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T. Shaska

Publications and source records attributed to T. Shaska.

At least 37 records · Page 2Linked to original sources
arXiv2013

Bielliptic curves of genus 3 in the hyperelliptic moduli

In this paper we study bielliptic curves of genus 3 defined over an algebraically closed field $k$ and the intersection of the moduli space $\M_3^b$ of such curves with the hyperelliptic moduli $\H_3$. Such intersection $§$ is an irreducible, 3-dimensional, rational algebraic variety. We determine the equation of this space in terms of the $Gl(2, k)$-invariants of binary octavics as defined in \cite{hyp_mod_3} and find a birational parametrization of $§$. We also compute all possible subloci of curves for all possible automorphism group $G$. Moreover, for every rational moduli point $\p \in §$, such that $| \Aut (\p) | > 4$, we give explicitly a rational model of the corresponding curve over its field of moduli in terms of the $Gl(2, k)$-invariants.

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arXiv2013

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 $σ$ 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.

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arXiv2013

Determining equations of families of cyclic curves

In previous work we determined automorphism groups of cyclic algebraic curves defined over fields of any odd characteristic. In this paper we determine parametric equations of families of curves for each automorphism group for such curves.

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arXiv2013

Degree 4 coverings of elliptic curves by genus 2 curves

Genus two curves covering elliptic curves have been the object of study of many articles. For a fixed degree $n$ the subloci of the moduli space $\mathcal M_2$ of curves having a degree $n$ elliptic subcover has been computed for $n=3, 5$ and discussed in detail for $n$ odd; see \cite{Sh1, SV2, Fr, FK}. When the degree of the cover is even the case in general has been treated in \cite{PRS}. In this paper we compute the sublocus of $\mathcal M_2$ of curves having a degree 4 elliptic subcover.

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arXiv2013

Thetanulls of cyclic curves of small genus

We study relations among the classical thetanulls of cyclic curves, namely curves $\mathcal X$ (of genus $g(\mathcal X)>1$) with an automorphism $σ$ such that $σ$ generates a normal subgroup of the group $G$ of automorphisms, and $g (\mathcal X/ < σ>) =0$. Relations between thetanulls and branch points of the projection are the object of much classical work, especially for hyperelliptic curves, and of recent work, in the cyclic case. We determine the curves of genus 2 and 3 in the locus $\mathcal M_g (G, \textbf{C})$ for all $G$ that have a normal subgroup $<\s>$ as above, and all possible signatures \textbf{C}, via relations among their thetanulls.

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arXiv2012

Theta functions and algebraic curves with automorphisms

Let $\X$ be an irreducible, smooth, projective curve of genus $g \geq 2$ defined over the complex field $\C.$ Then there is a covering $π: \X \longrightarrow ¶^1,$ where $¶^1$ denotes the projective line. The problem of expressing branch points of the covering $π$ in terms of the transcendentals (period matrix, thetanulls, e.g.) is classical. It goes back to Riemann, Jacobi, Picard and Rosenhein. Many mathematicians, including Picard and Thomae, have offered partial treatments for this problem. In this work, we address the problem for cyclic curves of genus 2, 3, and 4 and find relations among theta functions for curves with automorphisms. We consider curves of genus $g > 1$ admitting an automorphism $σ$ such that $\X^σ$ has genus zero and $σ$ generates a normal subgroup of the automorphism group $Aut(\X)$ of $\X$. To characterize the locus of cyclic curves by analytic conditions on its Abelian coordinates, in other words, theta functions, we use some classical formulas, recent results of Hurwitz spaces, and symbolic computations, especially for genera 2 and 3. For hyperelliptic curves, we use Thomae's formula to invert the period map and discover relations among the classical thetanulls of cyclic curves. For non hyperelliptic curves, we write the equations in terms of thetanulls. Fast genus 2 curve arithmetic in the Jacobian of the curve is used in cryptography and is based on inverting the moduli map for genus 2 curves and on some other relations on theta functions. We determine similar formulas and relations for genus 3 hyperelliptic curves and offer an algorithm for how this can be done for higher genus curves. It is still to be determined whether our formulas for $g=3$ can be used in cryptographic applications as in $g=2.$

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arXiv2012

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 $ψ: C \to E$ 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 $Ł_n$, is an algebraic subvariety of the moduli space $\M_2$. The space $Ł_2$ was studied in Shaska/Völklein and Gaudry/Schost. The space $Ł_3$ 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 $Ł_n$ for a general $n$ and then focus on small $n$. We describe some of the computational details which were skipped in Shaska/Völklein 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.

