Duke Mathematical Journal

Equidistribution in Bun2(P1)

Vivek Shende and Jacob Tsimerman

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Abstract

Fix a finite field. The set of PGL2 bundles over P1 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 π:CP1 determines another measure, given by counting the number of line bundles over C whose image on P1 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 PGL2 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 PGL2 bundles induced by taking a fixed line bundle L over C, and looking at the distribution of pairs (πM,π(LM)). As in the number field setting, ergodic theory classifies limiting measures for sufficiently special 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.

Article information

Source
Duke Math. J., Volume 166, Number 18 (2017), 3461-3504.

Dates
Received: 11 February 2015
Revised: 24 April 2017
First available in Project Euclid: 17 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1510887861

Digital Object Identifier
doi:10.1215/00127094-2017-0025

Mathematical Reviews number (MathSciNet)
MR3732881

Zentralblatt MATH identifier
06837465

Subjects
Primary: 11G20: Curves over finite and local fields [See also 14H25]
Secondary: 14H51: Special divisors (gonality, Brill-Noether theory)

Keywords
equidistribution ergodic theory Duke’s theorem

Citation

Shende, Vivek; Tsimerman, Jacob. Equidistribution in $\operatorname{Bun}_{2}(\mathbb{P}^{1})$. Duke Math. J. 166 (2017), no. 18, 3461--3504. doi:10.1215/00127094-2017-0025. https://projecteuclid.org/euclid.dmj/1510887861


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