# 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} }

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…

## 26 Citations

### D ec 2 01 3 Large 3-groups of automorphisms of algebraic curves in characteristic 3

- Mathematics
- 2021

Let S be a p-subgroup of the K-automorphism group Aut(X ) of an algebraic curve X of genus g ≥ 2 and p-rank γ defined over an algebraically closed field K of characteristic p ≥ 3. In this paper we…

### Bound on the order of the decomposition groups of an algebraic curve in positive characteristic

- MathematicsFinite Fields Their Appl.
- 2021

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

- Mathematics
- 2002

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…

### Doubly isogenous genus-2 curves with $D_4$-action

- Mathematics
- 2021

We study the zeta functions of curves over finite fields. Suppose C and C′ are curves over a finite field K, with K-rational base points P and P′, and let D and D′ be the pullbacks (via the…

### The twisting representation of the $L$-function of a curve

- Mathematics
- 2010

Let C be a smooth projective curve defined over a number field and let C' be a twist of C. In this article we relate the l-adic representations attached to the l-adic Tate modules of the Jacobians of…

### Fields of definition of elliptic k-curves and the realizability of all genus 2 sato–tate groups over a number field

- Mathematics
- 2015

Let $A/\mathbb{Q}$ be an abelian variety of dimension $g\geq 1$ that is isogenous over $\overline{\mathbb{Q}}$ to $E^g$, where $E$ is an elliptic curve. If $E$ does not have complex multiplication…

### Honda Theory for Formal Groups of Abelian Varieties over Q of GL2-Type

- Mathematics
- 2014

Honda proved that two formal groups attached to an elliptic curve E over Q are strongly isomorphic over Z, where one of them is obtained from the formal completion along the zero section of the Neron…

### The Vector Decomposition Problem for Elliptic and Hyperelliptic Curves

- Mathematics, Computer ScienceIACR Cryptol. ePrint Arch.
- 2005

This work shows that under certain conditions the vector decomposition problem (VDP) on a two-dimensional vector space is at least as hard as the computational Diffie-Hellman problem on a one-dimensional subspace, and presents a family of hyperelliptic curves of genus two that are suitable for the VDP.

## 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

- Mathematics
- 2000

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

- Mathematics
- 2004

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

- MathematicsANTS
- 1996

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

- Philosophy
- 1991

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…