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My question is about p-adic Hodge-Tate theory and p-adic Galois representation.

One of the important semi-linear object in p-adic Galois representation is the $\text{Breuil Module}$. There are examples of Breuil modules.

My question-

Is it easy to give or find examples of Breuil Modules?

Would it be a little but good exercise to construct or find out examples of Breuil modules ?

Any comment will be helpful to because I am new in this area.

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    $\begingroup$ It might help to give a link to the exact definition of Breuil module that you have in mind (there are several variants). $\endgroup$
    – David Loeffler
    Commented Jan 7, 2020 at 9:25
  • $\begingroup$ @DavidLoeffler, I did not quiet understand what you are meaning. But if i change some pre-materials to define Breuil modules, such as some restriction or addition to the divisible module. Are you saying something like this? $\endgroup$
    – MAS
    Commented Jan 7, 2020 at 11:41
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    $\begingroup$ I am saying that you should add to your question a reference (or better still a web link) which gives the definition of a Breuil module. $\endgroup$
    – David Loeffler
    Commented Jan 7, 2020 at 15:22
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    $\begingroup$ I can confirm David's feeling (in his answer) that it is fun to try to translate the paper of Berger, Li, and Zhu into the language of Breuil-Kisin modules. See arxiv.org/abs/1908.09036. Please let me know (here or in email) if you have any question. $\endgroup$
    – jfb
    Commented Jan 10, 2020 at 17:45
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    $\begingroup$ @JohnBergdall Aha, I didn't know that you had already worked out the "exercise" I suggested! Your paper looks to be a fantastic source for the OP (and others) to learn how to work with integral p-adic Hodge theory explicitly. $\endgroup$
    – David Loeffler
    Commented Jan 13, 2020 at 9:50

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It might be worth distinguishing here between two different but related constructions:

  • Breuil--Kisin modules, which are finite free modules over a relatively simple base ring, namely $\mathfrak{S} = W[[u]]$ where $W$ is the Witt vectors of the residue field;

  • Breuil modules, which are finite free modules over a rather more complicated ring $S$ containing $\mathfrak{S}$ (obtained from $\mathfrak{S}$ by some divided-power envelope construction).

B-K modules are simpler and easier to write down, and you can get a Breuil module from a B-K module by base-extension; so you might be well-advised to start by writing down some examples of Breuil--Kisin modules.

A nice exercise might be to try to write down some Breuil--Kisin modules of rank 2. There are some very nice examples of explicit rank 2 Wach modules (which are in many ways analogous to Breuil--Kisin modules, but only work when the base field is unramified) in a paper of Berger, Li and Zhu from 2004; it might be fun to try to translate their examples into the language of Breuil--Kisin modules, and see if you can extend them to some ramified base fields.

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  • $\begingroup$ Thank you very much for valuable answer $\endgroup$
    – MAS
    Commented Jan 7, 2020 at 16:17

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