# On the moments of torsion points modulo primes and their applications

## TL;DR

This paper investigates the distribution of torsion points modulo primes in algebraic groups over number fields, computing moments and orbit counts using advanced number theory tools, and explores their applications in understanding group actions.

## Contribution

It provides explicit calculations of the moments of torsion point counts and interprets these as orbit counts, offering new methods for analyzing group actions on algebraic structures.

## Key findings

- Computed the $k$-th moment limits for torsion points of algebraic groups.
- Connected moment limits to orbit counts via Galois group actions.
- Demonstrated asymptotic distribution for point counts on zero-dimensional algebraic sets.

## Abstract

Let $\mathbb{A}[n]$ be the group of $n$-torsion points of a commutative algebraic group $\mathbb{A}$ defined over a number field $F$. For a prime ideal $\mathfrak{p}$, we let $N_{\mathfrak{p}}(\mathbb{A}[n])$ be the number of $\mathbb{F}_\mathfrak{p}$-solutions of the system of polynomial equations defining $\mathbb{A}[n]$ when reduced modulo $\mathfrak{p}$. Here, $\mathbb{F}_{\mathfrak{p}}$ is the residue field at $\mathfrak{p}$. Let $\pi_F(x)$ denote the number of primes $\mathfrak{p}$ of $F$ whose norm $N(\mathfrak{p})$ do not exceed $x$. We then, for algebraic groups of dimension one, compute the $k$-th moment limit $$M_k(\mathbb{A}/F, n)=\lim_{x\rightarrow \infty} \frac{1}{\pi_F(x)} \sum_{N(\mathfrak{p}) \leq x} N_{\mathfrak{p}}^k(\mathbb{A}[n])$$ by appealing to the prime number theorem for arithmetic progressions and more generally the Chebotarev density theorem. We further interpret this limit as the number of orbits of the action of the absolute Galois group of $F$on $k$ copies of $\mathbb{A}[n]$ by an application of Burnside's Lemma. These concrete examples suggest a possible approach for determining the number of orbits of a group acting on $k$ copies of a set. We also show that for an algebraic set $Y$ of dimension zero, the corresponding arithmetic function $N_\mathfrak{p}(Y)$, defined on primes $\mathfrak{p}$ of $F$, has an asymptotic limiting distribution.