#### Vol. 1, No. 1, 2013

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Elliptic factors in Jacobians of hyperelliptic curves with certain automorphism groups

### Jennifer Paulhus

Vol. 1 (2013), No. 1, 487–505
##### Abstract

We decompose the Jacobian varieties of hyperelliptic curves up to genus $\mathfrak{2}\mathfrak{0}$, defined over an algebraically closed field of characteristic zero, with reduced automorphism group ${A}_{\mathfrak{4}}$, ${S}_{\mathfrak{4}}$, or ${A}_{\mathfrak{5}}$. Among these curves is a genus-$\mathfrak{4}$ curve with Jacobian variety isogenous to ${E}_{\mathfrak{1}}^{\mathfrak{2}}×{E}_{\mathfrak{2}}^{\mathfrak{2}}$ and a genus-$\mathfrak{5}$ curve with Jacobian variety isogenous to ${E}^{\mathfrak{5}}$, for $E$ and ${E}_{i}$ elliptic curves. These types of results have some interesting consequences for questions of ranks of elliptic curves and ranks of their twists.

##### Keywords
Jacobian varieties, hyperelliptic curves, automorphism groups of Riemann surfaces
##### Mathematical Subject Classification 2010
Primary: 14H40
Secondary: 11G30, 14H37
##### Milestones
Published: 14 November 2013
##### Authors
 Jennifer Paulhus Department of Mathematics and Statistics Grinnell College Grinnell, IA 50112 United States