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In his famous book "Selmer complexes", J.Nekovar, speaking about Greenberg conditions, wrote the following sentence (see 0.8.1 p.10)

these are the only local conditions that can be handled by elementary methods; the general case would require a heavy dose of crystalline machinery, which is not yet available.

Even though I know all the concepts involved in this sentence, I have absolutely no idea what it means. More precisely

  1. What does he mean by “the only local conditions that can be handled by elementary methods" ? EDIT as pointed out in the comments "elementary methods" are methods used in the book but the question is rather why Greenberg's conditions are the only ones to which these methods apply.

  2. What was not available in crystalline cohomology at the time he wrote this sentence?

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  • $\begingroup$ I think the "elementary methods" refer to the content of the book itself. Nekovář uses the local conditions at places above $p$ to be the Greenberg conditions. I don't know what $p$-adic Hodge theory was/is missing to replace them by $H_f$ conditions in general. $\endgroup$
    – Chris Wuthrich
    Commented Jul 25 at 12:35
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    $\begingroup$ See arxiv.org/pdf/1404.7386 $\endgroup$
    – coLaideronnette
    Commented Jul 26 at 0:11
  • $\begingroup$ @ChrisWuthrich Yes you right this was obvious but the second question is less obvious I think. $\endgroup$
    – Marsault Chabat
    Commented Jul 26 at 1:51
  • $\begingroup$ @coLaideronnette thank you very much, but could you be a little bit more precise, where do you think I might find an answer in this paper? $\endgroup$
    – Marsault Chabat
    Commented Jul 26 at 1:52
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    $\begingroup$ Oh I found it @coLaideronnette it allows us to connect the Greenberg Selmer group and that of Bloch-Kato'! It's really enlightening, thank you very much! $\endgroup$
    – Marsault Chabat
    Commented Jul 26 at 4:16

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