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As a finitely generated abelian group, the Mordell-Weil group of an abelian variety $A$ over a number field $K$ can be explicitly described by giving a $\Z$-basis for the group, 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 can choose $D$ of the form $D_0-D_\infty$, where $D_0$ is a rational effective divisor of degree at most $g$ with support disjoint from the line $z=0$ at infinity, and $D_\infty$ is a rational effective divisor supported entirely on the line at infinity.

The divisor $D_\infty$ is uniquely determined by $D_0$; it is zero if $D_0$ is zero, a suitable multiple of the (necessarily unique) rational Weierstrass point at infinity if one exists, and the intersection of $X$ with the line at infinity otherwise (in which case $D_0$ and $D_\infty$ both have degree 2).

The divisor $D_0$ can be described by a pair of homogeneous $a,b\in \Q[x,z]$, with $a(x,z)$ of degree at most $2$ and $b(x,z)$ of degree $3$. The divisor $D_0$ is then 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 $y^2-yz=x^5z$, the Mordell-Weil generator $x=0,y=z^3$ specifies the class of the divisor $(0:1:1)-(0:1:0)$.

Knowl status:
  • Review status: reviewed
  • Last edited by John Voight on 2020-01-07 13:02:26
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