Equidistribution on the space of rank two vector bundles over the projective line

Abstract

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.