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The Arithmetic of Genus Two Curves with (4,4)-Split Jacobians

  Published:2011-06-19
 Printed: Oct 2011
  • Nils Bruin,
    Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6
  • Kevin Doerksen,
    Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6
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Abstract

In this paper we study genus $2$ curves whose Jacobians admit a polarized $(4,4)$-isogeny to a product of elliptic curves. We consider base fields of characteristic different from $2$ and $3$, which we do not assume to be algebraically closed. We obtain a full classification of all principally polarized abelian surfaces that can arise from gluing two elliptic curves along their $4$-torsion, and we derive the relation their absolute invariants satisfy. As an intermediate step, we give a general description of Richelot isogenies between Jacobians of genus $2$ curves, where previously only Richelot isogenies with kernels that are pointwise defined over the base field were considered. Our main tool is a Galois theoretic characterization of genus $2$ curves admitting multiple Richelot isogenies.
Keywords: Genus 2 curves, isogenies, split Jacobians, elliptic curves Genus 2 curves, isogenies, split Jacobians, elliptic curves
MSC Classifications: 11G30, 14H40 show english descriptions Curves of arbitrary genus or genus $
Jacobians, Prym varieties [See also 32G20]
11G30 - Curves of arbitrary genus or genus $
14H40 - Jacobians, Prym varieties [See also 32G20]
 

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