Let X ⊂ Y be smooth, projective manifolds. Assume that ι : X P s is the zero locus of a generic section of V + = ⊕ i∈I ᏻ(k i ), where all the k i 's are positive. Assume furthermore that ᏺ X/Y = ι * (V − ), where V − = ⊕ j∈J ᏻ(−l j ) and all the l j 's are negative. We show that under appropriate restrictions, the generalized Gromov-Witten invariants of X inherited from Y can be calculated via a modified Gromov-Witten theory on P s . This leads to local mirror symmetry on the A-side.

Let C be a genus 2 curve defined over a field K, char K = p ≥ 0, and Jac(C, ι) its Jacobian, where ι is the principal polarization of Jac(C) attached to C. Assume that Jac(C) is (n, n)-geometrically reducible with E 1 and E 2 its elliptic components. We prove that there are only finitely many curves C (up to isomorphism) defined over K such that E 1 and E 2 are N -isogenous for n = 2 and N = 2, 3, 5, 7 with Aut (Jac C) ∼ = V 4 or n = 2, N = 3, 5, 7 with Aut (Jac C) ∼ = D 4 . The same holds if n = 3 and N = 5. Furthermore, we determine the Kummer and the Shioda-Inose surfaces for the above Jac C and show how such results in positive characteristic p > 2 suggest nice applications in cryptography.

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We connect the Aspinwall-Morrison calculation to Gromov-Witten theory.

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