# Characteristic polynomials of automorphisms of hyperelliptic curves

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Let alpha be an automorphism of a hyperelliptic curve C of genus g,

and let alpha' be the automorphism of P^1 induced by alpha.

Let n be the order of alpha and let n' be the order of alpha'.

We show that the triple (g,n,n') completely determines the

characteristic polynomial of the automorphism alpha^* 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.