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In different articles I have seen different definitions of Barsotti-Tate representations. I am wondering if and how these definitions are equivalent.

In Section 1.1 of Conrad-Diamond-Taylor they say that, for $K/\mathbb{Q}_\ell$ a finite extension and $E\subset \overline{\mathbb{Q}}_\ell$ some coefficient field and $V$ a 2-dimensional vector space over $E$, a Galois representation $$ \rho: G_K \to GL(V) $$ is Barsotti-Tate if there exists an $\ell$-divisible group $\Gamma_{/\mathcal{O}_K}$ such that $\rho$ acting on a stable $\mathcal{O}_E$-lattice inside $V$ is isomorphic as a $\mathbb{Z}_\ell[G_K]$-module to the generic fibre of $\Gamma_{/\mathcal{O}_K}$ with its action by $G_K.$ (This definition explains the name.)

In Section 2.3.1 of Gee-Liu-Savitt they say that the representation $\rho$ as above is Barsotti-Tate if for all embeddings $\tau:K\to \overline{K}$ we have that the $\tau$-labelled Hodge-Tate weights are $HT_\tau(V)=\{0,1\}.$

It is not obvious to me that these definitions are equivalent. Is this easy to see? Is a reference for this fact available?

Thanks.

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    $\begingroup$ Isn't it more or less Théorème 1.4 from Breuil's paper GROUPES p - DIVISIBLES, GROUPES FINIS ET MODULES FILTRES (Annals of Math. 151, 2000, 489-549. ? (here is a link from Breuil's webpage : math.u-psud.fr/~breuil/PUBLICATIONS/p-divisibles.pdf ) $\endgroup$
    – user65490
    Dec 19, 2017 at 15:18
  • $\begingroup$ Ah, great! I didn't know about this reference before, but it seems to do the trick. I'd still be interested to hear from someone if there is an intuitive reason why one would expect this to be true. Also, is this result known for p=2 or are there some problems in that case? $\endgroup$
    – Misja
    Dec 19, 2017 at 22:02
  • $\begingroup$ For $p=2$ the theorem is proved in a paper of Kisin. $\endgroup$ Jan 5, 2018 at 9:17

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