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 $E_{\mathfrak{1}}^{\mathfrak{2}}\times E_{\mathfrak{2}}^{\mathfrak{2}}$ 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.

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Mathematical Subject Classification 2010

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