(A preprint is available online.)

Let α be an automorphism of a hyperelliptic curve *C* of genus *g*,
and let α' be the automorphism of **P**^{1} induced by α.
Let *n* be the order of α and let *n*' be the order of α'.
We show that the triple (*g,n,n'*) completely determines the
characteristic polynomial of the automorphism α^{*} of the
Jacobian of *C*, unless *n* is even, *n=n'*, and (*2g+2*)/*n* is even,
in which case there are two possibilities. We give explicit
formulas for the characteristic polynomial in all cases.