An abelian variety $A$ defined over a field $K$ is said to split or decompose (over $K$) if it is isogenous to a product of abelian varieties $B_1\times B_2$ of lower positive dimension; the abelian varieties $B_1$ and $B_2$ are also defined over $K$, as is the isogeny from $A$ to $B_1\times B_2$. An abelian variety that does not decompose is said to be simple. A simple abelian variety may decompose over a finite extension $L/K$ (meaning that its base change to $L$ decomposes), unless it is geometrically simple.
When $A$ is the Jacobian of a genus 2 curve, the abelian varieties $B_1$ and $B_2$ are necessarily elliptic curves. If $B_1$ and $B_2$ are isogenous elliptic curves, then $A$ is isogenous to the square of $B_1$ (the abelian variety $B_1\times B_1$).
- Review status: reviewed
- Last edited by John Cremona on 2018-05-23 16:35:04
- 2018-05-23 16:35:04 by John Cremona (Reviewed)