Here are two questions I had about the past two number theory seminars. I haven’t had the opportunity to think about either of them seriously, so they may be easy (or more likely stupid).
Anthony Várilly-Alvarado: Honestly, I’ve never quite forgiven this guy for his behavior as an undergraduate. He was my TA when I taught Complex Analysis at Harvard, and he had the bad manners to do an absolutely wonderful job and be beloved by all the students. Nothing makes a (first time) professor look worse than a good TA. (It means I can’t even take any credit for the students in the class who became research mathematicians). Anyway, Tony gave a talk on his joint work with Dan Abramovich about the relation between Vojta’s conjecture and the problem of uniform bounds on torsion for abelian varieties. (Spoiler: one implies the other.) More specifically, assuming Vojta’s conjecture, there a universal bound on \(m\) (depending only on \(g\) and \(K)\) beyond which no abelian variety of dimension \(g\) over \(K\) can have full level structure.
If one wanted to prove this (say) for elliptic curves, and one was allowed to use any conjecture you pleased, you could do the following. Assume that \(E[m] = \mu_m \oplus \mathbf{Z}/m \mathbf{Z}\) for some large integer m. One first observes (by Neron-Ogg-Shafarevich plus epsilon) that E has to have semi-stable reduction at primes dividing N_E. Then the discriminant \(\Delta\) must be an \(m\)th power, and then Szpiro’s Conjecture (which is the same as the ABC conjecture) implies the desired result.
If you try to do the same thing in higher dimensions, you similarly deduce that A must have semi-stable reduction at primes dividing N_E. edit: some nonsense removed. One then gets implications on the structure of the Neron model at these bad primes, which one can hope to parlay in order to get information about local quantities associated to A analogous to the discriminant being a perfect power. But I’m not sure what generalizations of Szpiro’s conjecture there are to abelian varieties. A quick search found one formulation attributed to Hindry in terms of Faltings height, but it was not immediately apparent if one could directly deduce the desired result from this conjecture, nor what the relationship was with these generalizations to either ABC or to Vojta’s conjecture.
Ilya Khayutin: Ilya mentioned Linnik’s theorem that, if one ranges over imaginary quadratic fields in which a fixed small prime is split, the CM j-invariants become equidistributed. The role of the one fixed prime is to allow one to use ergodic methods relative to this prime. My naive question during the talk: given p is split, let \(\mathfrak{p}\) be a prime above p. Now one can take the subgroup of the class group corresponding to the powers of \(\mathfrak{p}.\) Do these equidistribute? The speaker’s response was along the lines that it would probably be quite easy to see this is false, but I didn’t have time after the talk to follow up. It’s certainly the case that, most of the time, the prime \(\mathfrak{p}\) will itself generate a subgroup of small index in the class group (the quotient will look like the random class group of a real quadratic field), but sometimes it will be quite large. For example, I guess one can take
\(\displaystyle{D = 2^n – 1, \qquad \mathfrak{p}^{n-2} = \left(\frac{1 + \sqrt{-D}}{2}\right)},\)
and the subgroup generated by this prime has order \(\log(D)\) compared to \(D^{1/2 + \epsilon}.\) So I decided (well, after writing this line in the blog I decided) to draw a picture for some choice of Mersenne prime. And then, after thinking a little how to draw the picture, realized it was unnecessary. The powers of \(\frak{p}\) in this case are given explicitly by
\(\displaystyle{\mathfrak{p}^m = \left(2^m, \frac{1 + \sqrt{-D}}{2}\right)},\)
It is transparent that for the first half of these classes, the first factor is much smaller than the second, but since the second term also has small real part, the ratio already lies inside the (standard) fundamental domain. Hence the corresponding points will lie far into the cusp. Similarly, the second half of the classes are just the inverses in the class group of the first half, and so will consist of the reflections of those points in \(x = 0\) and so also be far into the cusp. So I guess the answer to my question is, indeed, a trivial no. So here is a second challenge: suppose that 2 AND 3 both split. Then do the CM points generated by \(\mathfrak{p}\) for primes above 2 AND 3 equidistribute? Actually, in this case, it’s not clear off the top of my head that one can easily write down discriminants for which the index of this group is large. But even if you can, sometimes \(\mathbf{Z}^2\) subgroups get you much closer to equidistribution than \(\mathbf{Z}!\)
There’s a generalisation of Szpiro’s conjeture to jacobians of hyperelliptic curves (due to Paul Lockhart) that might be more along the lines of what you’re looking for:
On the discriminant of a hyperelliptic curve. (English summary)
Trans. Amer. Math. Soc. 342 (1994), no. 2, 729–752.
Interesting, thanks! That paper does point to the fact, however, that the relationship between Szpiro and ABC is somewhat special, and it’s probably correct to think (for the problem at hand) in terms of Vojta’s conjectures rather than ABC.