# Curves of genus 2 with group of automorphisms isomorphic to $D_8$ or $D_{12}$

@article{Cardona2002CurvesOG,
title={Curves of genus 2 with group of automorphisms isomorphic to \$D\_8\$ or \$D\_\{12\}\$},
author={Gabriel Cardona and Jordi Quer},
journal={Transactions of the American Mathematical Society},
year={2002},
volume={359},
pages={2831-2849}
}
• Published 17 March 2002
• Mathematics
• Transactions of the American Mathematical Society
The classification of curves of genus 2 over an algebraically closed field was studied by Clebsch and Bolza using invariants of binary sextic forms, and completed by Igusa with the computation of the corresponding three-dimensional moduli variety M 2 . The locus of curves with group of automorphisms isomorphic to one of the dihedral groups D 8 or D 12 is a one-dimensional subvariety. In this paper we classify these curves over an arbitrary perfect field k of characteristic chark ≠ 2 in the Ds…

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## References

SHOWING 1-10 OF 15 REFERENCES

### On curves of genus 2 with Jacobian of GL2-type

• Mathematics
• 1999
Abstract:Ribet [Ri] has generalized the conjecture of Shimura–Taniyama–Weil to abelian varieties defined over Q,giving rise to the study of abelian varieties of GL2-type. In this context, all curves

### Q‐Curves and Abelian Varieties of GL2‐Type

The relation between Q‐curves and certain abelian varieties of GL2‐type was established by Ribet (‘Abelian varieties over Q and modular forms’, Proceedings of the KAIST Mathematics Workshop (1992)

### Rational and Elliptic Parametrizations ofQ-Curves

• Mathematics
• 1998
Abstract We describe explicit parametrizations of the rational points of X *( N ), the algebraic curve obtained as quotient of the modular curve X 0 ( N ) by the group B ( N ) generated by the

### Abelian Varieties over Q and Modular Forms

Let C be an elliptic curve over Q. Let N be the conductor of C. The Taniyama conjecture asserts that there is a non-constant map of algebraic curves X 0 (N) — C which is defined over Q. Here, X o (N)

### Computational Aspects of Curves of Genus at Least 2

This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of

### Construction de courbes de genre 2 à partir de leurs modules

Soient A la variete des modules des courbes de genre 2, R la surface de A correspondant aux courbes ayant une involution autre que l’involution hyperelliptique, et P un point de A — R defini sur un