$p$-adic Manifold Learning and Benchmark Tasks from Impartial Games
We introduce $p$-adic manifold learning, propose an algorithm to solve it, and propose benchmark tasks from impartial games.
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Tomoki Mihara
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Published work
8 published item(s)
preprint2016arXiv
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 $\overline{k}$.
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preprint2015arXiv
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.
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preprint2015arXiv
Spectral Theory for $p$-adic Operators
We establish an algorithm for a criterion of the diagonalisability of a matrix over a local field by a unitary matrix. For this sake, we define the notion of normality of a $p$-adic operator, and give several criteria for the normality. We study the relation between the normality and the reduction. In the finite dimensional case, the normality of an operator is equivalent to the diagonalisability of a matrix by a unitary matrix. Therefore we also study the relation between the diagonalisability and the reduction. For example, we show that the diagonalisation of the reduction gives a partition of unity corresponding to the reduction of the spectrum, which gives a functorial lift of the eigenspace decomposition of the reduction.
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preprint2014arXiv
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.
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preprint2013arXiv
Iwasawa Theory for Locally Profinite Groups
We establish duality theory of p-adic unitary Banach representations of locally profinite groups. This is an extension of Iwasawa theory for profinite groups by P. Schneider and J. Teitelbaum. We also establish a criterion for an irreducibility of a unitary representation by certain simpleness of the Iwasawa module dual to it. Through the duality, the continuous induction of a unitary Banach representation of a closed subgroup is interpreted as a certain induction of Iwasawa module. It gives an explicit description of the dual of a continuous parabolic induction on GL_n(Q_p).
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preprint2013arXiv
Semisimplicity and Reduction of p-adic Representations of Topological Monoids
We give a criterion of the semisimplicity of a p-adic unitary representation of a topological monoid by the reduction of the associated operator algebra.
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preprint2013arXiv
Set Theory and p-adic Algebras
We verified that the existence of a maximal ideal of height 0 in a p-adic algebra in a certain class is independent of the axiom of ZFC. We established the theory on a P-point in the boundary of a topological space in the universal totally disconnected Hausdorff compactification. It is quite similar with the theory on a P-point in the boundary of a topological space in the Stone-Cech compactification. The latter theory relies on the real analysis, and the reason why the real analysis works for it is because the Stone-Cech compactification has the lifting property for a real bounded continuous function. On the other hand, the universal totally disconnected Hausdorff compactification does not have the lifting property for a real bounded continuous function in general, and hence the same technique with the real analysis is not valid for the former theory. We applied the p-adic analysis instead, and it yields a relation with a P-point in the boundary in the universal totally disconnected Hausdorff compactification and a maximal ideal of height 0 in the corresponding p-adic algebra.
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