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

Published online by Cambridge University Press:  20 November 2018

Nils Bruin  and
Kevin Doerksen
Nils Bruin
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6email: nbruin@sfu.cakdoerkse@sfu.ca
Kevin Doerksen
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6email: nbruin@sfu.cakdoerkse@sfu.ca
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Abstract

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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

11G3014H40genus 2 curvesisogeniessplit Jacobianselliptic curves
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 5 , 18 October 2011 , pp. 992 - 1024
Copyright
Copyright © Canadian Mathematical Society 2011

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