A deformation of the Tate module (Q1577630)

Let \(p\geq 7\) be a prime, \(\Lambda = {\mathbb{Z}}_p[[X]]\), \(K\) a number field, \(\widetilde K\) the field generated over \(K\) by adjoining all \(p\)-power roots of unity, \(\overline K\) an algebraic closure of \(K\). Let \(E\) be an elliptic curve over \(K\) with \(j(E) \neq 0, 1728.\) Building on and extending his results from [J. Algebra 229, 249-279 (2000; Zbl 1001.12004)] the author constructs a representation \(\rho: \text{Gal} (\overline K/\widetilde K) \rightarrow \text{SL}(2,\Lambda)\) such that the following conditions hold for some thin set \(W\subset K\), which does not depend on \(E\): The specialization of \(\rho\) at \(X=0\) is the usual Galois-representation on the Tate module \(T_p(E)\); if \(j(E)\not\in W\), then \(\rho\) is surjective; the fixed field of ker\((\rho)\) is Galois over \(K\), but for \(j(E)\not\in W\) \(\rho\) does not extend to Gal\((\overline K /K)\); \(\rho\) is unramified outside places which divide \(p\) or where \(E\) has bad reduction. This result is derived by specializing a result for transcendental \(j\)-invariant to the number field case -- where \(W\) enters via Hilbert's irreducibility theorem. As a consequence, there does exist not only the true division tower of \(E\) over \(K\), but also a lot of ``false'' division towers, having the same Galois group SL\((2,{\mathbb{Z}}_p)\).