8 research outputs found
Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group
This paper is devoted to the explicit description of the Galois descent
obstruction for hyperelliptic curves of arbitrary genus whose reduced
automorphism group is cyclic of order coprime to the characteristic of their
ground field. Along the way, we obtain an arithmetic criterion for the
existence of a hyperelliptic descent.
The obstruction is described by the so-called arithmetic dihedral invariants
of the curves in question. If it vanishes, then the use of these invariants
also allows the explicit determination of a model over the field of moduli; if
not, then one obtains a hyperelliptic model over a degree 2 extension of this
field.Comment: 35 pages; improve the readability of the pape
On explicit descent of marked curves and maps
We revisit a statement of Birch that the field of moduli for a marked
three-point ramified cover is a field of definition. Classical criteria due to
D\`ebes and Emsalem can be used to prove this statement in the presence of a
smooth point, and in fact these results imply more generally that a marked
curve descends to its field of moduli. We give a constructive version of their
results, based on an algebraic version of the notion of branches of a morphism
and allowing us to extend the aforementioned results to the wildly ramified
case. Moreover, we give explicit counterexamples for singular curves.Comment: 35 page