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Tomoki Mihara

Publications and source records attributed to Tomoki Mihara.

At least 19 recordsLinked to original sources
arXiv2026

Transcendence of Tonelli--Shanks Power Modulo Infinitely Large Primes

For a prime number $p$, we denote by $k_p$ the greatest odd divisor of $p-1$. Tonelli--Shanks algorithm is a classical algorithm to compute a square root of a quadratic residue $n \in \mathbb{Z}$ modulo $p$ as the product of $n^{\frac{k_p+1}{2}}$, which we call {\it Tonelli--Shanks power of $n$}, and a correction term given as a power of a quadratic non-residue modulo $p$. We prove the transcendence over $\mathbb{Q}$ of the images of the following in the ring $\mathscr{A}$ of integers modulo infinitely large primes: the greatest odd divisor $k_p$ of $p-1$, the composite $f(\frac{p - 1}{k_p})$ of any injective map $f \colon \mathbb{Z} \to \mathbb{Z}$ and the greatest $2$-power $\frac{p - 1}{k_p}$ dividing $p-1$, Tonelli--Shanks power $n^{\frac{_p+1}{2}}$ of any $n \in \mathbb{Z} \setminus \{-1,0,1\}$, and the correction term of Tonelli--Shanks algorithm for such $n$.

math.NT↗
arXiv2026

Generalisation of Baker's Forcing Method to Arbitrary Prime and NP-hardness of Several $p$-adic Optimisations

G.\ D.\ Baker formulated a forcing method to interpret integer optimisation problem into $2$-adic linear regression, and proved the NP-hardness of $2$-adic linear regression. We generalise the forcing method to a wider class of $p$-adic optimisation for the case where $p$ is not necessarily $2$, and prove the NP-hardness of $p$-adic linear regression, the NP-hardness of $2$-adic dynamic neural network by S.\ Albeverio, A.\ Khrennikov, and B.\ Tirrozi, and the NP-hardness of a partial generalisation of the $p$-adic optimisation problem associated to van der Put neural network by G.\ L.\ R.\ N'guessan.

cs.CC↗
arXiv2026

Schneider--Teitelbaum Duality over a Non-spherically Complete Field

We formulate Schneider--Teitelbaum duality between wide classes of Banach $k$-linear representations of $G$ and left $O_k[[G]]$-modules for a non-spherically complete field $k$, e.g.\ $\mathbb{C}_p$, and a profinite group $G$. We interpret a topological notion of a weak variant of irreducibility of a Banach $k$-linear representation of $G$ into a purely algebraic notion of a certain simplicity of the dual left $O_k[[G]]$-module. As applications, we give two $p$-adic families of infinite dimensional Banach $\mathbb{C}_p$-linear representations of a $p$-adic Lie group satisfying the weak irreducibility.

math.NT↗
arXiv2026

Notes on Algebraic Properties and Non-Standard Analysis of the Ring of Integers Modulo Infinitely Large Primes

As applications of non-standard analysis to the study of transcendence in the ring $\mathscr{A}$ of integers modulo infinitely large primes, we extend transcendence criteria by Anzawa--Funakura and Matsusaka--Seki. Using the extension of the criterion by Anzawa--Funakura, we observe conditional relation between asymptotic behaviour of prime gaps and transcendence in $\mathscr{A}$. We also summarise known algebraic and model theoretic results on $\mathscr{A}$ for number theorists, and share topics in transcendental number theory with algebraists and model theorists.

math.NT↗
arXiv2026

Almost Free Non-Archimedean Banach Spaces and Relation to Large Cardinals

Let $k$ be a complete valuation field. We formulate a free Banach $k$-vector space as a Banach $k$-vector space with an orthonormal Schauder basis, and an almost free Banach $k$-vector space as a non-Archimedean analogue of an almost free Abelian group. As non-Archimedean analogues of the classical facts that an almost free Abelian group is free under the assumption of the $\aleph_1$-strong compactness or the weak compactness of the cardinality, we show that an almost free Banach $k$-vector space is free under similar assumptions.

