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Asking for the moon, in my view. Here are 10 "heuristics" that try to place the theory. NB that many people stop at #1, as if this were enough. None of these points is particularly easy to track in th …

Armand Borel's Bourbaki Seminar 121 Groupes algébriques is from 1955, and uses "drapeau" (page 7). (It's online at archive.numdam.org.) This may not be the earliest occurrence, but there is a good rea …

Here are three points, and you'd have to care about at least one of them, I think. (1) A (co)homology class is better understood if it is represented geometrically in some way. This point really bel …

The "classical" example is surely duality of abelian varieties. If you want this duality to work over finite fields (or in characteristic p generally), it becomes apparent that you can't work with var …

Topics in Complex Function Theory, Abelian Functions and Modular Functions of Several Variables by C. L. Siegel is a standard reference using complex function theory. There are older works (e.g. H. F. …

The grandfather of all examples is by Gauss: http://en.wikipedia.org/wiki/Weil_conjectures#Background_and_history Of course Gauss didn't mention finite fields other than the prime field. I think it …

As you say, projective modules correspond in a highly moral way to vector bundles. Bundles pull back but do not push forward, in topologists' terms. This might be a good point at which to start. Sheav …

The first (set theory) formula is generalised in categorical logic to what is called "Frobenius reciprocity" there, and is then part of the handling of the existential quantifier (a natural way to go …

You have a typical recursively enumerable set S of integers, and a set X of lattice points cut out by a multivariate polynomial. We are talking about S being the projection (onto one axis) of X. Given …

I think your statement could usefully be sharpened in a couple of ways. Firstly, the state-of-the-art is that true statements for function fields are expected to have analogues for number fields. Th …

Well, if you are interested in counting solutions mod p, you could note that the "good" formulae are indeed related to the homogeneous approach/projective space. It is not just a question of restoring …

This should follow from the Theorem of Frobenius mentioned on p. 7 (PDF numbering) of http://websites.math.leidenuniv.nl/algebra/chebotarev.pdf.

Abelian varieties.

To discuss generally first, the book was written up by C. P. Ramanujam, and he was more conscientious than usual in trying to tie down Mumford's lectures to existing references. Still, it is quite har …

You didn't mention Weil, Basic Number Theory, where the case of a finite field of constants is handled, really only using Pontryagin duality. There is an elegant theory of John Tate that seems somewha …