Yoshihiro Ônishi

Abelian Functions for Trigonal Curves of Genus Three

We develop the theory of generalized Weierstrass sigma- and \wp-functions defined on a trigonal curve of genus three. In particular we give a list of the associated partial differential equations satisfied by the \wp-functions, a proof that the coefficients of the power series expansion of the sigma-function are polynomials of moduli parameters, and the derivation of two addition formulae.

Abelian functions for cyclic trigonal curves of genus 4

Abstract We discuss the theory of generalized Weierstrass σ and ℘ -functions defined on a trigonal curve of genus 4, following earlier work on the genus 3 case. The specific example of the “purely trigonal” (or “cyclic trigonal”) curve y 3 = x 5 + λ 4 x 4 + λ 3 x 3 + λ 2 x 2 + λ 1 x + λ 0 is discussed in detail, including a list of some of the associated partial differential equations satisfied by the ℘ -functions, and the derivation of addition formulae.

DETERMINANT EXPRESSIONS FOR HYPERELLIPTIC FUNCTIONS

In this paper we give an elegant generalization of the formula of Frobenius–Stickelberger from elliptic curve theory to all hyperelliptic curves. A formula of Kiepert type is also obtained by a limiting process from this generalization. In the appendix a determinant expression of D. G. Cantor is also derived.

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

Berry and Klein [J. Mod. Opt.43, 2139-2164 (1997)] showed that the Talbot effects in classical optics are naturally expressed by Gauss sums in number theory. Their result was obtained by a computation of Helmholtz equation. In this article, we calculate the effects using Fresnel integral and show that the result is also represented by Gauss sums. However function forms of these two computational results are apparently different. We show that the reciprocity law of Gauss sums connects these results and both completely agree with. The Helmholtz equation can be regarded as an equation based upon wavy nature in optics whereas the Fresnel integral is defined by a sum over the paths based upon a particle picture in optics. Thus the agreement of these two computational results could be interpreted in terms of the concept of the wave-particle complementarity, though the concept is for quantum mechanical phenomenon. This interpretation leads us to a relation between the reciprocity of Gauss sums in number theory and the wave-particle complementarity in wave physics. We discuss it in detail.

Abelian functions associated with genus three algebraic curves

We develop the theory of Abelian functions associated with algebraic curves. The growth in computer power and an advancement of efficient symbolic computation techniques has allowed for recent progress in this area. In this paper we focus on the genus three cases, comparing the two canonical classes of hyperelliptic and trigonal curves. We present new addition formulae, derive bases for the spaces of Abelian functions and discuss the differential equations such functions satisfy.

On the Moduli of a Quantized Elastica in ℙ and KdV Flows: Study of Hyperelliptic Curves as an Extension of Euler's Perspective of Elastica I

Quantization needs evaluation of all of states of a quantized object rather than its stationary states with respect to its energy. In this paper, we have investigated moduli of a quantized elastica, a quantized loop with an energy functional associated with the Schwarz derivative, on a Riemann sphere ℙ. Then it is proved that its moduli space is decomposed to a set of equivalent classes determined by flows obeying the Korteweg-de Vries (KdV) hierarchy which conserve the energy. Since the flow obeying the KdV hierarchy has a natural topology, it induces topology in the moduli space . Using the topology, is classified. Studies on a loop space in the category of topological spaces Top are well-established and its cohomological properties are well-known. As the moduli space of a quantized elastica can be regarded as a loop space in the category of differential geometry DGeom, we also proved an existence of a functor between a triangle category related to a loop space in Top and that in DGeom using the induced topology. As Euler investigated the elliptic integrals and its moduli by observing a shape of classical elastica on , this paper devotes relations between hyperelliptic curves and a quantized elastica on ℙ as an extension of Eulers perspective of elastica.

Addition formulae over the Jacobian pre-image of hyperelliptic Wirtinger varieties

Abstract Using Frobenius-Stickelberger-type relations for hyperelliptic curves (Y. Ônishi, Proc. Edinb. Math. Soc. (2) 48 (2005), 705–742), we provide certain addition formulae for any symmetric power of such curves, which hold on the strata Wk , the pre-images in the Jacobian of the classical Wirtinger varieties. In an appendix, we give similar relations for a trigonal curve y 3 = (x – b 1)(x – b 2)(x – b 3)(x – b 4).

Determinant expressions for abelian functions in genus two

In this paper we generalize the formula of Frobenius-Stickelberger and the formula of Kiepert type to the genus-two case.

Addition formulae for Abelian functions associated with specialized curves

We discuss a family of multi-term addition formulae for Weierstrass functions on specialized curves of low genus with many automorphisms, concentrating mostly on the case of genus 1 and 2. In the genus 1 case, we give addition formulae for the equianharmonic and lemniscate cases, and in genus 2 we find some new addition formulae for a number of curves.We discuss a family of multi-term addition formulae for Weierstrass functions on specialized curves of low genus with many automorphisms, concentrating mostly on the case of genus one and two. In the genus one case we give addition formulae for the equianharmonic and lemniscate cases, and in genus two we find some new addition formulae for a number of curves.

Determinant Formulae in Abelian Functions for a General Trigonal Curve of Degree Five

This paper gives a natural generalization of the Frobenius-Stickelberger formula and the Kiepert formula for elliptic functions [4, 5] to the curve of genus four defined by