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p-div gps over a finite field (e.g., $\mathbb F_p$) are classified by Dieudonne modules.

There are also results for p-div gps over integers of local fields (e.g., over $\mathbb Z_p$).

However, are there results for p-div gps over local fields (e.g.,$\mathbb Q_p$)? I couldn't find any references.. Or more generally, p-div gps over other fields?

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    $\begingroup$ Over a field of characteristic zero all formal groups are isomorphic. It is an elementary exercise to show the existence of a logarithm, i.e an isomorphism with the additive formal group. This is found, for example, in Silverman's Arithmetic of Elliptic Curves. $\endgroup$ Nov 11, 2016 at 13:03
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    $\begingroup$ A $p$-divisible group over a field of characteristic $\ne p$ is exactly equivalent to the data of a continuous representation of the absolute Galois group on a finite free $\mathbf{Z}_p$-module. In that generality these are total beasts, and there's not much one can say that is interesting. Over $p$-adic fields and number fields one can say much more by bringing in techniques of $p$-adic Hodge theory, but that is a very long story. $\endgroup$
    – nfdc23
    Nov 11, 2016 at 14:25
  • $\begingroup$ @nfdc23 "but that is a very long story": Can you point some references? or names of mathematicians? thanks! $\endgroup$
    – guestguest
    Nov 11, 2016 at 21:13
  • $\begingroup$ The range of topics around Galois representations and the Langlands program that have been influenced by $p$-adic Hodge theory is staggeringly vast (involving also Sato-Tate, Fermat's Last Theorem, etc.). Just Google "$p$-adic Hodge theory" or "Fontaine-Mazur Conjecture", for example. But this is getting way too far from what seems to have been the intent of your question, and $p$-divisible groups over fields not of char. $p$ (without reference to integral structure) are just $p$-adic Galois representations by another name; it is not a "good" way to view the latter. $\endgroup$
    – nfdc23
    Nov 12, 2016 at 1:04
  • $\begingroup$ @nfdc23 I see...so p-div gp over Q_p are really just Z_p representation of Gal(Q_P), right? $\endgroup$
    – guestguest
    Nov 12, 2016 at 9:03

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