On the Symplectic Type of Isomorphisms of the P-Torsion of Elliptic Curves

On the Symplectic Type of Isomorphisms of the P-Torsion of Elliptic Curves Synopsis

"Let be a prime. Let and be elliptic curves with isomorphic torsion modules and . Assume further that either every modules isomorphism admits a multiple with preserving the Weil pairing; or no isomorphism preserves the Weil pairing. This paper considers the problem of deciding if we are in case . Our approach is to consider the problem locally at a prime . Firstly, we determine the primes for which the local curves and contain enough information to decide between . Secondly, we establish a collection of criteria, in terms of the standard invariants associated to minimal Weierstrass models of and , to decide between . We show that our results give a complete solution to the problem by local methods away from . We apply our methods to show the non-existence of rational points on certain hyperelliptic curves of the form and where is a prime; we also give incremental results on the Fermat equation . As a different application, we discuss va