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1 December 2017 Equidistribution in Bun2(P1)
Vivek Shende, Jacob Tsimerman
Duke Math. J. 166(18): 3461-3504 (1 December 2017). DOI: 10.1215/00127094-2017-0025

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.

Citation

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Vivek Shende. Jacob Tsimerman. "Equidistribution in Bun2(P1)." Duke Math. J. 166 (18) 3461 - 3504, 1 December 2017. https://doi.org/10.1215/00127094-2017-0025

Information

Received: 11 February 2015; Revised: 24 April 2017; Published: 1 December 2017
First available in Project Euclid: 17 November 2017

zbMATH: 06837465
MathSciNet: MR3732881
Digital Object Identifier: 10.1215/00127094-2017-0025

Subjects:
Primary: 11G20
Secondary: 14H51

Keywords: Duke’s theorem , equidistribution , ergodic theory

Rights: Copyright © 2017 Duke University Press

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Vol.166 • No. 18 • 1 December 2017
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