Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group

Lercier, Reynald; Ritzenthaler, Christophe; Sijsling, Jeroen

doi:10.1090/mcom3032

Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group
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by Reynald Lercier, Christophe Ritzenthaler and Jeroen Sijsling PDF

Math. Comp. 85 (2016), 2011-2045 Request permission

Abstract:

This paper is devoted to the study of the Galois descent obstruction for hyperelliptic curves of arbitrary genus whose reduced automorphism groups are cyclic of order coprime to the characteristic of their ground field. We give an explicit and effectively computable description of this obstruction. Along the way, we obtain an arithmetic criterion for the existence of a so-called hyperelliptic descent.

We define homogeneous dihedral invariants for general hyperelliptic curves, and show how the obstruction can be expressed in terms of these invariants. If this obstruction vanishes, then the homogeneous dihedral invariants can also be used to explicitly construct a model over the field of moduli of the curve; if not, then one still obtains a hyperelliptic model over a degree $2$ extension of the field of moduli.

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