Genus two curves covering elliptic curves: a computational approach

A genus 2 curve $C$ has an elliptic subcover if there exists a degree $n$ maximal covering $\psi: C \to E$ to an elliptic curve $E$. Degree $n$ elliptic subcovers occur in pairs $(E, E')$. The Jacobian $J_C$ of $C$ is isogenous of degree $n^2$ to the product $E \times E'$. We say that $J_C$ is $(n, n)$-split. The locus of $C$, denoted by \L_n, is an algebraic subvariety of the moduli space \M_2. The space \L_2 was studied in Shaska/V\"olklein and Gaudry/Schost. The space \L_3 was studied in Shaska (2004) were an algebraic description was given as sublocus of \M_2. In this survey we give a brief description of the spaces \L_n for a general $n$ and then focus on small $n$. We describe some of the computational details which were skipped in Shaska/V\"olklein and Shaska (2004). Further we explicitly describe the relation between the elliptic subcovers $E$ and $E'$. We have implemented most of these relations in computer programs which check easily whether a genus 2 curve has $(2, 2)$ or $(3, 3)$ split Jacobian. In each case the elliptic subcovers can be explicitly computed.Comment: arXiv admin note: substantial text overlap with arXiv:1209.043

Subvarieties of the hyperelliptic moduli determined by group actions

Let $\mathcal H_g$ be the moduli space of genus $g$ hyperelliptic curves. In this note, we study the locus $\mathcal L$ in $\mathcal H_g$ of curves admitting a $G$-action of given ramification type $\sigma$ and inclusions between such loci. For each genus we determine the list of all possible groups, the inclusions among the loci, and the corresponding equations of the generic curve in $\mathcal L$. The proof of the results is based solely on representations of finite subgroups of $PGL_2 (\mathbb C)$ and the Riemann-Hurwitz formula

Curves of genus 2 with (n, n)-decomposable jacobians

Let $C$ be a curve of genus 2 and \psi_1:C \lar E_1 a map of degree $n$, from $C$ to an elliptic curve $E_1$, both curves defined over \bC. This map induces a degree $n$ map \phi_1:\bP^1 \lar \bP^1 which we call a Frey-Kani covering. We determine all possible ramifications for $\phi_1$. If \psi_1:C \lar E_1 is maximal then there exists a maximal map \psi_2:C\lar E_2, of degree $n$, to some elliptic curve $E_2$ such that there is an isogeny of degree $n^2$ from the Jacobian $J_C$ to $E_1 \times E_2$. We say that $J_C$ is $(n,n)$-decomposable. If the degree $n$ is odd the pair $(\psi_2, E_2)$ is canonically determined. For $n=3, 5$, and 7, we give arithmetic examples of curves whose Jacobians are $(n,n)$-decomposable

Some special families of hyperelliptic curves

Let \L_g^G denote the locus of hyperelliptic curves of genus $g$ whose automorphism group contains a subgroup isomorphic to $G$. We study spaces \L_g^G for G \iso \Z_n, \Z_2{\o}\Z_n, \Z_2{\o}A_4, or $SL_2(3)$. We show that for G \iso \Z_n, \Z_2{\o}\Z_n, the space \L_g^G is a rational variety and find generators of its function field. For G\iso \Z_2{\o}A_4, SL_2(3) we find a necessary condition in terms of the coefficients, whether or not the curve belongs to \L_g^G. Further, we describe algebraically the loci of such curves for $g\leq 12$ and show that for all curves in these loci the field of moduli is a field of definition

Genus 3 hyperelliptic curves with (2, 4, 4)-split Jacobians

We study degree 2 and 4 elliptic subcovers of hyperelliptic curves of genus 3 defined over $\mathbb C$. The family of genus 3 hyperelliptic curves which have a degree 2 cover to an elliptic curve $E$ and degree 4 covers to elliptic curves $E_1$ and $E_2$ is a 2-dimensional subvariety of the hyperelliptic moduli $\mathcal H_3$. We determine this subvariety explicitly. For any given moduli point $\mathfrak p \in \mathcal H_3$ we determine explicitly if the corresponding genus 3 curve $\mathcal X$ belongs or not to such family. When it does, we can determine elliptic subcovers $E$, $E_1$, and $E_2$ in terms of the absolute invariants $t_1, \dots, t_6$ as in \cite{hyp_mod_3}. This variety provides a new family of hyperelliptic curves of genus 3 for which the Jacobians completely split. The sublocus of such family when $E_1$ is isomorphic to $E_2$ is a 1-dimensional variety which we determine explicitly. We can also determine $\mathcal X$ and $E$ starting form the $j$-invariant of $E_1$

On the automorphism groups of some AG-codes based on Ca,bC_{a, b} curves

We study $C_{a, b}$ curves and their applications to coding theory. Recently, Joyner and Ksir have suggested a decoding algorithm based on the automorphisms of the code. We show how $C_{a, b}$ curves can be used to construct MDS codes and focus on some $C_{a, b}$ curves with extra automorphisms, namely $y^3=x^4+1, y^3=x^4-x, y^3-y=x^4$. The automorphism groups of such codes are determined in most characteristics.Comment: A recent update of an old paper from 2006. Some connections to superelliptic curves are explored and references update

Heights on algebraic curves

In these lectures we cover basics of the theory of heights starting with the heights in the projective space, heights of polynomials, and heights of the algebraic curves. We define the minimal height of binary forms and moduli height for algebraic curves and prove that the moduli height of superelliptic curves $\mathcal H (f) \leq c_0 \tilde H (f)$ where $c_0$ is a constant and $\tilde H$ the minimal height of the corresponding binary form. For genus $g=2$ and 3 such constant is explicitly determined. Furthermore, complete lists of curves of genus 2 and genus 3 hyperelliptic curves with height 1 are computed.Comment: Fixes some minor typos and corrected some missing references from the first version. Information security, coding theory and related combinatorics, 165-202, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., IOS, Amsterdam, 201

Quantum codes from superelliptic curves

Let \X be an algebraic curve of genus $g \geq 2$ defined over a field \F_q of characteristic $p > 0$. From \X, under certain conditions, we can construct an algebraic geometry code $C$. If the code $C$ is self-orthogonal under the symplectic product then we can construct a quantum code $Q$, called a QAG-code. In this paper we study the construction of such codes from curves with automorphisms and the relation between the automorphism group of the curve \X and the codes $C$ and $Q$.Comment: The article has been posted without journal's permission and must be withdraw

Hyperelliptic curves with reduced automorphism group A5

We study genus $g$ hyperelliptic curves with reduced automorphism group $A_5$ and give equations $y^2=f(x)$ for such curves in both cases where $f(x)$ is a decomposable polynomial in $x^2$ or $x^5$. For any fixed genus the locus of such curves is a rational variety. We show that for every point in this locus the field of moduli is a field of definition. Moreover, there exists a rational model $y^2=F(x)$ or $y^2=x F(x)$ of the curve over its field of moduli where $F(x)$ can be chosen to be decomposable in $x^2$ or $x^5$. While similar equations have been given in Bujalance, Cirre, Gamboa and Gromadzki (2001) over $\mathbb R$, this is the first time that these equations are given over the field of moduli of the curve

The 2-Weierstrass points of genus 3 hyperelliptic curves with extra automorphisms

For each group $G$, $(|G| > 2)$ \, which acts as a full automorphism group on a genus 3 hyperelliptic curve, we determine the family of curves which have 2-Weierstrass points. Such families of curves are explicitly determined in terms of the absolute invariants of binary octavics. The 1-dimensional families that we discover have the property that they contain only genus 0 components