Computational Aspects of Hyperelliptic Curves

We introduce a new approach of computing the automorphism group and the field of moduli of points \p=[C] in the moduli space of hyperelliptic curves \H_g. Further, we show that for every moduli point \p \in \H_g(L) such that the reduced automorphism group of \p has at least two involutions, there exists a representative $C$ of the isomorphism class \p which is defined over $L$

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

Families of genus two curves with many elliptic subcovers

We determine all genus 2 curves, defined over $\mathbb C$, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in $\mathcal M_2$. For each component we find a rational parametrization and construct the equation of the corresponding genus 2 curve and its elliptic subcovers in terms of the parameterization. Such families of genus 2 curves are determined for the first time. Furthermore, we prove that there are only finitely many genus 2 curves (up to $\mathbb C$-isomorphism) defined over $\mathbb Q$, which have degree 2 and 3 elliptic subcovers also defined over $\mathbb Q$

Equations of curves with minimal discriminant

In this paper we give an algorithm of how to determine a Weierstrass equation with minimal discriminant for superelliptic curves generalizing work of Tate for elliptic curves and Liu for genus 2 curves

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

Genus 2 fields with degree 3 elliptic subfields

In this paper we study genus 2 function fields K with degree 3 elliptic subfields. We show that the number of Aut(K)-classes of such subfields of K is 0,1,2, or 4. Also we compute an equation for the locus of such K in the moduli space of genus 2 curves

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

Superelliptic curves with minimal weighted moduli height

For a superelliptic curve $\mathcal X$, defined over $\mathbb Q$, let $\mathfrak p$ denote the corresponding moduli point in the weighted moduli space. We describe a method how to determine a minimal integral model of $\mathcal X$ such that: i) the corresponding moduli point $\mathfrak p$ has minimal weighted height, ii) the equation of the curve has minimal coefficients. Part i) is accomplished by reduction of the moduli point which is equivalent with obtaining a representation of the moduli point $\mathfrak p$ with minimal weighted height, as defined in [5], and part ii) by the classical reduction of the binary forms.Comment: The accepted version is updated. The paper will appear in Contemporary Math., (AMS), 202

Some remarks on the non-real roots of polynomials

Let $f \in { \mathbb R} ( t) [x]$ be given by $f(t, x) = x^n + t \cdot g(x)$ and $\beta_1 < \dots < \beta_m$ the distinct real roots of the discriminant $\Delta_{(f, x)} (t)$ of $f(t, x)$ with respect to $x$. Let $\gamma$ be the number of real roots of $g(x)=\sum_{k=0}^s t_{s-k} x^{s-k}$. For any $\xi > | \beta_m |$, if $n-s$ is odd then the number of real roots of $f(\xi, x)$ is $\gamma+1$, and if $n-s$ is even then the number of real roots of $f(\xi, x)$ is $\gamma$, $\gamma+2$ if $t_s>0$ or $t_s < 0$ respectively. A special case of the above result is constructing a family of totally complex polynomials which are reducible over $\mathbb Q$.Comment: 23 page