Statistics for traces of cyclic trigonal curves over finite fields

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.

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

References in corpus (1)

• The integral monodromy of hyperelliptic and trielliptic curves

Cited by in corpus (11)

• The Mean Value of $L(\frac{1}{2},\chi )$ in the Hyperelliptic Ensemble
• Conjectures for the integral moments and ratios of L-functions over function fields
• The fluctuations in the number of points of smooth plane curves over finite fields
• Semiample Bertini theorems over finite fields
• Homology of braid groups, the Burau representation, and ${\mathbb{F}}_q$-points on superelliptic curves
• Picard curves over Q with good reduction away from 3
• A heuristic for the distribution of point counts for random curves over a finite field
• Distributions of traces of Frobenius for smooth plane curves over finite fields
• Proportion of ordinarity in some families of curves over finite fields
• Statistics of the Jacobians of hyperelliptic curves over finite fields
• Distribution of Points on Abelian Covers Over Finite Fields