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arXiv2012

Hyperelliptic curves of genus 3 with prescribed automorphism group

We study genus 3 hyperelliptic curves which have an extra involution. The locus $Ł_3$ of these curves is a 3-dimensional subvariety in the genus 3 hyperelliptic moduli $\H_3$. We find a birational parametrization of this locus by affine 3-space. For every moduli point $\p \in \H_3$ such that $|\Aut (\p)|>2$, the field of moduli is a field of definition. We provide a rational model of the curve over its field of moduli for all moduli points $\p \in \H_3$ such that $|\Aut(\p)|>4$. This is the first time that such a rational model of these curves appears in the literature.

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arXiv2012

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.

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arXiv2012

Some special families of hyperelliptic curves

Let $Ł_g^G$ denote the locus of hyperelliptic curves of genus $g$ whose automorphism group contains a subgroup isomorphic to $G$. We study spaces $Ł_g^G$ for $G \iso \Z_n, \Z_2ø\Z_n, \Z_2øA_4$, or $SL_2(3)$. We show that for $G \iso \Z_n, \Z_2ø\Z_n$, the space $Ł_g^G$ is a rational variety and find generators of its function field. For $G\iso \Z_2øA_4, SL_2(3)$ we find a necessary condition in terms of the coefficients, whether or not the curve belongs to $Ł_g^G$. 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.

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arXiv2012

Some remarks on the hyperelliptic moduli of genus 3

In 1967, Shioda \cite{Shi1} determined the ring of invariants of binary octavics and their syzygies using the symbolic method. We discover that the syzygies determined in \cite{Shi1} are incorrect. In this paper, we compute the correct equations among the invariants of the binary octavics and give necessary and sufficient conditions for two genus 3 hyperelliptic curves to be isomorphic over an algebraically closed field $k$, $\ch k \neq 2, 3, 5, 7$. For the first time, an explicit equation of the hyperelliptic moduli for genus 3 is computed in terms of absolute invariants.

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arXiv2012

On the generic curve of genus 3

We study genus $g$ coverings of full moduli dimension of degree $d=[\frac {g+3} 2]$. There is a homomorphism between the corresponding Hurwitz space $\H$ of such covers to the moduli space $\M_g$ of genus $g$ curves. In the case $g=3$, using the signature of such covering we provide an equation for the generic ternary quartic. Further, we discuss the degenerate subloci of the corresponding Hurwitz space of such covers from the computational group theory viewpoint. In the last section, we show that one of these degenerate loci corresponds to the locus of curves with automorphism group $C_3$. We give necessary conditions in terms of covariants of ternary quartics for a genus 3 curve to belong to this locus.

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arXiv2012

Genus 2 curves that admit a degree 5 map to an elliptic curve

We continue our study of genus 2 curves $C$ that admit a cover $ C \to E$ to a genus 1 curve $E$ of prime degree $n$. These curves $C$ form an irreducible 2-dimensional subvariety $Ł_n$ of the moduli space $\M_2$ of genus 2 curves. Here we study the case $n=5$. This extends earlier work for degree 2 and 3, aimed at illuminating the theory for general $n$. We compute a normal form for the curves in the locus $Ł_5$ and its three distinguished subloci. Further, we compute the equation of the elliptic subcover in all cases, give a birational parametrization of the subloci of $Ł_5$ as subvarieties of $\M_2$ and classify all curves in these loci which have extra automorphisms.

