15
$\begingroup$

Let $f(x) \in \mathbb Z[x]$ be a monic polynomial of degree $g$ with all roots having absolute value $\sqrt{q}$. How many principally polarized abelian varieties over $\mathbb F_q$ have $f(x)$ as the characteristic polynomial of Frobenius?

By "how many" I mean the groupoid cardinality - the sum over all isomorphism classes of $1$ over the order of the group of polarization-preservering automorphisms.

Is there a simple formula for it, like there is in the $g=1$ case (at least if you consider $L$-functions simple)?

I found some description of this set in the literature but I wasn't able to extract a formula from it.

Even if no formula exists, I would be happy to have an upper bound. Is it $O(q^{g(g+1)/4 + \epsilon})$? Note that there are $\approx q^{g(g+1)/2}$ ppavs of genus $g$ and $\approx q^{g(g+1)/4}$ polynomials, so this would say that the number doesn't vary much from its average.

Improve this question
$\endgroup$
4
  • 2
    $\begingroup$ Is there a missing square root, as in $\sqrt{q}$? $\endgroup$
    – Vesselin Dimitrov
    Commented Sep 11, 2015 at 21:27
  • 2
    $\begingroup$ It's a (Kronecker) class number. For elliptic curves, this is (of course) in Deuring. Good references are Schoof J. Comb. Th. 1987, for elliptic curves and Waterhouse, Ann. Sci ENS 1969, in general (but maybe the polarization is an issue in dim > 1). $\endgroup$
    – Felipe Voloch
    Commented Sep 11, 2015 at 22:32
  • $\begingroup$ @FelipeVoloch Thanks for the reference! Unfortunately I think the polarization is crucial because that's where the unit group will come in - the class group is not a nice count because you should really divide by the order of the automorphism group, which is infinite. I think that's related to why there isn't as nice an analytic formula as in the imaginary quadratic case - the regulator appears in the class number formula. $\endgroup$
    – Will Sawin
    Commented Sep 12, 2015 at 15:19
  • $\begingroup$ For $g=2$ there are papers of Howe, Maisner, Nart, Ritzenthaler which may help. $\endgroup$
    – Felipe Voloch
    Commented Sep 12, 2015 at 15:38

0

Reset to default

You must log in to answer this question.