= Involution of a degree-two map from a genus one curve
{title2=$\iota(p)=s-p$}
Over an algebraically closed field of characteristic different from two, a degree-two map from a smooth <genus one curve> to the <projective line> has a deck involution. After choosing an origin, all its fiber <divisors on an algebraic curve> are linearly equivalent and have the same group sum $s$, by the <Abel-Jacobi map of a genus-one curve>. The involution is consequently $p\mapsto s-p$, including its fixed ramification points. Composing two such involutions gives a <translation on an elliptic curve>.
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