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 H1rmdR(E,K) in eigenspaces for the natural Ftimes-action coincides with the Hodge decomposition of H1rmdR(E,BbbC) and (for ordinary good reduction at p) with the Dwork-Katz decomposition of H1rmdR(E)otimesB for p-adic algebras B.
Then, he asks for a converse statement. Namely, is it true that if the Hodge decomposition of H1rmdR(E,BbbC), 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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