This file contains the text of an email I sent to Nick Katz in late 
October of the year 2000. I have corrected two typographical errors, but
have left the letter otherwise unchanged. The math discussed in the 
letter became the basis for the paper "Variations in the distribution of
principally polarized abelian varieties among isogeny classes," 
described here: http://ewhowe.com/papers/paper50.html

Note that the URL and the affiliation in my .sig are now both outdated.

                                 --- Everett Howe, May 2020
                                 
========================================================================                                 


Date: Thu, 26 Oct 2000 11:48:55 -0700 (PDT)
From: "Everett W. Howe" <however@alumni.caltech.edu>
To: Nick Katz <nmk@math.princeton.edu>
Subject: Frobenius eigenvalue distribution

Dear Nick,

I've come up with a pretty simple heuristic argument "explaining"
the limiting distribution of Frobenius eigenvalues for principally
polarized abelian varieties over finite fields, and I wonder whether
you and Sarnak had thought of this as well.  I'm not optimistic about
the chances of turning the argument into an actual proof --- it uses
the Brauer-Siegel theorem at a certain point, and loses a lot of
accuracy at that point --- but it probably can be made to prove a
weaker version of your theorem with Sarnak.

Of course you know of the following heuristic for elliptic curves:
The number of elliptic curves with trace t over F_q is equal to
the class number of the imaginary quadratic order of discriminant
Delta = t^2 - 4*q, and you expect that the class number will be
about Sqrt(|Delta|), so you expect to get the familiar semi-circular
distribution of traces.  What I can do is generalize this heuristic
to higher dimensions.

Here's the argument:

Consider an isogeny class C of n-dimensional abelian varieties over
a finite field F_q.  Let's assume that C is *ordinary* and that C is
*simple*.  Then the Weil polynomial associated to C is an irreducible
polynomial f of degree 2n.  Let pi be a root of f in Qbar, let K be
the CM-field Q(pi), and let K+ be the maximal real subfield of K.
Let pibar be the complex conjugate of pi.

Consider the order R = Z[pi, pibar] of K and the order R+ = Z[pi+pibar]
of K+.  I want to make the following assumptions:

    1. R is the maximal order of K 
        (which implies that R+ is the maximal order of K+);
    2. K is ramified over K+ at a finite prime;
    3. The unit group of R is equal to the unit group of R+.

(It will still be possible to say something when these assumptions
are not met, but they make life easier, and they are not too 
unreasonable.)

Now, the abelian varieties in the isogeny class C correspond (via
Deligne's paper [Invent. Math. 8 1969 238--243]) to the isomorphism
classes of those finitely-generated R-modules that can be embedded 
as lattices in K.  Our first assumption implies that the isomorphism 
classes of these R-modules are simply the ideal classes of K.

My thesis [Trans. Amer. Math. Soc. 347 (1995) 2361--2401] shows that
there is an ideal class J in the narrow class group of K+ with the
following property:  An ideal class I of K corresponds to an abelian
variety that has a principal polarization if and only if N(I) = J,
where N is the norm from the class group of K to the narrow class group
of K+.

Our second assumption shows that this norm map is surjective, so we
find that the number of abelian varieties in C that have a principal
polarization is equal to the quotient h(K) / h+(K+) of the class number
of K by the narrow class number of K+.

Now, if an abelian variety A in C has at least one principal
polarization, then the number of non-isomorphic principal polarizations
on A is equal to the index

   [(totally positive units of K+) : (norms of units of K)]

and our third assumption shows that this index is equal to

   [(totally positive units of K+) : (squares of units of K+)],

and this index is equal to the quotient of the narrow class number
of K+ by the regular class number of K+.

Thus, the number of principally-polarized varieties (A,lambda) with
A in C is equal to the quotient h(K)/h(K+).


Now let's use Brauer-Siegel.  It is easy to show that the discriminant
of the ring R = Z[pi,pibar] is equal (up to sign) to the norm from K
to Q of (pi - pibar) times the square of the discriminant of R+.
Thus, the ratio of Delta(R) to Delta(R+) is equal to

   N(pi-pibar)*Delta(R+).

Let pi_1, ..., pi_n, pibar_1, ..., pibar_n be the images of pi in the
complex numbers, and let theta_1, ..., theta_n be the corresponding
arguments.  Then N(pi-pibar) is equal to a certain power of q times a 
certain power of 2 times the product

   \prod_{i} (\sin\theta_i)^2

and the discriminant of R+ = Z[pi+pibar] is equal to a certain power of
q times a certain power of 2 times the product

   \prod_{i<j} (\cos\theta_i - \cos\theta_j)^2 .

Now, from Brauer-Siegel we *expect* that 

   h(K)/h(K+) is about Reg(K+)/Reg(K) * Sqrt(Delta(K)/Delta(K+)).

Now, Reg(K+)/Reg(K) is just 2^(1-n) because of our third assumption,
so up to a certain power of q and a certain power of 2, we have

   h(K)/h(K+) is about Sqrt(N(pi-pibar)*Delta(R+))

        which is about the absolute value of

   \prod_{i} (\sin\theta_i) * \prod_{i<j} (\cos\theta_i - \cos\theta_j).

========================================================================

OK.  So the above function is giving us the distribution of PPAVs with
respect to Lebesgue measure on the "coefficient space" of the first n
coefficients of the characteristic polynomials of Frobenius.  It's a
nice little exercise to show that changing from this measure to Lebesgue 
measure on the space of Frobenius angles theta_i introduces a Jacobian 
factor proportional to

   \prod_{i} (\sin\theta_i) * \prod_{i<j} (\cos\theta_i - \cos\theta_j).

So, taking the product of the Jacobian factor with the distribution
function given above, we find that the distribution of Frobenius angles
with respect to Lebesgue measure on the angle-space is given by 
something proportional to

 \prod_{i} (\sin\theta_i)^2 * \prod_{i<j} (\cos\theta_i - \cos\theta_j)^2.


========================================================================


As I mentioned, it doesn't seem likely that this could be made more
rigorous (because of the Brauer-Siegel step), but probably it could
be made to prove something about (say) the log of the Frobenius angle
distribution.

In any case, I think it does shed some light on where the distribution
is coming from --- the distribution is simply reflecting the relative 
sizes of a couple of discriminants that naturally occur when considering
abelian varieties.

========================================================================

What do you think of all this?

All the best,

Everett


________________________________________________________________________
Everett Howe                          Center for Communications Research
however@alumni.caltech.edu                           4320 Westerra Court
http://alumni.caltech.edu/~however                  San Diego, CA  92121