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I've been working with the Hodge-Tate map in the context of modular forms, but I don't understand really what it is. For an elliptic curve $E$ over $K$, a number field, the Hodge-Tate decomposition is its Hodge-Tate decomposition as a Galois representation

$$ H^1(E,\mathbb{C}) = H^1(E,\mathcal{O}_K) \otimes_{\mathcal{O}_K} \mathbb{C} = H^0(E,\Omega^1_{E}) \oplus H^1(E,\mathcal{O}_E). $$

(Or something close to that.) There is an analog of this decomposition when we move to $\mathbb{C}_p$, the completion of $\overline{\mathbb{Q}}_p$, involving Tate twists to make it Galois equivariant.

Somehow the "Hodge-Tate map" $HT$ is the more natural inclusion of one of these pieces into the whole (I think $H^0(E,\Omega^1_E) \to H^1(E,\mathbb{C})$), while its dual $HT^\vee$ is the more natural projection. Most sources I've seen are very careful to only talk about the Hodge-Tate map for elliptic curves over $\mathbb{C}$ (or $\mathbb{C}_p$), and not for families of elliptic curves; while some others say that $HT$ can be thought of as the map in the Hodge filtration

$$ 0 \to \omega \xrightarrow{HT} \mathcal{H} \xrightarrow{HT^\vee} \omega^\vee \to 0. $$

Here $\omega$ and $\mathcal{H} = \mathcal{H}^\vee$ are sheaves over the moduli space $X$ whose fibers over $x$ are the relevant cohomology groups of the elliptic curve $E$ corresponding to $x$. This interpretation pretty directly puts $HT$ as existing for families of elliptic curves.

My question is this: when we talk about the "Hodge-Tate map" $HT$ in the context of modular forms, what exactly do we mean, and when exactly is it defined? Is it defined in families? Is it algebraic? If so, what rings is it defined over?

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