Equidistribution in Bun2(P1)

Abstract

Fix a finite field. The set of ${PGL}_2$ bundles over ${\mathbb{P}}^1$ is in bijection with the natural numbers, and carries a natural measure assigning to each bundle the inverse of the number of automorphisms. A branched double cover $\pi :C\to {\mathbb{P}}^1$ determines another measure, given by counting the number of line bundles over $C$ whose image on ${\mathbb{P}}^1$ has a given sheaf of endomorphisms. We show the measures induced by a sequence of such hyperelliptic curves tends to the canonical measure on the space of ${PGL}_2$ bundles.

This is a function field analogue of Duke’s theorem on the equidistribution of Heegner points, and can be proven similarly. Our real interest is the corresponding analogue of the “Mixing Conjecture” of Michel and Venkatesh. This amounts to considering measures on the space of pairs of ${PGL}_2$ bundles induced by taking a fixed line bundle $\mathcal{L}$ over $C$, and looking at the distribution of pairs $({\pi }_{\ast }\mathcal{M},{\pi }_{\ast }(\mathcal{L}\otimes \mathcal{M}\left)\right)$. As in the number field setting, ergodic theory classifies limiting measures for sufficiently special $\mathcal{L}$.

The heart of this work is a geometric attack on the general case. We 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: in high degree, the cohomologies of these loci match the cohomology of the Jacobian. The second, we establish in characteristic zero and conjecture in characteristic $p$: the cohomology of these spaces grows at most exponentially in the genus of the curve $C$.