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thanks. I didn't know integral powers of the cyclotomic characters satisfy all those properties (except Hodge-Tate !!!). Now I can understand why Berger required conditions on the Hodge-Tate weights of the $\rho_n$.
Dear Joel. A while ago, I also heard a mathematician mentioning the thesis of a student of Coleman. As little as I know, he tried to construct some (symmetric tensor ???) L-function on the eigencurve. But there are serious mistake(s) in his work, and you are looking at it, according to that mathematician. Does your paper "critical p-adic L-function" address that problem, or is it just another project?
Dear Matt. Thanks for your answer. I'm not very clear with all the cohomology theories that people are using. A few months ago, I tried to read Deligne's paper (in 1969) that constructed l-adic representation from eigenforms, and then Langlands' one in the Antwerp II. Both attempts failed miserably...
Dear David, I take it back, because: 1) Emerton used the non-compactified (instead of the compactified) modular curve to define his completed cohomology group. 2) As Joel and Kevin pointed below, even we somehow have a natural map of C_p vector spaces, it's still not a good reason to believe there's a natural map between the Z_p modules.
Here's a brief reason for my "belief", hope there's no mistake: Let N and p be nice, and we define the space V as in Gouvea's book. Instead of taking the whole completed cohomology as in p.45 of Emerton's paper, we choose the "smaller piece" H^1(K^p)_A in p.44, where K^p is the Gamma_1(N)-problem, and A is Z_p. I think there's an embedding after tensoring both of these Z_p-modules by C_p.
