On the distribution of torsion points modulo primes

Authors

  • Yen-Mei Julia Chen National Central University
  • Yen-Liang Kuan National Central University

Keywords:

Number fields, Algebraic groups, Torsion points, Elliptic curves Complex multiplication.

Abstract

Let $\Bbb A$ be a commutative algebraic group defined over a number field $K$. For a prime $\wp$ in $K$ where $\Bbb A$ has good reduction, let $N_{\wp,n}$ be the number of $n$-torsion points of the reduction $\tilde\Bbb A$ of $\Bbb A$ modulo $\wp$ in $\Bbb F_{\wp}$, where $\Bbb F_{\wp}$ denotes the residue field and $n$ is a positive integer. When $\Bbb A$ is of dimension one and $n$ is relative prime to a fixed finite set of primes depending on $\Bbb A_{/K}$, we determine the asymptotic average values of $N_{\wp, n}$ as the prime $\wp$ varies. This average value as function of n always agrees with a divisor function. DOI: 10.1017/S0004972712000019

Published

2012-08-27

Issue

Section

Articles