Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 057, 14 pages      arXiv:2004.05099      https://doi.org/10.3842/SIGMA.2020.057

On Frobenius' Theta Formula

Alessio Fiorentino and Riccardo Salvati Manni
Sapienza Università di Roma, Italy

Received April 14, 2020, in final form June 11, 2020; Published online June 17, 2020

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.

Key words: hyperelliptic curves; theta functions; Jacobians of hyperelliptic curves; Kummer variety.

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