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Yoshihiro Ônishi

Publications and source records attributed to Yoshihiro Ônishi.

12 recordsLinked to original sources
arXiv2021

Arithmetic over the Gaussian Number Field on a Certain Family of Elliptic Curves with Complex Multiplication

This work is a sequel of a previous work of one of the authors (Y.Ô), which treated certain congruence relation between an elliptic Gauss sum and a coefficient of power series expansion at the origin of the lemniscate sine function. We extend the previous result (in \cite{O}) which concerned only for non-vanishing elliptic Gauss sums. We give new congruence relations between power series coefficients of the lemniscate cosine function, which hold if and only if the corresponding elliptic Gauss sum vanishes.

math.NT↗
arXiv2015

Hurwitz integrality of power series expansion of the sigma function for a plane curve

This paper shows Hurwitz integrality of the coefficients of expansion at the origin of the sigma function \(σ(u)\) associated to a certain plane curve which should be called a plane telescopic curve. For the prime \(2\), the expansion of \(σ(u)\) is not Hurwitz integral, but \(σ(u)^2\) is. This paper clarifies the precise structure of this phenomenon. Throughout the paper, computational examples for the trigonal genus three curve (\((3,4)\)-curve) \(y^3+(μ_1x+μ_4)y^2+(μ_2x^2+μ_5x+μ_8)y=x^4+μ_3x^3+μ_6x^2+μ_9x+μ_{12}\) (\(μ_j\) are constants) are given.

math.NT↗
arXiv2014

Some New Addition Formulae for Weierstrass Elliptic Functions

We present new addition formulae for the Weierstrass functions associated with a general elliptic curve. We prove the structure of the formulae in n-variables and give the explicit addition formulae for the 2- and 3-variable cases. These new results were inspired by new addition formulae found in the case of an equianharmonic curve, which we can now observe as a specialisation of the results here. The new formulae, and the techniques used to find them, also follow the recent work for the generalisation of Weierstrass' functions to curves of higher genus.

math.AG↗
arXiv2004

Theory of The Generalized Bernoulli-Hurwitz Numbers for The Algebraic Functions of Cyclotomic Type and The Universal Bernoulli Numbers

Hurwitz numbers are the Laurent coefficients of an elliptic function $\wp(u)$ of cyclotomic type, and they are natural generalization of the Bernoulli numbers. This paper gives new generalization of Bernoulli and Hurwitz numbers for higher genus cases. They satisfy completely von Staudt-Clausen type theorem, an extension of von Staudt second theorem, and Kummer type congruence relation. The present paper is revised and combined version of math.NT/0304377 and math.NT/0312178 containing many numerical examples.

math.NT↗
arXiv2002

Wave-Particle Complementarity and Reciprocity of Gauss Sums on Talbot Effects

Berry and Klein (J. Mod. Opt. (1997) 43 2139-2164) showed that the Talbot effects in classical optics are naturally explained by Gauss sums studied in number theory. Their result was based on Helmholtz equation. In this article, we explain the effect based on Fresnel integral also by Gauss sums. These two explanations are shown to agree with by the reciprocity law of Gauss sums. The relation between this agreement and the wave-particle complementarity is also discussed.

math-ph↗