Let #(u) and #(u) be the usual functions in the theory of elliptic functions. The following two formulae were found in the nineteenth-century. First one is

HYPERELLIPTICFUNCTIONSINGENUSTHREE7

Lemma2.3.Let beanelementofΛ.Thefunctionu→σ(u)onC3satis esthetranslationalformula

σ(u+ )=χ( )σ(u)expL(u+ , ),

whereχ( )=±1isindependentofu,L(u,v)isaformwhichisbilinearoverthe√real eldandC-linearwithrespecttothe rstvariableu,andL( 1, 2)is2π

u(j) u(k)logσ(u), jk···r(u)=

uσ(u),σjk···r(u)=(j)