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Field of moduli and field of definition for curves of genus 2
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Let M_2 be the moduli space that classifies genus 2 curves. If a curve C is defined over a field k, the corresponding moduli point P=[C] is defined over k. Mestre solved the converse problem for curves with Aut(C) isomorphic to C_2. Given a moduli point defined over k, Mestre finds an obstruction to the existence of a corresponding curve defined over k, that is an element in Br_2(k) not always trivial. In this paper we prove that for all the other possibilities of Aut(C), every moduli point defined over k is represented by a curve defined over k. We also give an explicit construction of such a curve in terms of the coordinates of the moduli point.
Forward citations
Cited by 2 Pith papers
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Genus 2 Supersingular Isogeny Oblivious Transfer
Extends Barreto-Oliveira-Benits supersingular isogeny oblivious transfer from elliptic curves to principally polarized supersingular abelian surfaces of genus 2.
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Undeniable signatures based on isogenies of supersingular hyperelliptic curves
This paper proposes an undeniable signature scheme built from isogenies of genus-2 supersingular hyperelliptic curves.
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