The distribution of \mathbbF_q-points on cyclic ℓ-covers of genus g

2015/05/26 by Bucur, Alina, David, Chantal, Feigon, Brooke +4
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1505.07136

Abstract

We study fluctuations in the number of points of ℓ-cyclic covers of the projective line over the finite field \mathbbF_q when q ≡ 1 \mod ℓ is fixed and the genus tends to infinity. The distribution is given as a sum of q+1 i.i.d. random variables. This was settled for hyperelliptic curves by Kurlberg and Rudnick, while statistics were obtained for certain components of the moduli space of ℓ-cyclic covers by Bucur, David, Feigon and Lalín. In this paper, we obtain statistics for the distribution of the number of points as the covers vary over the full moduli space of ℓ-cyclic covers of genus g. This is achieved by relating ℓ-covers to cyclic function field extensions, and counting such extensions with prescribed ramification and splitting conditions at a finite number of primes.

Related

• Distribution of Points on Cyclic Curves over Finite Fields
• Statistics for biquadratic covers of the projective line over finite fields
• The distribution of points on superelliptic curves over finite fields
• Number of Points on the Full Moduli Space of Curves over Finite Fields
• Weighted distribution of points on cyclic covers of the projective line over finite fields