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The Mordell-Weil group of an abelian variety $A$ over a number field $K$ is a finitely generated group, explicitly described by giving a $\Z$-basis for the group, or equivalently a (minimal) set of Mordell-Weil generators.

When $A$ is the Jacobian of a genus 2 curve $X$ over $\Q$, elements of $A(\Q)$ can be specified as the linear equivalence class of a rational divisor $D$ on $X$ of degree zero. If $X$ is defined by the (homogeneous) minimal equation $y^2 + h(x,z)y = f(x,z),$ we may express $D$ in the form $D_0-nD_\infty$, where $D_0$ is a rational effective divisor of degree at most $g=2$ and $D_\infty$ is the sum of the points in the intersection of the curve with the line $z=0$ at infinity, with $n=\deg(D_0)/\deg(D_\infty)\in \{0,1,2\}$.

The divisor $D_0$ can be described by a pair of homogeneous polynomials $a,b\in \Q[x,z]$, with $a(x,z)$ of degree at most $2$ and $b(x,z)$ of degree $3$: the divisor $D_0$ is the intersection of $X$ with the locus of $a(x,z)=0$ and $y=b(x,z)$. When $h=0$ and $\deg(f)=5$ this is equivalent to the Mumford representation of $D_0$ as $(u,v)$ with $u(x)=a(x,1)$ and $v(x)=b(x,1)$.

For example, on the curve $X \colon y^2 - y = x^5$ with homogenization $y^2-yz^3=x^5z$, the Mordell-Weil generator $x=0,\ y=z^3$ specifies the class of the divisor $(0:1:1)-(0:1:0)$.

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• Review status: reviewed
• Last edited by John Voight on 2020-01-28 12:59:05
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