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Homology of braid groups, the Burau representation, and points on superelliptic curves over finite fields

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Abstract

The (reduced) Burau representation V n of the braid group B n is obtained from the action of B n on the homology of an infinite cyclic cover of the disc with n punctures. In this paper, we calculate H *(B n ; V n ). Our topological calculation has the following arithmetic interpretation (which also has different algebraic proofs): the expected number of points on a random superelliptic curve of a fixed genus over F q is q.

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Authors and Affiliations

  1. Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL, 60637, USA

    Weiyan Chen

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  1. Weiyan Chen
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Correspondence to Weiyan Chen.

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Chen, W. Homology of braid groups, the Burau representation, and points on superelliptic curves over finite fields. Isr. J. Math. 220, 739–762 (2017). https://doi.org/10.1007/s11856-017-1534-7

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  • DOI: https://doi.org/10.1007/s11856-017-1534-7

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