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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