arXiv subjects

Kei Hagihara

Publications and source records attributed to Kei Hagihara.

7 recordsLinked to original sources

arXiv2023

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.

math.NT↗

arXiv2023

The Hodge Realization of the Polylogarithm and the Shintani Generating Class for Totally Real Fields

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.

math.NT↗

arXiv2022

$p$-adic Polylogarithms and $p$-adic Hecke $L$-functions for Totally Real Fields

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.

math.NT↗

arXiv2020

Category of mixed plectic Hodge structures

The purpose of this article is to investigate the properties of the category of mixed plectic Hodge structures defined by Nekovář and Scholl. We give an equivalent description of mixed plectic Hodge structures in terms of the weight and partial Hodge filtrations. We also construct an explicit complex calculating the extension groups in this category.

math.AG↗

arXiv2020

Canonical Equivariant Cohomology Classes Generating Zeta Values of Totally Real Fields

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 canonical 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, 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 Shintani 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.

math.NT↗

arXiv2020

The Hodge realization of the polylogarithm on the product of multiplicative groups

The purpose of this article is to describe explicitly the polylogarithm class in absolute Hodge cohomology of a product of multiplicative groups, in terms of the Bloch-Wigner-Ramakrishnan polylogarithm functions. We will use the logarithmic Dolbeault complex defined by Burgos to calculate the corresponding absolute Hodge cohomology groups.

math.AG↗

arXiv2019

The global optimum of shallow neural network is attained by ridgelet transform

We prove that the global minimum of the backpropagation (BP) training problem of neural networks with an arbitrary nonlinear activation is given by the ridgelet transform. A series of computational experiments show that there exists an interesting similarity between the scatter plot of hidden parameters in a shallow neural network after the BP training and the spectrum of the ridgelet transform. By introducing a continuous model of neural networks, we reduce the training problem to a convex optimization in an infinite dimensional Hilbert space, and obtain the explicit expression of the global optimizer via the ridgelet transform.

stat.ML↗