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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 …

Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2 by Cassels and Flynn may help. That is, depending on whether the genus 2 case is enough to get started, and what type of questions you have …

The papers by Roquette http://www.rzuser.uni-heidelberg.de/~ci3/rv.pdf, http://www.rzuser.uni-heidelberg.de/~ci3/rv2.pdf, http://www.rzuser.uni-heidelberg.de/~ci3/rv3.pdf and http://www.rzuser.uni- …

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 …

In the case of Hilbert modular surfaces, continued fractions appeared in the work of Hirzebruch on their singularities. This is easy to find online. This generalises somewhat in Shintani's work, as wa …

I was once told, by someone who would likely be right about such things, that the version of Weil reciprocity for abelian varieties (as in Lang, Abelian Varieties) should come out of consideration of …

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 …

Readers of MO are probably aware of the pedagogic need that would provoke such a query. It's around 60 years since Serre's FAC, and I imagine some people would say "you still have to read the original …

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 …

Edited: One point is that Hodge's original version of the conjecture was wrong, and in a couple of ways. You do need rational coefficients (integral is too much to ask for, see ref below). Also a mor …

Doesn't this follow quite quickly by setting one variable equal to 0? Edit: I was thinking this way. Factors of homogeneous polynomials are homogeneous. Setting the final variable $x_n$ to 0 therefo …

The reason Kummer theory is involved is that the Galois covering of an elliptic curve E created by multiplication by n, assuming that n is prime to the characteristic of the base field K, has Galois g …

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 …

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 …

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 …