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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, $C:y^2=f(x)$ 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 identity element of $J$). It is well known that for each $mathfrak{b} in J(K)$ there are exactly $2^{2g}$ elements $mathfrak{a} in J(K)$ such that $2mathfrak{a}=mathfrak{b}$. M. Stoll constructed an algorithm that provides Mumford representations of all such $mathfrak{a}$, in terms of the Mumford representation of $mathfrak{b}$. The aim of this paper is to give explicit formulas for Mumford representations of all such $mathfrak{a}$, when $mathfrak{b}in J(K)$ is given by $P=(a,b) in C(K)subset J(K)$ in terms of coordinates $a,b$. We also prove that if $g>1$ then $C(K)$ does not contain torsion points with order between $3$ and $2g$.
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, $C: y^2=f(x)$ the corresponding genus $g$ hyperelliptic curv
We prove that the jacobian of a hyperelliptic curve $y^2=(x-t)h(x)$ has no nontrivial endomorphisms over an algebraic closure of the ground field $K$ of characteristic zero if $t in K$ and the Galois group of the polynomial $h(x)$ over $K$ is very bi
Let $K$ be an algebraically closed field of characteristic different from $2$, $g$ a positive integer, $f(x)in K[x]$ a degree $2g+1$ monic polynomial without repeated roots, $C_f: y^2=f(x)$ the corresponding genus g hyperelliptic curve over $K$, and
We approach the Torelli problem of recostructing a curve from its Jacobian from a computational point of view. Following Dubrovin, we design a machinery to solve this problem effectively, which builds on methods in numerical algebraic geometry. We ve
Let $K$ be a field of characteristic different from $2$, $bar{K}$ its algebraic closure. Let $n ge 3$ be an odd prime such that $2$ is a primitive root modulo $n$. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without r