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Masanori Asakura

Publications and source records attributed to Masanori Asakura.

At least 19 recordsLinked to original sources
arXiv2026

On the adelic Gaussian hypergeometric function

We define the adelic hypergeometric function of special Gaussian type by means of a tower of hypergeometric curves. This function takes values in an adelic completed group ring and interpolates all the hypergeometric functions of the same type over all finite fields. It specializes at the unit argument to the adelic beta function of Ihara and Anderson. We prove some transformation formulas and a summation formula for the adelic hypergeometric function, which are known classically for complex hypergeometric functions.

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arXiv2026

Periods of Limiting Mixed Hodge Structures of Projective Hypersurfaces

For a generic one-parameter degeneration of projective hypersurfaces, we show that the periods of the limiting mixed Hodge structure are generated by certain special values of logarithm, Gamma and Dirichlet $L$-functions. Our proof is based on the analytic continuation of solutions to the GKZ system.

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arXiv2025

Frobenius structure on hypergeometric equations, p-adic polygamma values and p-adic L-values

Recently, Kedlaya proves certain formula describing explicitly the Frobenius structure on a hypergeometric equation. In this paper, we give a generalization of it. In our case, the Frobenius matrix is no longer described by p-adic gamma function, and then we describe it by the p-adic polygamma functions. Since the p-adic polygamma values are linear combinations of p-adic L-values of Dirichlet characters, it turns out that the Frobenius matrix is described by p-adic L-values. Our result has an application to the study on Frobenius on p-adic cohomology. We show that, for a projective smooth family such that the Picard-Fuchs equation is a hypergeometric equation, the Frobenius matrix on the log-crystalline cohomology is described by some values of the logarithmic function and p-adic L-functions of Dirichlet characters.

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arXiv2023

New p-adic hypergeometric functions and syntomic regulators

We introduce new p-adic convergent functions, which we call the p-adic hypergeometric functions of logarithmic type. The first main result is to prove the congruence relations that are similar to Dwork's. The second main result is that the special values of our new functions appear in the syntomic regulators for hypergeometric curves, Fermat curves and some elliptic curves. According to the p-adic Beilinson conjecture by Perrin-Riou, they are expected to be related with the special values of p-adic L-functions. We provide one example for this.

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arXiv2022

A generalization of the Ross symbols in higher K-groups and hypergeometric functions II

This is a sequel of the paper "A generalization of the Ross symbols in higher K-groups and hypergeometric functions I" where we introduced higher Ross symbols in higher $K$-groups of the hypergeometric schemes, and discussed the Beilinson regulators. In this paper we give its p-adic counterpart and an application to the $p$-adic Beilinson conjecture for K3 surfaces of Picard number 20.

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arXiv2022

Milnor K-theory, F-isocrystals and Syntomic Regulators

We introduce a category of filtered F-isocrystals and construct a symbol maps on Milnor K-theory which is compatible with the syntomic symbol maps to the log syntomic cohomology. These are fundamental materials in our applications on syntomic regulators which we work in other papers.

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arXiv2021

A generalization of the Ross symbols in higher K-groups and hypergeometric functions I

The Ross symbol is defined to be an element {1-z,1-w\} in K_2 of a Fermat curve z^n+w^m=1. Ross showed that it is non-torsion by computing the Beilinson regulator. In this paper, we introduce a generalization of the Ross symbols in K_{d+1} of a variety (1-x_0^{n_0})\cdots(1-x_d^{n_d})=t. The main result is that the Beilinson regulator is described by the hypergeometric functions {}_{d+3}F_{d+2}'s.

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arXiv2021

Zeta functions of certain K3 families : application of the formula of Clausen

Based on the theory of rigid cohomology, we provide an explicit formula of zeta functions of certain K3 families, which we call the hypergeometric type. The central point of our argument is the comparison between the 2nd rigid cohomology of a K3 and the symmetric product of an elliptic curve, that is brought from the classical formula of Clausen.

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arXiv2020

An algorithm of computing special values of Dwork's p-adic hypergeometric functions in polynomial time

Dwork's $p$-adic hypergeometric function is defined to be a ratio ${}_sF_{s-1}(t)/{}_sF_{s-1}(t^p)$ of hypergeometric power series. Dwork showed that it is a uniform limit of rational functions, and hence one can define special values on $|t|_p=1$. However to compute the value modulo $p^n$ in the naive method, the bit complexity increases by exponential when $n\to\infty$. In this paper we present a certain algorithm whose complexity increases at most $O(n^4(\log n)^3)$.

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arXiv2019

Chern class and Riemann-Roch theorem for cohomology theory without homotopy invariance

In this paper, we formulate axioms of certain graded cohomology theory for which Chern class maps from higher K-theory are defined, following the method of Gillet [Gi1]. We will not include homotopy invariance nor purity in our axioms. It will turn out that the Grothendieck-Riemann-Roch theorem and the Riemann-Roch theorem without denominators hold for our higher Chern classes. We will also give two applications of our Riemann-Roch results.

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arXiv2019

Explicit logarithmic formulas of special values of hypergeometric functions 3F2

In a joint paper [4] by Otsubo, Terasoma and the first author, we proved that the special value 3F2(a,b,q;a+b,q;1) of the generalized hypergeometric function is a linear combination of log of algebraic numbers if the triplet (a,b,q) of rational numbers satisfies a certain numerical condition. However there remains a question how to obtain explicit descriptions of the values. In this paper, we give a method to do this, which is a further development of the technique in [4].

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arXiv2017

Regulators of K_2 of Hypergeometric Fibrations

We discuss Beilinson's regulator on K_2 of certain fibrations of algebraic varieties which we call the hypergeomtric fibrations. The main result is to describe regulators via the hypergeometric functions 3F2 or 4F3. We also discuss the Beilinson conjecture on the special values of L-functions.

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arXiv2017

Regulators of K_1 of Hypergeometric Fibrations

We study a deformation of what we call hypergeometric fibrations. Its periods and K_1-regulators are described in terms of hypergeometric functions 3F2 in a variable given by the deformation parameter.

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arXiv2016

CM periods, CM regulators and hypergeometric functions, I

We study the $H^2$ of certain surfaces with complex multiplication by a cyclotomic field. The periods are written in terms of values of the gamma function and the conjecture of Gross-Deligne is verified. The regulators of certain $K_1$-elements are written in terms of values of hypergeometric functions ${}_3F_2$, and we prove their non-vanishing.

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