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In his 1976 classical Annals paper on $p$-adic interpolation, N. Katz uses the fact that if $E_{/K}$ is an elliptic curve with complex multiplications in the quadratic field $F$, up to a suitable tensoring the decomposition of the algebraic $H_{\rm dR}^{1}(E,K)$ in eigenspaces for the natural $F^\times$-action coincides with the Hodge decomposition of $H_{\rm dR}^{1}(E,{\Bbb C})$ and (for ordinary good reduction at $p$) with the Dwork-Katz decomposition of $H_{\rm dR}^{1}(E)\otimes B$ for $p$-adic algebras $B$.

Then, he asks for a converse statement. Namely, is it true that if the Hodge decomposition of $H_{\rm dR}^{1}(E,{\Bbb C})$, where $E_{/K}$ is an elliptic curve, is induced by a splitting of the algebraic de Rham, then $E$ has complex multiplications?

The question is left unanswered in that paper. Does anyone know if the question has been answered since?

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What is $K$ in this question? Too lazy to find Katz' paper in the mess behind me :-) To compute the de Rham cohomology with coefficients in $C$ you'll need a map $K\to C$ right? Or are we considering all such maps? Or have I misunderstood the question? –  Kevin Buzzard Feb 2 '10 at 11:49
Ok, I guess I was a bit lazy myself too..... :-) $K$ is a finite extension of $\Bbb Q$ over which the complex multiplications are defined, that is $F\subseteq K$. The comparison between algebraic $H^1$ and the Rham over $\Bbb C$ is allowed by choosing (and fixing) an embedding $K\rightarrow{\Bbb C}$. –  Andrea Mori Feb 2 '10 at 14:09

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