In previous work we determined automorphism groups of cyclic algebraic curves defined over fields... more 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.
We study relations among the classical thetanulls of cyclic curves, namely curves $\mathcal X$ (o... more We study relations among the classical thetanulls of cyclic curves, namely curves $\mathcal X$ (of genus $g(\mathcal X)>1$) with an automorphism $\sigma$ such that $\sigma$ generates a normal subgroup of the group $G$ of automorphisms, and $g (\mathcal X/ < \sigma>) =0$. 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.
ABSTRACT We investigate the decomposition of Jacobians of superelliptic curves based on their aut... more ABSTRACT We investigate the decomposition of Jacobians of superelliptic curves based on their automorphisms. For curve with equation $y^n=f(x^m)$ we provide an necessary and sufficient condition in terms of $m$ and $n$ for the decomposition of the Jacobian induced by the automorphisms of the curve. Moreover, we generalize a construction in \cite{Ya} of a family of non-hyperelliptic curves $\X_{r,s} $ and determine arithmetic conditions on $r$ and $s$ that the Jacobians $\J (\X_{r, s})$ decomposes.
We introduce a new approach of computing the automorphism group and the field of moduli of points... more 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 Hg. Further, we show that for every moduli point p ∈ Hg(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.
In the process of computing the Galois group of a prime degree polynomial f (x) over Q we suggest... more In the process of computing the Galois group of a prime degree polynomial f (x) over Q we suggest a preliminary checking for the existence of non-real roots. If f (x) has non-real roots, then combining a 1871 result of Jordan and the classification of transitive groups of prime degree which follows from CFSG we get that the Galois group of f (x) contains Ap or is one of a short list. Let f (x) ∈ Q[x] be an irreducible polynomial of prime degree p ≥ 5 and r = 2s be the number of non-real roots of f (x). We show that if s satisfies s (s log s + 2 log s + 3) ≤ p then Gal(f ) = Ap, Sp.
... 234 TANUSH SHASKA AND JENNIFER L. THOMPSON It is not clear whether the tuple (a, b, c, d, e, ... more ... 234 TANUSH SHASKA AND JENNIFER L. THOMPSON It is not clear whether the tuple (a, b, c, d, e, f) is determined uniquely once we fix the moduli point ... 2. Covers of full moduli dimension Let Mg be the moduli space of curves of genus g&amp;gt; 2 and P1= P1 (C) the Riemann sphere ...
In previous work we determined automorphism groups of cyclic algebraic curves defined over fields... more 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.
We study relations among the classical thetanulls of cyclic curves, namely curves $\mathcal X$ (o... more We study relations among the classical thetanulls of cyclic curves, namely curves $\mathcal X$ (of genus $g(\mathcal X)>1$) with an automorphism $\sigma$ such that $\sigma$ generates a normal subgroup of the group $G$ of automorphisms, and $g (\mathcal X/ < \sigma>) =0$. 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.
ABSTRACT We investigate the decomposition of Jacobians of superelliptic curves based on their aut... more ABSTRACT We investigate the decomposition of Jacobians of superelliptic curves based on their automorphisms. For curve with equation $y^n=f(x^m)$ we provide an necessary and sufficient condition in terms of $m$ and $n$ for the decomposition of the Jacobian induced by the automorphisms of the curve. Moreover, we generalize a construction in \cite{Ya} of a family of non-hyperelliptic curves $\X_{r,s} $ and determine arithmetic conditions on $r$ and $s$ that the Jacobians $\J (\X_{r, s})$ decomposes.
We introduce a new approach of computing the automorphism group and the field of moduli of points... more 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 Hg. Further, we show that for every moduli point p ∈ Hg(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.
In the process of computing the Galois group of a prime degree polynomial f (x) over Q we suggest... more In the process of computing the Galois group of a prime degree polynomial f (x) over Q we suggest a preliminary checking for the existence of non-real roots. If f (x) has non-real roots, then combining a 1871 result of Jordan and the classification of transitive groups of prime degree which follows from CFSG we get that the Galois group of f (x) contains Ap or is one of a short list. Let f (x) ∈ Q[x] be an irreducible polynomial of prime degree p ≥ 5 and r = 2s be the number of non-real roots of f (x). We show that if s satisfies s (s log s + 2 log s + 3) ≤ p then Gal(f ) = Ap, Sp.
... 234 TANUSH SHASKA AND JENNIFER L. THOMPSON It is not clear whether the tuple (a, b, c, d, e, ... more ... 234 TANUSH SHASKA AND JENNIFER L. THOMPSON It is not clear whether the tuple (a, b, c, d, e, f) is determined uniquely once we fix the moduli point ... 2. Covers of full moduli dimension Let Mg be the moduli space of curves of genus g&amp;gt; 2 and P1= P1 (C) the Riemann sphere ...
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