Field of moduli and field of definition for curves of genus 2

By Gabriel Cardona, J. Quer. 2002

83 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/734b416ecd8fb2e79e1f6669623d17b111f4ac99

How researchers see this paper

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In Algebra and Number Theory

1. T. Streng
2. Christophe Ritzenthaler
3. Reynald Lercier

In Geometry and Topology

1. T. Streng
2. Christophe Ritzenthaler
3. Reynald Lercier

In Mathematical Physics

1. T. Streng
2. Christophe Ritzenthaler
3. Reynald Lercier

Citations in the corpus

• Complex multiplication of abelian surfaces (2010, background): “In the literature, one finds various sets of absolute Igusa invariants [12, 35, 50, 45, 61, 98].”
• Fields of moduli of hyperelliptic curves (2004, unknown): “E-mail address: bhuggins@math.berkeley.edu”
• Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects (2011, background): “Similar issues for genus 2 are completely solved by [11] (see also [10, Th.”