On Frobenius' Theta Formula

Abstract

Mumford's well-known characterization of the hyperelliptic locus of the moduli space of ppavs in terms of vanishing and non-vanishing theta constants is based on Neumann's dynamical system. Poor's approach to the characterization uses the cross ratio. A key tool in both methods is Frobenius' theta formula, which follows from Riemann's theta formula. In a 2004 paper, Grushevsky gives a different characterization in terms of cubic equations in second-order theta functions. In this note, we first show the connection between the methods by proving that Grushevsky's cubic equations are strictly related to Frobenius' theta formula, and we then give a new proof of Mumford's characterization via Gunning's multisecant formula.