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.

It is well known that every finite subgroup of GLd(Q[ell ]) is conjugate to a subgroup of GLd(Z[ell ]). However, this does not remain true if we replace general linear groups by symplectic groups. We say that G is a group of inertia type of G is a finite group which has a normal Sylow-p-subgroup with cyclic quotient. We show that if [ell ]>d+1, and G is a subgroup of Sp2d(Q[ell ]) of inertia type, then G is conjugate in GL2d(Q[ell ]) to a subgroup of Sp2d(Z[ell ]). We give examples which show that the bound is sharp. We apply these results to construct, for every odd prime [ell ], isogeny classes of Abelian varieties all of whose polarizations have degree divisible by [ell ]2. We prove similar results for Euler characteristic of invertible sheaves on Abelian varieties over fields of positive characteristic.