Halves of points of an odd degree hyperelliptic curve in its jacobian

2018/07/18 by Zarhin, Yuri G.
#11G10 #14G27 #14H40 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1807.07008

Abstract

Let f(x) be a degree (2g+1) monic polynomial with coefficients in an algebraically closed field K with char(K)≠ 2 and without repeated roots. Let \mathfrakR⊂ K be the (2g+1)-element set of roots of f(x). Let C: y²=f(x) be an odd degree genus g hyperelliptic curve over K. Let J be the jacobian of C and J[2]⊂ J(K) the (sub)group of its points of order dividing 2. We identify C with the image of its canonical embedding into J (the infinite point of C goes to the identity element of J). Let P=(a,b)∈ C(K)⊂ J(K) and M_1/2,P⊂ J(K) the set of halves of P in J(K), which is J[2]-torsor. In a previous work we established an explicit bijection between M_1/2,P and the set of collections of square roots \mathfrakR_1/2,P:=\\mathfrakr: \mathfrakR → K| \mathfrakr(α)²=a-α ∀ α∈\mathfrakR; ∏_α∈\mathfrakR \mathfrakr(α)=-b\. The aim of this paper is to describe the induced action of J[2] on \mathfrakR_1/2,P (i.e., how signs of square roots \mathfrakr(α)=√(a-α) should change).

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