Abstract
AbstractLet A be the product of an abelian variety and a torus over a number field K, and let $$m \ge 2$$ be a square-free integer. If $\alpha \in A(K)$ is a point of infinite order, we consider the set of primes $\mathfrak p$ of K such that the reduction $(\alpha \bmod \mathfrak p)$ is well defined and has order coprime to m. This set admits a natural density, which we are able to express as a finite sum of products of $\ell$ -adic integrals, where $\ell$ varies in the set of prime divisors of m. We deduce that the density is a rational number, whose denominator is bounded (up to powers of m) in a very strong sense. This extends the results of the paper Reductions of points on algebraic groups by Davide Lombardo and the second author, where the case m prime is established.
Highlights
This article is the continuation of the paper Reductions of points on algebraic groups by Davide Lombardo and the second author [4]
We refer to this other work for the history of the problem, which started in the 1960s with work of Hasse on the multiplicative orders of rational numbers modulo primes
Let A be the product of an abelian variety and a torus over a number field K, and let m 2 be a square-free integer
Summary
This article is the continuation of the paper Reductions of points on algebraic groups by Davide Lombardo and the second author [4]. We refer to this other work for the history of the problem, which started in the 1960s with work of Hasse on the multiplicative orders of rational numbers modulo primes. If α ∈ A(K) is a point of infinite order, we consider the set of primes p of K such that the reduction (α mod p) is well defined and has order coprime to m. This set admits a natural density (see Theorem 7), which we denote by Densm(α)
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