Cohomological Characterization of CM Curves

This question asks about a specific open problem posed by Nicholas Katz in his 1976 Annals paper on p-adic interpolation, concerning the relationship between complex multiplication and the algebraic origin of the Hodge decomposition.

Background

For an elliptic curve E with complex multiplication by a quadratic field F, Katz observed that:

• The action of F× on the algebraic de Rham cohomology H¹ₚₘ(E,K) induces a decomposition
• This algebraically-defined decomposition coincides with the Hodge decomposition H¹ₚₘ(E,ℂ) = H¹⁰ ⊕ H⁰¹
• For ordinary good reduction at p, it also aligns with the Dwork-Katz decomposition

The Question

Katz asked for a converse: If the Hodge decomposition of H¹ₚₘ(E,ℂ) for an elliptic curve E is induced by a splitting of the algebraic de Rham cohomology, must E have complex multiplication?

Status of the Problem

This question has indeed been answered affirmatively. The result follows from more general theory regarding Hodge structures:

1. For an elliptic curve E defined over a number field, if the Hodge decomposition of H¹ₚₘ(E,ℂ) is defined over the base field (i.e., comes from an algebraic splitting), then E must have complex multiplication.

2. This is a special case of a more general phenomenon: for a variety X, if the Hodge decomposition of H^n(X,ℂ) is "algebraic" in this sense, then the corresponding Hodge structure has a large endomorphism algebra.

The proof relies on the fact that for a non-CM elliptic curve, the Mumford-Tate group is GL₂, which doesn't preserve any non-trivial decomposition of the cohomology. Only when the Mumford-Tate group is smaller (which happens precisely in the CM case) can the Hodge decomposition have an algebraic origin.

This result appears in work by Deligne on absolute Hodge cycles and has been further developed in the theory of motives and Hodge structures.