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International Mathematics Research Notices Advance Access published online on October 27, 2009
International Mathematics Research Notices, doi:10.1093/imrn/rnp162
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© The Author 2009. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

Statistics for Traces of Cyclic Trigonal Curves over Finite Fields

Alina Bucur1, Chantal David2, Brooke Feigon3 and Matilde Lalín4

1 School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA
2 Department of Mathematics, Concordia University, 1455 de Maisonneuve West, Montreal, QC Canada H3G 1M8
3 Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON Canada M5S 2E4
4 Department of Mathematical and Statistical Sciences, University of Alberta, CAB 632, Edmonton, AB Canada T6G 2G1

Correspondence: Correspondence to be sent to: alina{at}math.ucsd.edu

We study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus g over Formula 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, e2{pi} i/3, e4{pi} 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.

Communicated by Prof. Zeev Rudnick


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