Product formulae on elliptic curves
Let \(E\) be an elliptic curve, \(D\) a divisor on \(E\) linearly equivalent to 0, and \(P\) a generic point of \(E\). Choosing a regular differential \(\omega\) on \(E\) allows one to normalize the choice of function with divisor \(D\). Call this function \(h_D (P, E, \omega)\). Let \(\varphi : E \to E'\) be a separable isogeny and let \(\omega'\) be such that \(\omega = \varphi^* \omega'\). Define \(\alpha (D,E, \omega, \varphi)\) by \[ \prod h_D (P - Q,E, \omega) = \alpha (D,E, \omega, \varphi) h_{\varphi (D)} (\varphi (P),\;E', \omega'), \] where the product is over \(Q \in \text{Ker} \varphi\). The purpose of the paper is to study \(\alpha (D,E, \omega, \varphi)\). The main result is that there is a monoid \({\mathcal M}\) of endomorphisms of \(E\) such that \(\alpha (\varphi)\) is a Hecke character for \(\varphi \in {\mathcal M}\). As an example, the author uses the curve \(E : y^2 = 4x^3 - 1\) and the divisor \(D\) of \(x\) to deduce cubic reciprocity. -- Special product formulae of the type studied in the present paper were used by \textit{C. R. Mathews} [Invent. Math. 52, 163-185 (1979; Zbl 0388.10025); 54, 23-52 (1979; Zbl 0414.10036)] in his work on cubic and quartic Gauss sums.
scientific article; zbMATH DE number 617413