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I want to learn about homotopy theory on number fields, and I heard that the theory of motives made it possible, so I want to know what is a good textbook for motive theory.

To be honest, I don’t know algebraic geometry (I only read Hartshorne). So please tell me what is needed to get some knowledge about Algebraic Geometry (SGA? Or other book?).

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    $\begingroup$ If you're asking for a roadmap to motives and also references for general algebraic geometry beyond Hartshorne's book, I think the question is too broad for MO. If you actually intend to ask only about motives, anyway, have you made a google search? A search within past MO questions (e.g. mathoverflow.net/questions/11932/references-for-artin-motives)? What does/does not satisfy you about that material? $\endgroup$
    – Qfwfq
    Commented Jul 23, 2018 at 8:34
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    $\begingroup$ One reference for motives is Andre, Y. Une Introduction aux Motifs (SMF 2004). $\endgroup$
    – user19475
    Commented Jul 23, 2018 at 8:53
  • $\begingroup$ I intend both but mainly I want to know about motives. Thank you for letting me know it, I saw it just now. I just started using MO, so I didn’t know how to use... I will search past MO questions. Thank you very much,Qfwfq. $\endgroup$
    – Gear
    Commented Jul 23, 2018 at 9:01
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    $\begingroup$ A reference for learning motives is Lectures on the Theory of Pure Motives by Murre, Nagel, Peters. It does not address homotopy theory however. $\endgroup$
    – François Brunault
    Commented Jul 23, 2018 at 11:28

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I would add this as a comment, but I do not have enough reputation to do so.

While there are certainly more contemporary references, Voevodsky's "Triangulated category of motives over a field" is a place where you can read about motives (https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/s5.pdf). His paper with Morel on $\mathbb{A}^{1}$-homotopy theory would be a source to address your interest in homotopy theory (http://www.math.ias.edu/vladimir/files/A1_homotopy_with_Morel_published.pdf). A classical reference is Manin's article "Correspondences, motifs, and monoidal transformations."

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A wealth of information is contained in: Uwe Jannsen, Steven L. Kleiman, Jean Pierre Serre, “Proceedings of Symposia in Pure Mathematics, Vol 55, Parts 1 and 2”. Amer Mathematical Society (February 1, 1994)

This is from before the Voevodsky era. But I think you should know about the classical story before plunging into the derived category of motives. See especially the paper on “Classical motives” by Scholl in the above reference. However, since you mention homotopy theory, you certainly should work your way towards $\mathbb{A}^1$-homotopy theory à la Voevodsky and others.

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