Hi KConrad; yes, the lcm of the denominators gets closer to 1 at each step (as Qiaochu said already in the question statement). The unanswered question is whether there is a conceptual proof of this that is any clearer than the proofs given so far. For example, can one make it conceptually clear why the relevant hypothesis on the quadratic form is that the absolute value of its value at the "error vector" in the lattice point approximation should be between 0 and 1?

Well, in any case, it's a nice solution.

I am reminded of Hendrik Lenstra's proof that there are infinitely many composite numbers: Suppose that there are only finitely many composite numbers. Multiply them together. DON'T add 1!

@Hunter: I see that the triviality of H^1(L,G)^{Gal(L/F)} is sufficient, but why is it necessary?

It would be better to write "strictly increasing" in place of "increasing", to guarantee that $f^{-1}$ exists.

I personally feel that almost every sentence in this solution could be improved by including the justifications for claims made, starting with the sentence beginning "Clearly...". I hope that Anton will choose to expand it!

@David: There are local obstructions in the bad reduction case. For example, $x^3+2y^3+4z^3=0$ has no $2$-adic point (consider the $2$-valuations of the three monomials!) Any cubic polynomial congruent to this one mod 8 defines a curve with no $2$-adic point. This shows that a positive fraction of the curves is excluded.

Regarding the answer to 2.i), I think the answer should be no for all N because a pair (E,C) with E an elliptic curve and C a cyclic subgroup always has at least the automorphism -1, no matter what N is. Maybe we are not all thinking about the same moduli problem?

@JSE: I think you meant to have 2g(X)-2 on the LHS of your Riemann-Hurwitz, in which case you need n greater than 2g+2 (the number of fixed points of the hyperelliptic involution on a hyperelliptic curve of genus g). As for algebraic space vs. scheme, it's going to be a scheme since if you include level structure what you have is quasi-projective.

Essentially the same question was asked earlier on this site: mathoverflow.net/questions/6676/…

It was Shafarevich who at the 1962 ICM asked whether this set was empty, so maybe he should get the mathoverflow "reputation". :) According to Parshin, Abrashkin proved the result independently of Fontaine at about the same time: see the addendum at the end of Fontaine's paper, and see Abrashkin, V. A. Galois modules of group schemes of period $p$ over the ring of Witt vectors. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 4, 691--736, 910; translation in Math. USSR-Izv. 31 (1988), no. 1, 1--46.

By the way, this covering is the same as the tower of modular curves X(7) --> X_1(7) --> X_0(7). In particular, it can be extended to an even larger tower X(7) --> X(1) with Galois group PSL_2(F_7), in which the upper triangular matrices form the order-21 group.