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Working from well-known results regarding l-adic Galois representations attached to elliptic curves arising from successive preimages of the identity, we consider a natural deformation. Given a non-zero point P on a curve, we investigate the Galois action on the splitting fields of preimages of P under multiplication-by-l maps. We give a group-theoretic structure theorem for the corresponding Galois group, and state a conjecture regarding composita of two such splitting fields.
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Hughes, Matthew Phillip, "Galois Representations From Non-Torsion Points on Elliptic Curves" (2013). Senior Projects Spring 2013. 317.