2009/07/31 by Alina Bucur, Chantal David, Brooke Feigon +2 · 2 citations
Mathematics · Social Sciences · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Vietnamese History and Culture Studies #math.NT #msc:11G20 #msc:11G25 #msc:11T55
paper · pdf · doi:10.1093/imrn/rnp162
published as Int. Math. Res. Not. IMRN 2010, no. 5, 932--967 · 30 pages, added statement and sketch of proof in Section 7 for generalization of results to p-fold covers of the projective line, the final version of this article will be published in International Mathematics Research Notices
arxiv created 2009/09/11 · openalex publication_date 2009/10/27 · arxiv updated 2010/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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.