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You have chosen an example where the moduli space (a projective space) is a homogeneous space. So, geometrically, all the objects it parametrises are "the same", and the nature of the moduli space mer …

The question about the projective line seems more fundamental. And geometrically this seems to be about the graphs of rational functions. Certainly the function field of V provides morphisms from a de …

I would expect not. The reason being that if this is anything like true (products of quadratic forms and and linear forms being dense in the ordinary "strong" or norm topology - presumably what you me …

Abelian varieties.

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 …

Talking about the arithmetic genus (http://en.wikipedia.org/wiki/Arithmetic_genus), it's the alternating sum of Hodge numbers all of which are 0. So, in short, yes.

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 …

This should work out for you, with a bit more geometry and perhaps a little trickery. You are asking for points over finite fields avoiding a given hypersurface, in affine space. It is conceptually a …

I'm not quite sure about the formulation of the question: but there is something worth saying anyway, since it isn't often emphasised in basic texts. The points of order 3 can be identified with the i …

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 …

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 …

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 …

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 …

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 …

If we think about Diophantine equations in general, the situation is "hopeless". That's a theorem. Nevertheless in number theory we want to study such equations, in special cases at least, so some ide …