Statistics for Traces of Cyclic Trigonal Curves over Finite Fields

Bucur, Alina, David, Chantal, Feigon, Brooke and Lalín, Matilde (2010) Statistics for Traces of Cyclic Trigonal Curves over Finite Fields. International Mathematics Research Notices, 2010 (5). pp. 932-967. ISSN 1073-7928

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Abstract

We study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus g over F q as the curve varies in an irreducible component of the moduli space. We show that for q fixed and g increasing, the limiting distribution of the trace of Frobenius equals the sum of q + 1 independent random variables taking the value 0 with probability 2/(q + 2) and 1, e2p i/3, e4p i/3 each with probability q/(3(q + 2)). This extends the work of Kurlberg and Rudnick who considered the same limit for hyperelliptic curves. We also show that when both g and q go to infinity, the normalized trace has a standard complex Gaussian distribution and how to generalize these results to p-fold covers of the projective line.

Item Type: Article
Faculty \ School: Faculty of Science > School of Mathematics
Depositing User: Vishal Gautam
Date Deposited: 18 Mar 2011 14:52
Last Modified: 21 Jul 2020 23:33
URI: https://ueaeprints.uea.ac.uk/id/eprint/18580
DOI: 10.1093/imrn/rnp162

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