Jacobi's formula for Hesse cubic curves
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)].
scientific article; zbMATH DE number 5723517