Hironori Shiga | Searcharxiv

arXiv2023

On Kummer-like surfaces attached to singularity and modular forms

We study a family of lattice polarized $K3$ surfaces which is an extension of the family of Kummer surfaces derived from principally polarized Abelian surfaces. Our family has two special properties. First, it is coming from a resolution of a simple $K3$ singularity. Second, it has a natural parametrization by Hermitian modular forms of four complex variables. In this paper, we show two results: (1) We determine the transcendental lattice and the Néron-Severi lattice of a generic member of our family. (2) We give a detailed description of the double covering structure associated with our $K3$ surfaces.

math.AG↗

arXiv2020

Geometric interpretation of Hermitian modular forms via Burkhardt invariants

We give a complete theta expression of a pair of Hermitian modular forms as an inverse period mapping of lattice polarized $K3$ surfaces. Our result gives a non-trivial relation among moduli of $K3$ surfaces, theta functions and the finite complex reflection group of rank $5$.

math.NT↗

arXiv2017

To the Hilbert class field from the hypergeometric modular function

In this article we make an explicit approach to the higher degree case of the problem: " For a given $CM$ field $M$, construct its maximal abelian extension $C(M)$ (i.e. the Hilbert class field) by the adjunction of special values of certain modular functions" in a restricted case. We make our argument based on Shimura's main result on the complex multiplication theory of his article in 1967. His main result is constructed for a quaternion algebra $B$ over a totally real number field $F$. We determine the modular function which gives the canonical model for the case $B$ is coming from an arithmetic triangle group. That is our main theorem. And we make an explicit case-study for $B$ corresponding to the triangle group $Δ(3,3,5)$. The corresponding canonical model appears as a restriction of the Appell's hypergeometric modular function on a 2-dimensional hyperball to a hyperplane section. That is a modular function for the family of the Koike pentagonal curves $w^5=z(z-1)(z-λ_1)(z-λ_2)$ with two parameters $λ_1,λ_2$. We use the result of K. Koike in 2003 to get an theta representation of the canonical model function. By using this expression, we show several examples of the Hilbert class fields of the $CM$ fields those are embedded in the above $B$.

math.NT↗