Jacobi's formula for Hesse cubic curves
type: Item
instance of: scholarly article
title: Jacobi's formula for Hesse cubic curves
zbMATH Open document ID: 1197.14032
author: J. Martínez
published in: Acta Mathematica Vietnamica
publication date: 2010-06-18T00:00:00Z
full work available at URL: Toc ACTA 1 35.htm
review text: Let \(\mu \in \mathbb{C} \setminus \{1,\omega,\omega^2\}\) where \(\omega=\exp(2i\pi/3)\) and \(C(\mu) : t_0^3+t_1^3+t_2^3-3 t_0 t_1 t_2=0\) be the Hesse model of an elliptic curve. The authors do for this model a classical and careful study of periods and relations to theta functions in analogy with the study on the Weierstrass model. They first express a basis of the periods \(\psi_A(\mu),\psi_B(\mu)\) in terms of hypergeometric and beta functions and study the monodromy. Then they reprove a result of [\textit{A. Krazer}, Lehrbuch der Thetafunktionen. New York: Chelsea Publishing Company (1970; Zbl 0212.42901)] relating \(\mu\) to quotients of theta functions with characteristics evaluated at \(\tau=\psi_A(\mu)/\psi_B(\mu)\). They use this result to derive an analogue of Thomae's formula (or Jacobi's formula) relating the value of the previous theta functions at \(\tau\), the periods and \(\mu\). As corollary, they get a relation between the value of theta function at \(\tau=\omega\) and the value of the Gamma function at \(1/3\) which can also be obtained from the Chowla-Selberg formula in [\textit{A. Chowla} and \textit{S. Selberg}, J. Reine Angew. Math. 227, 86--110 (1967; Zbl 0166.05204)].
reviewed by: Christophe Ritzenthaler
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14H42
zbMATH DE Number: 5723517
zbMATH Keywords: periods
zbMATH Keywords: elliptic curve
zbMATH Keywords: monodromy
zbMATH Keywords: cubic curve
zbMATH Keywords: Thomae formula
zbMATH Keywords: theta function
zbMATH Keywords: hypergeometric function
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5723517