We consider the l adic representation of a number field K constructed from an elliptic curve E over K with given rational point 0. So this is basically the natural representation R:Gal(Q^{alg}/K) to Aut(V_l(E)), automorphisms of the vector space obtained by tensoring the rank 2 free Z_l Tate module T_l(E) with Q_l. I suppose that this is unrammified at the primes which P do not lie above l (which is to say that R(I_w)=1 for any prime w in Q^{alg} dividing P) but how do I show this?
What about l adic reps associated with varieties in general (the action of Gal(Q^{alg}/K) is on the etale-cohomology modules, in fact the representation described above may be recovered in this way)?
* For an elliptic curve E, it is shown in Serre's book, IV 5, that at a prime P not above l, the l-adic representation in question is unramified iff E has good reduction at P. For P above l, I don't know whether such a general criterion is available, but in Iwasawa theory for instance, people usually restrict themselves in replacing the absolute Galois group G by its quotient G_S ("unramified outside S"), where S is a finite set of primes containing the primes of bad reduction and the primes above l. As for the étale cohomology groups that you refer to, at least with the usual coefficients (Z_l, Q_l , Q_l/Z_l, etc. and their Tate twists), they coincide with the Galois cohomology groups of G_S (with the same coefficients)
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