Statistics for traces of cyclic trigonal curves over finite fields

TL;DR

This paper investigates the distribution of the Frobenius trace for cyclic trigonal curves over finite fields, revealing its limiting behavior as genus and field size grow, extending previous hyperelliptic curve results.

Contribution

It extends the understanding of Frobenius trace distributions to cyclic trigonal curves and generalizes results to p-fold covers, providing new probabilistic models.

Findings

Limiting distribution of Frobenius trace as genus increases at fixed q

Normalized trace approaches a standard complex Gaussian when both g and q grow

Distribution of trace values described by a specific probabilistic model

Abstract

We study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus g over a field of q elements 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 the Frobenius equals the sum of q+1 independent random variables taking the value 0 with probability 2/(q+2) and 1, e^{(2pi i)/3}, e^{(4pi 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.