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N-covers of hyperelliptic curves

By N. Bruin and E. V. Flynn


For a hyperelliptic curve C of genus g with a divisor class of order n=g+1, we shall consider an associated covering collection of curves D$_\delta$, each of genus g$^2$. We describe, up to isogeny, the Jacobian of each D$_\delta$ via a map from D$_\delta$ to C, and two independent maps from D$_\delta$ to a curve of genus g(g-1)/2. For some curves, this allows covering techniques that depend on arithmetic data of number fields of smaller degree than standard 2-coverings; we illustrate this by using 3-coverings to find all Q-rational points on a curve of genus 2 for which 2-covering techniques would be impractical

Topics: Algebraic geometry, Number theory
Year: 2003
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