An Explicit Theorem of the Square for Hyperelliptic Jacobians

LetA be an abelian variety over a field k, D a symmetric divisor onA, s and d the sum and difference maps fromA×A intoA, andp1 andp2 the projections onto the first and second factors. The theorem of the square and the seesaw principle [M1, Secs. 5, 6] guarantee that there exists a function f(u, v) on A×A (determined up to constant multiples) with divisor s∗D+ d ∗D− 2p∗ 1D− 2p∗2D. Since this function encodes all the information about the group morphism on A, it is useful to know f(u, v) explicitly. Indeed, if a, b, c ∈A and if Dc is the image of D under the translation-by-cmap, then the divisor of f ( u− a+b 2 ,− a+b 2 )/ f ( u− a+b 2 , −a+b 2 ) isDa+b+D−Da−Db,which is the theorem of the square forD. If k is the complex numbers, then the construction of f is classical. One merely takes a theta function θ with divisor D (see e.g. [La]); then f(u, v) = θ(u+ v)θ(u− v)/θ(u)θ(v),