in Lubin-Tate extensions

Let $K$ and $k$ be $p$-adic fields. Let $L$ be the composite field of $K$ and a certain Lubin-Tate extension over $k$ (including the case where $L=K(μ_{p^{\infty}})$). In this paper, we show that there exists an explicitly described constant $C$, depending only on $K,k$ and an integer $g \ge 1$, which satisfies the following property: If $A_{/K}$ is a $g$-dimensional CM abelian variety, then the order of the $p$-torsion subgroup of $A(L)$ is bounded by $C$. We also give a similar bound in the case where $L=K(\sqrt[p^{\infty}]{K})$. Applying our results, we study bounds of orders of torsion subgroups of some CM abelian varieties over number fields with values in full cyclotomic fields.

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arXiv2022

Bounds on torsion of CM abelian varieties over a $p$-adic field with values in a field of $p$-power roots

Let $p$ be a prime number and $M$ the extension field of a $p$-adic field $K$ obtained by adjoining all $p$-power roots of all elements of $K$. In this paper, we show that there exists a constant $C$, depending only on $K$ and an integer $g>0$, which satisfies the following property:If $A_{/K}$ is a $g$-dimensional CM abelian variety, then the order of the torsion subgroup of $A(M)$ is bounded by $C$.

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arXiv2021

Torsion of algebraic groups and iterate extensions associated with Lubin-Tate formal groups

We show finiteness results on torsion points of commutative algebraic groups over a $p$-adic field $K$ with values in various algebraic extensions $L/K$ of infinite degree. We mainly study the following cases: (1) $L$ is an abelian extension which is a splitting field of a crystalline character (such as a Lubin-Tate extension). (2) $L$ is a certain iterate extension of $K$ associated with Lubin-Tate formal groups, which is familiar with Kummer theory.

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arXiv2020

A note on highly Kummer-faithful fields

We introduce a notion of highly Kummer-faithful fields and study its relationship with the notion of Kummer-faithful fields. We also give some examples of highly Kummer-faithful fields. For example, if $k$ is a number field of finite degree over $\mathbb{Q}$, $g$ is an integer $>0$ and $\mathbf{m}=(m_p)_p$ is a family of non-negative integers, where $p$ ranges over all prime numbers, then the extension field $k_{g,\mathbf{m}}$ obtained by adjoining to $k$ all coordinates of the elements of the $p^{m_p}$-torsion subgroup $A[p^{m_p}]$ of $A$ for all semi-abelian varieties $A$ over $k$ of dimension at most $g$ and all prime numbers $p$, is highly Kummer-faithful.

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arXiv2018

Torsion of abelian varieties and Lubin-Tate extensions

We show that, for an abelian variety defined over a $p$-adic field $K$ which has potential good reduction, its torsion subgroup with values in the composite field of $K$ and a certain Lubin-Tate extension over a $p$-adic field is finite.

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arXiv2017

Lattices in crystalline representations and Kisin modules associated with iterate extensions

Cais and Liu extended the theory of Kisin modules and crystalline representations to allow more general coefficient fields and lifts of Frobenius. Based on their theory, we classify lattices in crystalline representations by Kisin modules with additional structures under a Cais-Liu's setting. Furthermore, we give a geometric interpretation of Kisin modules of height one in terms of Dieudonné crystals of $p$-divisible groups, and show a full faithfulness theorem for a restriction functor on torsion crystalline representations.

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arXiv2015

Lattices in potentially semi-stable representations and weak $(φ,\hat{G})$-modules

Let $p$ be a prime number and $r$ a non-negative integer. In this paper, we prove that there exists an anti-equivalence between the category of weak $(φ,\hat{G})$-modules of height $r$ and a certain subcategory of the category of Galois stable lattices in potentially semi-stable $p$-adic representations with Hodge-Tate weights in $[0,r]$. This gives an answer to a Tong Liu's question about the essential image of a functor on weak $(φ,\hat{G})$-modules. For a proof, following Liu's methods, we construct linear algebraic data which classify lattices in potentially semi-stable representations.

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arXiv2014

On Galois equivariance of homomorphisms between torsion potentially crystalline representations

Let K be a complete discrete valuation field of mixed characteristic (0,p) with perfect residue field. Let (π_n)_{n\ge 0} be a system of p-power roots of a uniformizer π=π_0 of K with π^p_{n+1}=π_n, and define G_s (resp.\ G_{\infty}) the absolute Galois group of K(π_s) (resp.\ K_{\infty}:=\bigcup_{n\ge 0} K(π_n)). In this paper, we study G_s-equivatiantness properties of G_{\infty}-equivariant homomorphisms between torsion (potentially) crystalline representations

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arXiv2013

On congruences of Galois representations of number fields

We give a criterion for two l-adic Galois representations of an algebraic number field to be isomorphic when restricted to a decomposition group, in terms of the global representations mod l. This is applied to prove a generalization of a conjecture of Rasmussen-Tamagawa under a semistablity condition, extending some results of one of the authors. It is also applied to prove a congruence result on the Fourier coefficients of modular forms.

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arXiv2012

Torsion representations arising from $(φ,\hat{G})$-modules

The notion of a $(φ,\hat{G})$-module is defined by Tong Liu in 2010 to classify lattices in semi-stable representations. In this paper, we study torsion $(φ,\hat{G})$-modules, and torsion p-adic representations associated with them, including the case where p=2. First we prove that the category of torsion p-adic representations arising from torsion $(φ,\hat{G})$-modules is an abelian category. Secondly, we construct a maximal (minimal) theory for $(φ,\hat{G})$-modules by using the theory of étale $(φ, \hat{G})$-modules, essentially proved by Xavier Caruso, which is an analogue of Fontaine's theory of étale $(φ,Γ)$-modules. Non-isomorphic two maximal (minimal) objects give non-isomorphic two torsion p-adic representations.

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arXiv2011

Non-existence of certain CM abelian varieties with prime power torsion

In this paper, we study a conjecture of Rasmussen and Tamagawa, on the finiteness of the set of isomorphism classes of abelian varieties with constrained prime power torsion. Our result is related with abelian varieties which have complex multiplication over their fields of definition.

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arXiv2010

Non-existence of certain Galois representations with a uniform tame inertia weight

In this paper, we prove the non-existence of certain semistable Galois representations of a number field. Our consequence can be applied to some geometric problems. For example, we prove a special case of a Conjecture of Rasmussen and Tamagawa, related with the finiteness of the set of isomorphism classes of abelian varieties with constrained prime power torsion.

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