On Jacobians of curves with superelliptic components

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 ${\mathcal{X}}_{r,s}$ and determine arithmetic conditions on $r$ and $s$ that the Jacobians $\mbox{Jac} (\mathcal X_{r, s})$ decomposes.

References in corpus (7)

• The arithmetic of genus two curves with (4,4)-split Jacobians
• Genus 2 fields with degree 3 elliptic subfields
• Determining equations of families of cyclic curves
• Bielliptic curves of genus 3 in the hyperelliptic moduli
• Families of genus two curves with many elliptic subcovers
• Genus 3 hyperelliptic curves with (2, 4, 4)-split Jacobians
• The arithmetic of genus two curves

Cited by in corpus (7)

• From hyperelliptic to superelliptic curves
• Weierstrass points of superelliptic curves
• Genus 3 hyperelliptic curves with (2, 4, 4)-split Jacobians
• On the automorphism groups of some AG-codes based on $C_{a,b}$ curves
• Heights on algebraic curves
• A correction to the paper "On Curves with Split Jacobian"
• Theta functions of superelliptic curves