I’m speaking on an “open problems” panel in honor of Dick Gross’s 60th birthday. I’ve got 10 minutes. I think I know what I’m going to say, but I was just informed that the panel is to be allowed to run late if audience interest demands it. So I thought it would be good to stock up a little more material. And it occurred to me this might be a good opportunity to blogsource! So, readers: if you were going to promote an open problem in front of Dick Gross and his wellwishers, what would it be?
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Quomodocumque 
I’ve been asking people the following question for a few months and I mentioned it to you in Tucson.
Let K be a number field and assume that, for all but finitely many places v of K, we are given an elliptic curve E_v/K_v with good reduction. Assume also that we are given, for each prime p, a representation f_p of the absolute Galois group of K to GL_2(Z_p). Now assume that, for all v and p, the restriction of f_p to a decomposition group of v is the same as the action of the absolute Galois group of K_v on T_p(E_v). Does it follow that there exists an elliptic curve E/K such that the f_p comes from T_p(E)?
I can do the case of K = rational numbers only.
Suppose E is a modular elliptic curve over a real quadratic field F, with associated Hilbert modular form f. Let X be the Hilbert modular surface on which f lives. There is a common submotive in H^2(X) and H^2(Res_{F/Q}(E)), so the Tate conjecture predicts an “interesting” codimension two cycle in the fourfold X \times Res_{F/Q}(E). What is this cycle?
How old were you when people stopped making fun of your name?
No, I’m just kidding — please don’t ask him this! Rather, if it’s not too late, please convey my warmest regards.