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arXiv2012

Codes over rings of size four, Hermitian lattices, and corresponding theta functions

Let $K=Q(\sqrt{-\ell})$ be an imaginary quadratic field with ring of integers $Ø_K$, where $\ell$ is a square free integer such that $\ell\equiv 3 \mod 4$ and $C=[n, k]$ be a linear code defined over $Ø_K/2Ø_K$. The level $\ell$ theta function $\Th_{Ł_{\ell} (C)} $ of $C$ is defined on the lattice $Ł_{\ell} (C):= \set {x \in Ø_K^n : ρ_\ell (x) \in C}$, where $ρ_{\ell}:Ø_K \rightarrow Ø_K/2Ø_K$ is the natural projection. In this paper, we prove that: % i) for any $\ell, \ell^\prime$ such that $\ell \leq \ell^\prime$, $\Th_{Λ_\ell}(q)$ and $\Th_{Λ_{\ell^\prime}}(q)$ have the same coefficients up to $q^{\frac {\ell+1}{4}}$, % ii) for $\ell \geq \frac {2(n+1)(n+2)}{n} -1$, $\Th_{Ł_{\ell}} (C)$ determines the code $C$ uniquely, % iii) for $\ell < \frac {2(n+1)(n+2)}{n} -1$ there is a positive dimensional family of symmetrized weight enumerator polynomials corresponding to $\Th_{\La_\ell}(C)$.

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arXiv2012

Codes over rings of size $p^2$ and lattices over imaginary quadratic fields

Let $\ell>0$ be a square-free integer congruent to 3 mod 4 and $Ø_K$ the ring of integers of the imaginary quadratic field $K=Q(\sqrt{-\ell})$. Codes $C$ over rings $Ø_K / p Ø_K$ determine lattices $Λ_\ell (C) $ over $K$. If $ p \nmid \ell$ then the ring $\R:=Ø_K / p Ø_K$ is isomorphic to $\F_{p^2}$ or $\F_p \times \F_p$. Given a code $C$ over $\R$, theta functions on the corresponding lattices are defined. These theta series $θ_{Λ_{\ell}(C)}$ can be written in terms of the complete weight enumerator of $C$. We show that for any two $\ell < \ell^\prime$ the first $\frac {\ell + 1} 4$ terms of their corresponding theta functions are the same. Moreover, we conjecture that for $\ell > \frac {p(n+1)(n+2)} 2$ there is a unique complete weight enumerator corresponding to a given theta function. We verify the conjecture for primes $p< 7$ and $\ell \leq 59$.

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arXiv2006

Hyperelliptic curves with extra involutions

The purpose of this paper is to study hyperelliptic curves with extra involutions. The locus $Ł_g$ of such genus $g$ hyperelliptic curves is a $g$-dimensional subvariety of the moduli space of hyperelliptic curves $\H_g$. We discover a birational parametrization of $Ł_g$ via dihedral invariants and show how these invariants can be used to determine the field of moduli of points $\p \in Ł_g$. We conjecture that for $\p\in \H_g$ with $|\Aut(\p)| > 2$ the field of moduli is a field of definition and prove this conjecture for any point $\p\in Ł_g$ such that the Klein 4-group is embedded in the reduced automorphism group of $\p$. Further, for $g=3$ we show that for every moduli point $\p \in \H_3$ such that $| \Aut (\p) | > 4$, the field of moduli is a field of definition and provide a rational model of the curve over its field of moduli.

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arXiv2003

Determining the automorphism group of a hyperelliptic curve

In this note we discuss techniques for determining the automorphism group of a genus $g$ hyperelliptic curve $\X_g$ defined over an algebraically closed field $k$ of characteristic zero. The first technique uses the classical $GL_2 (k)$-invariants of binary forms. This is a practical method for curves of small genus, but has limitations as the genus increases, due to the fact that such invariants are not known for large genus. The second approach, which uses dihedral invariants of hyperelliptic curves, is a very convenient method and works well in all genera. First we define the normal decomposition of a hyperelliptic curve with extra automorphisms. Then dihedral invariants are defined in terms of the coefficients of this normal decomposition. We define such invariants independently of the automorphism group $\Aut (\X_g)$. However, to compute such invariants the curve is required to be in its normal form. This requires solving a nonlinear system of equations. We find conditions in terms of classical invariants of binary forms for a curve to have reduced automorphism group $A_4$, $S_4$, $A_5$. As far as we are aware, such results have not appeared before in the literature.

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