math.LO↗
arXiv2026

Structural Hierarchy of Reid Class of non-Archimedean Banach Spaces

Let $k$ be a complete valuation field. We formulate a class $\mathscr{R}$ of Banach $k$-vector spaces analogous to Reid class of Abelian groups. We formulate an analogue of the hierarchy of Reid class introduced by K.\ Eda, and verify a counterpart of the classification theorem of Reid class by K.\ Eda. As an application, we verify that the Banach $\mathbb{C}_p$-vector spaces \begin{eqnarray*} & & \ell^{\infty}(\mathbb{N},\mathbb{C}_p), \text{\rm C}_0(\mathbb{N},\mathbb{C}_p), \ell^{\infty}(\mathbb{N},\text{\rm C}_0(\mathbb{N},\mathbb{C}_p)), \text{\rm C}_0(\mathbb{N},\ell^{\infty}(\mathbb{N},\mathbb{C}_p)), \\ & & \ell^{\infty}(\mathbb{N},\text{\rm C}_0(\mathbb{N},\ell^{\infty}(\mathbb{N},\mathbb{C}_p))), \text{\rm C}_0(\mathbb{N},\ell^{\infty}(\mathbb{N},\text{\rm C}_0(\mathbb{N},\mathbb{C}_p))), \end{eqnarray*} and so on are all distinct, the Banach $\mathbb{C}_p$-vector space of bounded continuous functions $\mathbb{Q} \to \mathbb{C}_p$ and its dual Banach $\mathbb{C}_p$-vector spaces cannot be expressed by iterated application of bounded direct product and completed direct sum, and there is no left adjoint functor of the forgetful functor from $\mathscr{R}$ to the category of Banach $\mathbb{C}_p$-vector spaces.

math.LO↗
arXiv2026

$p$-adic Character Neural Network

We propose a new frame work of $p$-adic neural network. Unlike the original $p$-adic neural network by S.\ Albeverio, A.\ Khrennikov, and B.\ Tirrozi using a family of characteristic functions indexed by hyperparameters of precision as activation functions, we use a single injective $p$-adic character on the topological Abelian group $\mathbb{Z}_p$ of $p$-adic integers as an activation function. We prove the $p$-adic universal approximation theorem for this formulation of $p$-adic neural network, and reduce it to the feasibility problem of polynomial equations over the finite ring of integers modulo a power of $p$.

math.NT↗
arXiv2026

Non-Archimedean Analogue of Chase's Lemma

We formulate and verify a non-Archimedean analogue of Chase's lemma. Following the framework by K.\ Eda removing restriction of cardinality from analogy on direct product between countability and non-$\omega_1$-measurability, we extend the non-Archimedean analogue of Chase's lemma to a non-Archimedean counterpart of the extension by K.\ Eda of the extension by M.\ Dugas and B.\ Zimmermann-Huisgen of Chase's lemma.

math.LO↗
arXiv2026

Bounded Additive Relation and Application to Finite Multiple Zeta Values

We formulate an algebraic problem to find a generating system of a finite subset of an Abelian group with respect to linear relations whose coefficients are bounded by a constant, and recall MITM algorithm for the problem. As an application of MITM algorithm for the Abelian group \begin{eqnarray*} \mathbb{Z}/106700590455862347842907841856033238416352421 \mathbb{Z} \end{eqnarray*} combined with Chinese remainder algorithm, we give a table of expected linear relations of finite multiple zeta values of weight $10$.

