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It is generally not known if a smooth variety over a perfect field embeds into a smooth proper variety.

Lemma 5.4 in Formes modulaires et représentations $\ell$-adiques provides such an embedding for some specific variety and this allows one to deduce the Ramanujan conjecture from Weil I.

If one also knows Weil II then the embedding is not necessary (or I guess if one knows alterations then the Weil II result follows from Weil I).

However if we keep playing geometry and forget about cohomology does Lemma 5.4 generalize in some way to Shimura varieties of PEL type?

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    $\begingroup$ I can't check the exact statement of Deligne's lemma 5.4 right now, but Kai-Wei Lan has a lot of work in defining toroidal compactifications of PEL Shimura varieties. See for instance his PhD thesis $\endgroup$
    – Wojowu
    Commented Sep 29, 2021 at 12:54
  • $\begingroup$ We would need a compactification of a "Kuga-Sato variety" (I'm not sure about the exact definition outside of the modular curve case). $\endgroup$
    – soft-drinks
    Commented Sep 29, 2021 at 12:56
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    $\begingroup$ Yes, there are compactifications of Kuga-Sato varieties, and Lan's work is the place to look for them. Check out Harris--Lan--Taylor--Thorne for a major application of these ideas to constructing Galois representations (hence rather in the spirit of Deligne's work) $\endgroup$
    – David Loeffler
    Commented Sep 29, 2021 at 12:58

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