what are the standard references for this subject, i.e., the realization of functions as traces of Frobenius acting on sheaves? Is there any motivation or philosophy coming from categorification, number theoretic analogies, or other directions?
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3 Answers
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Frenkel's Recent Advances in the Langlands Program and his book on loop groups have some introductory discussion of Grothendieck's function-sheaf dictionary. Earlier references (in French) include sections 1.1 and 1.2 of Laumon's 1987 paper Transformation de Fourier, constants d'equations fonctionelles, et conjecture de Weil, SGA 4.5, and (the first reference) SGA 5, Exp 12 and 15.
The short answer to your last question is "yes". One reason to use sheaves instead of functions is that you can do more with them, i.e., they have more functorial descriptive power. There are three operations on functions: upper star yields pull-back, derived tensor product yields multiplication, and lower shriek (some say "bang" or "surprise") yields integration along fibers. Unlike functions, l-adic sheaves also have a dualizing object, which gives lower star, upper shriek, and internal hom.
answered Nov 16, 2009 at 18:32
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My preferred reference is Kiehl and Weissauer's book (very complete, and in English).
Of course, if you want motivation, you can read my blog post (this the companion to the one Thomas linked to).
answered Nov 16, 2009 at 22:42
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The secret blogging seminar contains some posts on that. I'd say that categorification as motivation for it works best, but the historical reason were the Weil conjectures. Shin's description , then the surveys of the geometric Langlands program in the arxiv contain often some description of it.
answered Nov 16, 2009 at 13:13