math.NT↗
arXiv2022

Banach halos and short isometries

The aim of this article is twofold. First, we develop the notion of a Banach halo, similar to that of a Banach ring, except that the usual triangular inequality is replaced by the inequality $|a + b| \leq (|a| , |b|)_p$ involving the p-norm for some $p \in]0, +\infty]$, or by the inequality $|a+b|\leq C\max(|a|,|b|)$. This allows us to have a flow of powers on Banach halos and to work, e.g., with the square of the usual absolute value on $\mathbb{Z}$. Then we define and study the group of short isometries of normed involutive coalgebras over a base commutative Banach halo. An aim of this theory is to define a representable group $K_n\subset {\rm GL}_n$ whose points with values in $\mathbb{R}$ give $O_n(\mathbb{R})$ and whose points with values in $\mathbb{Q}_p$ give GL$_n(\mathbb{Z}_p)$, giving to the analogy between these two groups a kind of geometric explanation.

math.AG↗
arXiv2021

Derived Analytic Geometry for Z-Valued Functions. Part I -- Topological Properties

We study the Banach algebras ${\rm C}(X, R)$ of continuous functions from a compact Hausdorff topological space $X$ to a Banach ring $R$ whose topology is discrete. We prove that the Berkovich spectrum of ${\rm C}(X, R)$ is homeomorphic to $\zeta(X) \times {\mathcal M}(R)$, where $\zeta(X)$ is the Banaschewski compactification of $X$ and ${\mathcal M}(R)$ is the Berkovich spectrum of $R$. We study how the topology of the spectrum of ${\rm C}(X, R)$ is related to the notion of homotopy Zariski open embedding used in derived geometry. We find that the topology of $\zeta(X)$ can be easily reconstructed from the homotopy Zariski topology associated to ${\rm C}(X, R)$. We also prove some results about the existence of Schauder bases on ${\rm C}(X, R)$ and a generalisation of the Stone--Weierstrass Theorem, under suitable hypotheses on $X$ and $R$.

math.AG↗
arXiv2021

Homotopy Epimorphisms and Derived Tate's Acyclicity for Commutative C*-algebras

We study homotopy epimorphisms and covers formulated in terms of derived Tate's acyclicity for commutative C*-algebras and their non-Archimedean counterparts. We prove that a homotopy epimorphism between commutative C*-algebras precisely corresponds to a closed immersion between the compact Hausdorff topological spaces associated to them, and a cover of a commutative C*-algebra precisely corresponds to a topological cover of the compact Hausdorff topological space associated to it by closed immersions admitting a finite subcover. This permits us to prove derived and non-derived descent for Banach modules over commutative C*-algebras.

math.AG↗
arXiv2015

Galois Representations Associated to $p$-adic Families of Modular Forms of Finite Slope

We define a pro-$p$ Abelian sheaf on a modular curve of a fixed level $N \geq 5$ divisible by a prime number $p \neq 2$. Every $p$-adic representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ associated to an eigenform is obtained as a quotient of its étale cohomology. For any compact $\mathbb{Z}_p[[1 + N \mathbb{Z}_p]]$-algebra $Λ_1$ satisfying certain suitable conditions, we construct a representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ over $Λ_1$ associated to a $Λ_1$-adic cuspidal eigenform of finite slope as a scalar extension of a quotient of the étale cohomology.

math.NT↗
arXiv2015

Hahn--Banach Theorem and Duality Theory on non-Archimedean Locally Convex Spaces

Let $k$ be a local field with valuation ring $O_k$ and residue field $\overline{k}$. We extend Hahn--Banach theorem for the class of seminormed $k$-vector spaces to several classes of locally convex spaces and subspaces over $k$, $O_k$, and $\overline{k}$. We establish analogues of Iwasawa-type duality for several classes of locally convex spaces over $k$, $O_k$, and $\overline{k}$.

math.NT↗
arXiv2014

Characterisation of the Berkovich Spectrum of the Banach Algebra of Bounded Continuous Functions

For a complete valuation field k and a topological space X, we prove the universality of the underlying topological space of the Berkovich spectrum of the Banach k-algebra Cbd(X,k) of bounded continuous k-valued functions on X. This result yields three applications: a partial solution to an analogue of Kaplansky conjecture for the automatic continuity problem over a local field, comparison of two ground field extensions of Cbd(X,k), and non-Archimedean Gel'fand theory.

math.NT↗