Statistics for Traces of Cyclic Trigonal Curves over Finite Fields

We study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus g over 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 o...

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Published in:International mathematics research notices 2010, Vol.2010 (5), p.932-967
Main Authors: Bucur, Alina, David, Chantal, Feigon, Brooke, Lalín, Matilde
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description We study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus g over 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, e2π i/3, e4π 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.
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title Statistics for Traces of Cyclic Trigonal Curves over Finite Fields
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