  Fix a finite field. A hyperelliptic curve determines a measure on the
discrete space of rank two bundles on the projective line: the mass of a given
vector bundle is the number of line bundles whose pushforward it is. In a
sequence of hyperelliptic curves whose genera tend to infinity, these measures
tend to the natural measure on the space of rank two bundles. This is a
function field analogue of Duke's theorem on the equidistribution of Heegner
points, and can be proven similarly: it follows from a manipulation of zeta
functions, plus the Riemann Hypothesis for curves.
  Likewise, a sequence of hyperelliptic curves equipped with line bundles gives
rise to a sequence of measures on the space of pairs of rank 2 bundles. We give
a conjectural classification of the possible limit measures which arise; this
is a function field analogue of the "Mixing Conjecture" of Michel and
Venkatesh. As in the number field setting, ergodic theory suffices when the
line bundle is sufficiently special.
  For the remaining bundles, we turn to geometry and count points on
intersections of translates of loci of special divisors in the Jacobian of a
hyperelliptic curve. To prove equidistribution, we would require two results.
The first, we prove: the upper cohomologies of these loci agree with the
cohomology of the Jacobian. The second, which we establish in characteristic
zero and conjecture in characteristic p, is that the sum of the Betti numbers
of these spaces grows at most as the exponential of the genus of the
hyperelliptic curve.
