Determinant formulae in abelian functions for a general trigonal curve of degree five
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 this result for matrices of arbitrary size as in the classical formula, but the entries of the matrices are monomials in \(x\) and \(y\) instead of derivatives. The author also introduces a formula for \(C\) that is similar to the Kiepert formula [\textit{L. Kiepert}, J. Reine Angew. Math. 76, 21--33 (1873; JFM 05.0259.01)].
scientific article; zbMATH DE number 6017913