Statistics for Traces of Cyclic Trigonal Curves over Finite Fields

Alina Bucur, Chantal David, Brooke Feigon, Matilde Lalín

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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.
Original languageEnglish
Pages (from-to)932-967
Number of pages36
JournalInternational Mathematics Research Notices
Volume2010
Issue number5
DOIs
Publication statusPublished - 27 Oct 2010

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