ON THE DISTRIBUTION OF TORSION POINTS MODULO PRIMES

Abstract

AbstractLet $\Bbb A$ be a commutative algebraic group defined over a number field K. For a prime ℘ in K where $\Bbb A$ has good reduction, let N℘,n be the number of n-torsion points of the reduction of $\Bbb A$ modulo ℘ where n is a positive integer. When $\Bbb A$ is of dimension one and n is relatively prime to a fixed finite set of primes depending on $\Bbb A_{/K}$, we determine the average values of N℘,n as the prime ℘ varies. This average value as a function of n always agrees with a divisor function.