Determinant formulae in abelian functions for a general trigonal curve of degree five
type: Item
instance of: scholarly article
title: Determinant formulae in abelian functions for a general trigonal curve of degree five
zbMATH Open document ID: 1282.11067
author: Yoshihiro Ônishi
published in: Computational Methods and Function Theory
publication date: 2012-03-22T00:00:00Z
review text: The classical Frobenius-Stickelberger formula [\textit{G. Frobenius} and \textit{L. Stickelberger}, J. Reine Angew. Math. 83, 175--179 (1877; JFM 09.0347.02)] computes the determinant of the following matrix as a product of values of Weierstrass sigma functions: \[ \begin{bmatrix} 1 & \wp(u_0) & \wp'(u_0) & \cdots & \wp^{(n-1)}(u_0) \\ \vdots & \vdots & \vdots & \cdots & \vdots\\ 1 & \wp(u_n) & \wp'(u_n) & \cdots & \wp^{(n-1)}(u_n) \end{bmatrix} \] where \(\wp\) is the Weierstrass \(\wp\)-function and \(u_k\) are complex variables. In this paper under review the author introduces a determinant formula, which is similar to the classical one, for the curve \(C\) given by \[ \begin{aligned} y^3 + &(\mu_2 x + \mu_5)y^2 +(\mu_4 x^2 + \mu_7 x + \mu_{10})y\\ &= x^5 + \mu_3 x^4 + \mu_6 x^3 + \mu_9 x^2 + \mu_{12} + \mu_{15} \end{aligned} \] where \(\mu_j\) are constants, and the indices of \(u_j\) are chosen according to the homogeneous weight of each term. Let \(W\) be the image of the Abel-Jacobi map in \(\mathbb C^4/\Lambda\) where \(\Lambda\) is a lattice, let \(k\) be the projection map: \(\mathbb C^4 \to \mathbb C^4/\Lambda\), and let \(\sigma : \mathbb C^4 \to \mathbb C\) be the sigma function attached to \(C\). For each \(v\) in \(k^{-1}(W)\), find \((x,y)\) in \(C\) such that \(v=\lambda +\int_\infty^{(x,y)} \omega\) for some \(\lambda\in\Lambda\), and denote \[ v^* = \int_\infty^{(x,y^*)} \omega;\quad v^{**} = \int_\infty^{(x,y^{**})} \omega \] where \((x,y)\), \((x,y^*)\), and \((x,y^{**})\) denote the three points on \(C\) with a common \(x\)-coordinate. The author proves that for each \(u\) and \(v\) in \(k^{-1}(W)\), \[ - \frac{ \sigma_3(u+v)\, \sigma_3(u+v^*)\, \sigma_3(u+v^{**}) } {\sigma_2(u)^3 \sigma_2(v) \sigma_2(v^*) \sigma_2(v^{**}) } = \det\begin{bmatrix} 1 & x(u) \\ 1 & x(v) \end{bmatrix} \] where \(\sigma_2\) and \(\sigma_3\) are the partial derivatives with respect to the 2nd and the 3rd entries of \(u\), respectively. The author also generalizes thi…
reviewed by: Sungkon Chang
Mathematics Subject Classification ID: 11G30
Mathematics Subject Classification ID: 14H45
zbMATH DE Number: 6017913
zbMATH Keywords: curves of high genus
zbMATH Keywords: abelian functions
zbMATH Keywords: Jacobian varieties
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6017913