研究者詳細 - 萩原　啓

萩原　啓（ハギハラ　ケイ）

Hagihara, Kei

写真a

所属（所属キャンパス）: 理工学研究科（矢上）

職名: 特任講師（有期）

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

Asakura M., Hagihara K.

Tunisian Journal of Mathematics 8 （ 3 ） 453 - 496 2026年

ISSN 25767658

　概要を見る

Recently, Kedlaya proved a formula which explicitly describes the Frobenius structure on certain p-adic hypergeometric equations. In this paper, we present a generalization of his formula, which is applicable to cases the original does not cover. A striking feature of our generalized formula is that, in these newly covered cases, the Frobenius matrix is expressed by the p-adic polygamma values and, consequently, by p-adic L-values for Dirichlet characters. As an application to p-adic geometry, we show that, for a projective smooth family whose Picard–Fuchs equation is a hypergeometric one, the Frobenius matrix on the corresponding log crystalline cohomology is described in terms of some values of the logarithmic function and p-adic L-functions.
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The Hodge realization of the polylogarithm and the Shintani generating class for totally real fields

Bannai K., Bekki H., Hagihara K., Ohshita T., Yamada K., Yamamoto S.

Advances in Mathematics 448 2024年06月

ISSN 00018708

　概要を見る

In this article, we construct the Hodge realization of the polylogarithm class in the equivariant Deligne–Beilinson cohomology of a certain algebraic torus associated to a totally real field. We then prove that the de Rham realization of this polylogarithm gives the Shintani generating class, a cohomology class generating the values of the Lerch zeta functions of the totally real field at nonpositive integers. Inspired by this result, we give a conjecture concerning the specialization of this polylogarithm class at torsion points, and discuss its relation to the Beilinson conjecture for Hecke characters of totally real fields.
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CANONICAL EQUIVARIANT COHOMOLOGY CLASSES GENERATING ZETA VALUES OF TOTALLY REAL FIELDS

Bannai K., Hagihara K., Yamada K., Yamamoto S.

Transactions of the American Mathematical Society Series B （Transactions of the American Mathematical Society Series B） 10 613 - 635 2023年

　概要を見る

It is known that the special values at nonpositive integers of a Dirichlet L-function may be expressed using the generalized Bernoulli numbers, which are defined by a generating function. The purpose of this article is to consider the generalization of this classical result to the case of Hecke L- functions of totally real fields. Hecke L-functions may be expressed canonically as a finite sum of zeta functions of Lerch type. By combining the non-canonical multivariable generating functions constructed by Shintani [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), pp. 393—417], we newly construct a canonical class, which we call the Shintani generating class, in the equivariant cohomology of an algebraic torus associated to the totally real field. Our main result states that the specializations at torsion points of the derivatives of the Shin- tani generating class give values at nonpositive integers of the zeta functions of Lerch type. This result gives the insight that the correct framework in the higher dimensional case is to consider higher equivariant cohomology classes instead of functions.References
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p-adic polylogarithms and p-adic Hecke L-functions for totally real fields

Bannai K., Hagihara K., Yamada K., Yamamoto S.

Journal fur die Reine und Angewandte Mathematik （Journal fur die Reine und Angewandte Mathematik） 2022 （ 791 ） 53 - 87 2022年10月

ISSN 00754102

　概要を見る

The purpose of this article is to newly define the p-adic polylogarithm as an equivariant class in the cohomology of a certain infinite disjoint union of algebraic tori associated to a totally real field. We will then express the special values of p-adic L-functions interpolating nonpositive values of Hecke L-functions of the totally real field in terms of special values of these p-adic polylogarithms.
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