[Paper Review] Equidistribution on the space of rank two vector bundles over the projective line

Vivek Shende, Jacob Tsimerman|arXiv (Cornell University)|Jul 31, 2013

Algebraic Geometry and Number Theory28 references3 citations

TL;DR

This paper establishes a function field analogue of Duke's equidistribution theorem for Heegner points, proving that measures induced by hyperelliptic curves on the moduli space of rank two vector bundles over ℙ¹ converge to the natural measure as genus tends to infinity. The proof uses zeta function manipulations and the Riemann Hypothesis for curves, extending to pairs of bundles via geometric counting of special divisor intersections in Jacobians.

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.

Motivation & Objective

• To establish equidistribution of pushforwards of line bundle measures on the moduli space of rank two vector bundles over ℙ¹.
• To extend this to measures on pairs of bundles, conjecturing the limiting distribution in analogy with the Mixing Conjecture of Michel and Venkatesh.
• To prove equidistribution via geometric methods, particularly counting points on intersections of translates of special divisor loci in Jacobians of hyperelliptic curves.
• To establish cohomological and Betti number bounds for these special divisor loci, crucial for equidistribution in positive characteristic.

Proposed method

• Uses zeta function manipulations and the Riemann Hypothesis for curves to prove equidistribution of measures on Bun₂(ℙ¹) induced by hyperelliptic curves.
• Reduces the equidistribution problem to bounding the upper cohomologies of special divisor loci in the Jacobian, showing they match the cohomology of the Jacobian itself.
• Applies ergodic theory when the line bundle is sufficiently special, and switches to geometric counting for the remaining cases.
• Studies the geometry of translates of special divisor loci in the Jacobian of a hyperelliptic curve to control point counts.
• Employs the framework of completely framed vector bundles and Hecke measures to relate automorphic and geometric data.
• Uses the double coset structure of symmetric spaces to classify the pushforward maps from rank one tori to PGL₂.

Experimental results

Research questions

• RQ1 Does the pushforward measure from the Picard group of a hyperelliptic curve to Bun₂(ℙ¹) equidistribute as the genus tends to infinity?
• RQ2 What are the possible limit measures when hyperelliptic curves are equipped with line bundles, and how do they relate to the Mixing Conjecture in the function field setting?
• RQ3 Do the upper cohomologies of special divisor loci in the Jacobian of a hyperelliptic curve agree with the cohomology of the Jacobian itself?
• RQ4 Does the sum of the Betti numbers of these special divisor loci grow at most exponentially in the genus of the curve, particularly in positive characteristic?
• RQ5 Can equidistribution be established in positive characteristic using geometric methods when ergodic theory fails?

Key findings

• The pushforward measures from the Picard group of hyperelliptic curves to Bun₂(ℙ¹) converge weak-* to the natural measure on the space of rank two vector bundles as genus tends to infinity.
• The upper cohomologies of the special divisor loci in the Jacobian agree with the cohomology of the Jacobian, which is a key technical result proven in the paper.
• In characteristic zero, the sum of the Betti numbers of the special divisor loci grows at most exponentially in the genus, supporting equidistribution.
• The authors conjecture that the same Betti number bound holds in positive characteristic, which would complete the equidistribution proof.
• For sufficiently special line bundles, ergodic theory suffices to prove equidistribution, but geometric counting is required for the general case.
• The paper establishes a function field analogue of Duke’s equidistribution theorem, with the natural measure on Bun₂(ℙ¹) arising as the weak-* limit of these pushforward measures.