Isogenous components of Jacobian surfaces

Let be a genus 2 curve defined over a field K, charK=p⩾0, and its Jacobian, where ι is the principal polarization of attached to . Assume that is (n, n)-geometrically reducible with E1 and E2 its elliptic components. We prove that there are only finitely many curves (up to isomorphism) defined over K such that E1 and E2 are N-isogenous for n=2 and N=2,3,5,7 with or n=2, N=3,5,7 with . The same holds if n=3 and N=5. Furthermore, we determine the Kummer and Shioda–Inose surfaces for the above and show how such results in positive characteristic p>2 suggest nice applications in cryptography.