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There is much important work written in Weil's language of algebraic geometry rather than schemes (besides Weil himself, I can think of Shimura, Neron immediately).

My question is: is it worth the effort (e.g. time) to read these? This, at least seems to me, to fall under the umbrella of "reading the masters".

I have realized that this question is a little broad, indeed much broader than some related questions like

Scheme theoretic interpretations of the Weil's foundations of algebraic geometry

Some arithmetic terminology: "universal domain", "specialization", "Chow point"

where some specific parts of Weil's language are discussed. But I do not want to miss the beautiful mathematics in these work, and I am not an expert to be sufficiently informed.

From the related questions above, I understand that at least parts of the Weil language have been subsumed by e.g. SGA. But how about other important work e.g. Shimura's? (again, not an expert myself, so I can only think of Shimura/Neron; and Neron models have been addressed in scheme language) Translating it on-demand also requires actually understanding the language.

Background: I might as well ask the question on math.stackexchange, but I am thinking maybe mathoverflow is a better venue given the different types of questions asked at the 2 places. I am in the learning process hoping to reach research-level in arithmetic geometry; in this direction: have gone through parts of Hartshorne/Mumford-Oda/Liu/Vakil's notes/Gortz-Wedhorn (yeah, AG is not easy for me, I have to learn from different sources), and some number theory from Neukirch's book, in the process of reading Silverman's books on elliptic curves.

Thank you very much!

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    $\begingroup$ "With all due respect to the role of Andre Weil in the development of algebraic geometry, nobody should ever again have to read Weil's "Foundations of algebraic geometry": EGA must be an adequate logical starting point for the subject. Hence, if there is an important, interesting, or useful theorem whose published proofs use pre-Grothendieck methods in such an essential way so as to render them impenetrable to later generations (or to me?), and if I have a need to understand why the theorem is true and consequently I figure out a scheme-theoretic proof, then I'll try to write it up." $\endgroup$
    – Paul Siegel
    Commented May 8, 2016 at 13:54
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    $\begingroup$ I bought a used copy of Weil's Foundations long ago as a student, but I found it pretty impenetrable. So for me the answer was no, but I don't want to discourage anyone else from trying. By the way Serre, doesn't using Weil's language in his book. His approach is closer to FAC i.e. it's sheaf theoretic. $\endgroup$
    – Donu Arapura
    Commented May 8, 2016 at 15:07
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    $\begingroup$ There are several important papers by Gerstenhaber on varieties of commuting matrices that use Weil's language, for example jstor.org/stable/1970336?seq=1#page_scan_tab_contents. $\endgroup$
    – Mark Wildon
    Commented May 8, 2016 at 15:45
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    $\begingroup$ @PaulSiegel: Conrad also wrote a book with Chai and Oort which explain in particular the main theorem of complex multiplication for abelian varieties in the schematic language. bookstore.ams.org/surv-195. See also math.stanford.edu/~conrad/vigregroup/vigre04/mainthm.pdf $\endgroup$
    – ACL
    Commented May 8, 2016 at 16:37
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    $\begingroup$ I would suggest reading a bit of one of the papers you find intriguing and decide for yourself. I.e. if a master writes a paper, it will contain insights, so it is up to you to overcome the burden of the language. I once read a few pages of Zariski on simple points, and benefited enormously. I never made headway on Weil's Foundations, but some have done so. A related example is Riemann's works in the original, and in translation. Everyone told me they were impenetrable, but they were not, and reading them rendered his work much more transparent than later versions. just read a few pages. $\endgroup$
    – roy smith
    Commented May 9, 2016 at 4:18

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