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Characteristic polynomials of automorphisms of hyperelliptic curves. (English) Zbl 1184.14047
Lachaud, Gilles (ed.) et al., Arithmetic, geometry, cryptography and coding theory. Proceedings of the 11th international conference, CIRM, Marseilles, France, November 5–9, 2007. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4716-9/pbk). Contemporary Mathematics 487, 101-111 (2009).
Let \(\alpha\) be an automorphism of a genus \(g\) curve \(C\) over a field \(k\) and let \(\alpha^*\) be the corresponding automorphism of the Jacobian of \(C\). Let \(n\) be the order of \(\alpha\) and let \(f\) be the characteristic polynomial of \(\alpha^*\). As the authors say in the introduction, the values \(g\) and \(n\) do not in general determine \(f\). If \(C\) is hyperelliptic and \(\iota\) the hyperelliptic involution, let \(\bar{n}\) the order of the automorphism induced by \(\alpha\) on \(C/\langle \iota \rangle\). The authors prove that, apart from the case \(n=\bar{n}\) is even and \((2g+2)/n\) is even, the triple \((g,n,\bar{n})\) determines \(f\) completely and give the possible \(f\).
The main idea of the proof is to compute the multiplicity of the roots of \(f\) (which are \(n\)-th roots of unity) by using the fact that the dimension of the part of the Jacobian of \(C\) on which a power of \(\alpha^*\) acts trivially is twice the genus of the quotient of \(C\) by this power of \(\alpha\). A technical lemma gives the possible genera.
For the entire collection see [Zbl 1166.11003].
14H37 Automorphisms of curves
14H40 Jacobians, Prym varieties
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