Suppose $K$ is an imaginary quadratic field (with class number 1, for simplicity), $p \ne 2$ a prime split in $K$, and $K_\infty$ the $\mathbb{Z}_p^2$-extension of $K$.
If $E / \mathbb{Q}$ is an elliptic curve with CM by $K$, then there is a construction (due to Katz) for a "two-variable $p$-adic $L$-function" attached to $E$, which is a $p$-adic measure on the Galois group $K_\infty / K$, interpolating $L$-values of the twists of the Groessencharacter of $K$ attached to $E$ by finite-order characters of p-power conductor. See e.g. de Shalit's book "Iwasawa theory of elliptic curves with complex multiplication" (Academic Press, 1987)
If $E / \mathbb{Q}$ is any elliptic curve with good ordinary reduction at $p$ (or more generally any ordinary modular form of weight $\ge 2$), but not necessarily with CM by $K$, there is also a construction of a two-variable $L$-function attached to $E$, written down by Perrin-Riou (J. London Math. Soc 38 (1988), 1-32) based on earlier work by Hida and others. This interpolates $L$-values of the twists of $E$ by certain 2-dimensional Artin representations of $\mathbb{Q}$, obtained by inducing up finite-order characters of $\operatorname{Gal}(K_\infty / K)$.
My question is this: if we apply Perrin-Riou's method to an $E$ which happens to have CM by $K$, then what is the relation between the $L$-functions coming from the two constructions?
(My impression is that Perrin-Riou's construction should give the product of Katz's $L$-function with its conjugate, corresponding to the decomposition of the Tate module of $E$ as a $\operatorname{Gal}(\overline{K} / K)$-representation into the direct sum of two conjugate characters. But I'm puzzled by the discrepancy of coefficient fields: Perrin-Riou's measure takes values in some finite extension of $\mathbb{Q}_p$, while Katz's lives in the completion of the unramified $\mathbb{Z}_p$-extension of $\mathbb{Q}_p$, which is far larger.)
Best Answer
The conditions on K in Perrin - Riou's paper (Heegner hypothesis) probably exclude the case that we can take K to be the field of complex multiplication.