Division by 2 on hyperelliptic curves and jacobians

2016/06/16 by Zarhin, Yuri G.
#11G10 #14G27 #14H40 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1606.05252

Abstract

Let K be an algebraically closed field of characteristic different from 2, g a positive integer, f(x) a degree (2g+1) polynomial with coefficients in K and without multiple roots, C: y²=f(x) the corresponding genus g hyperelliptic curve over K and J the jacobian of C. We identify C with the image of its canonical embedding into J (the infinite point of C goes to the zero point of J). For each point P=(a,b)∈ C(K) there are 2^2g points (1)/(2)P ∈ J(K). We describe explicitly the Mumford represesentations of all (1)/(2)P. The rationality questions for (1)/(2)P are also discussed.

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