"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 variants of a question raised by Mazur concerning the existence of symplectic isomorphisms between the torsion of two non-isogenous elliptic curves defined over "--
Freitas and Kraus show the non-existence of rational points on certain hyper-elliptic curves of the form y2 = x(su)p - l and y2 = x(su)p - 2l where l is a prime. They also give incremental results of the Fermat equation x2 + y(su)3 = zp. Their topics are motivation and results, the existence of local symplectic criteria, the criterion in the case of good reduction, elliptic curves with potentially good reduction, the morphism E, proof of the criteria, and applications. Annotation ©2022 Ringgold, Inc., Portland, OR (protoview.com)
