ag.algebraic geometry - morphisms from abelian varieties to rational curves.

Let E be an elliptic curve and let x,y be points on it. Then the divisor

[x] + [-x] + [y] - 2[2y] - [-3y]

is principal, so is div(f) for some function f:E -> P^1; this function sends x and y to the same point of P^1 (namely, 0) but sends -x and -y to different points (unless you were very unlucky in your choices) so [-1] can't descend along f.