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A new proof for the non-degeneracy of the Frey-Rück pairing and a connection to isogenies over the base field

Frey and Rück have described a non-degenerate bilinear pairing on the Jacobians of curves; this includes elliptic curves. We present a new mathematical foundation for this pairing and use it to give a different proof of its non-degeneracy. We then present yet another method of looking at this pairing using isogenies.