Robert M. Guralnick and Everett W. Howe: Characteristic polynomials of automorphisms of hyperelliptic curves, pp. 101–112 in: Arithmetic, Geometry, Cryptography and Coding Theory (G. Lachaud, C. Ritzenthaler, and M. A. Tsfasman, eds.), Contemporary Mathematics 487, American Mathematical Society, Providence, RI, 2009, MR 2010j:14060, Zbl 1184.14047.
(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 P1 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.