[Paper Review] The distribution of the number of points on trigonal curves over $\F_q$

Melanie Matchett Wood|arXiv (Cornell University)|Aug 11, 2011

Algebraic Geometry and Number Theory4 references5 citations

TL;DR

This paper determines the limiting distribution of the number of $ℚ_q$-rational points on random trigonal curves over $ℚ_q$ as the genus $g \to \infty$, showing that the expected number of points is $q+2 - \frac{1}{q^2+q+1}$, exceeding $q+1$—a contrast to many other curve families. The method uses cubic extensions of function fields and adapts Datskovsky-Wright counting techniques, with fiberwise independence over $ℚ_q$-points of $\mathbb{P}^1$.

ABSTRACT

We give a short determination of the distribution of the number of $\F_q$-rational points on a random trigonal curve over $\F_q$, in the limit as the genus of the curve goes to infinity. In particular, the expected number of points is $q+2-\frac{1}{q^2+q+1}$, contrasting with recent analogous results for cyclic $p$-fold covers of $\mathbb P^1$ and plane curves which have an expected number of points of $q+1$ (by work of Kurlberg, Rudnick, Bucur, David, Feigon and Lalín) and curves which are complete intersections which have an expected number of points $

Motivation & Objective

• To determine the limiting distribution of the number of $\mathbb{F}_q$-rational points on random trigonal curves over $\mathbb{F}_q$ as the genus $g \to \infty$.
• To explain why the expected number of points on such curves exceeds $q+1$, contrasting with other curve families like cyclic $p$-fold covers and complete intersections.
• To generalize the analysis to $n$-gonal curves with full $S_n$ monodromy using function field analogs of Bhargava's heuristics.
• To provide a conjectural framework for the expected number of points on $n$-gonal curves with full symmetric monodromy based on local densities of étale algebras.

Proposed method

• Relates trigonal curves to cubic extensions of function fields over $\mathbb{F}_q$, using the correspondence between degree-3 covers of $\mathbb{P}^1$ and cubic extensions.
• Applies the Datskovsky-Wright counting method for cubic extensions to enumerate curves with specified fiberwise behavior over each $\mathbb{F}_q$-point of $\mathbb{P}^1$.
• Computes the limiting probability distribution of the number of $\mathbb{F}_q$-rational points in the fiber over each $\mathbb{F}_q$-point of $\mathbb{P}^1$, showing independence across points.
• Derives the expected number of points as $q+2 - \frac{1}{q^2+q+1}$ by summing the expected fiber sizes over the $q+1$ points of $\mathbb{P}^1(\mathbb{F}_q)$.
• Extends the method to arbitrary smooth curves $E$ over $\mathbb{F}_q$ in place of $\mathbb{P}^1$, computing the expected number of points as $\#E(\mathbb{F}_q)\left(1 + \frac{q}{q^2+q+1}\right)$.
• Uses function field analogs of Bhargava's heuristics to conjecture the expected fiber size for $n$-gonal curves with full $S_n$ monodromy, based on conjugacy class statistics and local densities of $k_v$-algebras.

Experimental results

Research questions

• RQ1 What is the limiting distribution of the number of $\mathbb{F}_q$-rational points on a random trigonal curve over $\mathbb{F}_q$ as the genus tends to infinity?
• RQ2 Why does the expected number of points on trigonal curves exceed $q+1$, unlike in many other curve families?
• RQ3 Can the distribution of rational points on $n$-gonal curves with full $S_n$ monodromy be predicted using function field analogs of Bhargava's heuristics?
• RQ4 What is the expected number of $\mathbb{F}_q$-points in the fiber over a fixed $\mathbb{F}_q$-point of the base curve for a random $n$-gonal curve?
• RQ5 How do the local behaviors of fibers over $\mathbb{F}_q$-points of the base curve contribute to the global point count on $n$-gonal curves?

Key findings

• The limiting distribution of the number of $\mathbb{F}_q$-rational points on a random trigonal curve is the sum of $q+1$ i.i.d. random variables $X_i$, each taking values in $\{0,1,2,3\}$ with probabilities $\frac{2q^2}{6q^2+6q+6}$, $\frac{3q^2+6}{6q^2+6q+6}$, $\frac{6q}{6q^2+6q+6}$, and $\frac{q^2}{6q^2+6q+6}$, respectively.
• The expected number of $\mathbb{F}_q$-rational points on a random trigonal curve is $q+2 - \frac{1}{q^2+q+1}$, which is strictly greater than $q+1$.
• The fiber sizes over the $q+1$ points of $\mathbb{P}^1(\mathbb{F}_q)$ are asymptotically independent, and the distribution of the number of points is the sum of $q+1$ i.i.d. copies of the fiber size distribution.
• For a smooth curve $E$ over $\mathbb{F}_q$, the expected number of $\mathbb{F}_q$-rational points on a random degree-3 cover of $E$ is $\#E(\mathbb{F}_q)\left(1 + \frac{q}{q^2+q+1}\right)$.
• The conjecture for $n$-gonal curves with full $S_n$ monodromy predicts the expected fiber size over each $\mathbb{F}_q$-point of the base curve to be $1 + \frac{1}{q} + O\left(\frac{1}{q^2}\right)$, derived from local densities of $k_v$-algebras.
• For $n=4$, the conjectured expected fiber size is $1 + \frac{q^2+q}{q^3+q^2+2q+1}$, and for $n=5$, it is $1 + \frac{q^3+2q^2+2q}{q^4+q^3+2q^2+2q+1}$, based on partition statistics and conjugacy class weights in $S_n$.