The arithmetic of genus two curves with (4,4)-split Jacobians

Details The arithmetic of genus two curves with (4,4)-split Jacobians The arithmetic of genus two curves with (4,4)-split Jacobians

2009-02-19

arxiv.org/abs/0902.3480v3

Canad. J. Math. 63(2011), 992-1021

Mathematics

Number Theory

30 pages

Scientific paper

10.4153/CJM-2011-039-3

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.

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