On Frobenius' Theta Formula

Symmetry Integrability and Geometry-Methods and Applications

article

Alessio Fiorentino¹  Riccardo Salvati Manni¹ 

[1]Sapienza Università di Roma

关键词: hyperelliptic curves;    theta functions;    Jacobians of hyperelliptic curves;    Kummer variety;   

DOI  :  10.3842/SIGMA.2020.057

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown