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Let E be a rational elliptic curve with CM by the ring of integers of a quadratic imaginary field, and let p be an odd prime of good ordinary reduction for E. The prime p splits in K. Then we have at least two essentially different ways to attach to E a p-adic avatar of its Hasse-Weil L-series (which we know it coincides with the Hecke L-series of its grossencharakter and that of its conjugate). In fact, working over Q, we have the p-adic L-function constructed by Mazur and Swinnerton-Dyer, and working over K, we can use Katz's work to attach to the grossencharakter of E an analogous one-variable p-adic L-function.

Do we know how these p-adic L-functions are related to each other?

(They are both conjecturelly related to the arithmetic of E via respective p-adic versions of the Birch and Swinnerton-Dyer conjecture, but I am wondering -- and this is the stress of the above question -- if the precise relation between the mere different constructions has been studied.)

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Katz discusses the relationship in section 9.5 of his paper "p-adic Interpolation of Real Analytic Eisenstein Series". See specifically his comparison theorem 9.5.25. –  Rob Harron Jan 3 '11 at 4:00
Thanks for the reference, Rob H. Also, I have just found that in the last section of Rubin's paper "The 'main conjectures' of Iwasawa theory for imaginary quadratic fields" he points out that this result of Katz allows one to show that "the restriction of the twist [defined above] of the [Katz] two-variable L-function to the 'cyclotomic line' is the Mazur and Swinnerton-Dyer L-function". –  monodromy Jan 4 '11 at 21:25
Now I wonder if one can hope for a similar relation for a CM parabolic eigenform $f$ of higher weight $k>2$ with rational $q$-expansion, where all the ingredients are still available: the work of Mazur-Tate-Teitelbaum attaches to such an $f$ a cyclotomic $p$-adic $L$-function, and one can use Katz's work as before, using now the grossencharaker of infinity type (k-1,0) of $f$, to attach another $p$-adic $L$-function to $f$... –  monodromy Jan 4 '11 at 21:34

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