Statistics for Traces of Cyclic Trigonal Curves over Finite Fields

By Alina Bucur, Chantal David, Brooke Feigon, Matilde Lal'in. 2009

49 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics, Sociology and Political Science

View Paper: https://www.semanticscholar.org/paper/7267d9f4876fa532c7d67ce39f634dfb086fc2d8

Citations in the corpus

• The mean value of L(12,χ) in the hyperelliptic ensemble (2012, unknown): “2 in [2] we have that ∑ D∈H2g+1,q (D,l)=1 1 = ∑ D∈V 2g+1 (D,l)=1 ∑ Q 2|D μ(Q ) = ∑”