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REDUCTIONS OF POINTS ON ALGEBRAIC GROUPS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 20 / Issue 5 / September 2021
- Published online by Cambridge University Press:
- 14 November 2019, pp. 1637-1669
- Print publication:
- September 2021
-
- Article
- Export citation
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Let
$A$ be the product of an abelian variety and a torus defined over a number field 
$K$. Fix some prime number 
$\ell$. If 
$\unicode[STIX]{x1D6FC}\in A(K)$ is a point of infinite order, we consider the set of primes 
$\mathfrak{p}$ of 
$K$ such that the reduction 
$(\unicode[STIX]{x1D6FC}\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{p})$ is well-defined and has order coprime to 
$\ell$. This set admits a natural density. By refining the method of Jones and Rouse [Galois theory of iterated endomorphisms, Proc. Lond. Math. Soc. (3)100(3) (2010), 763–794. Appendix A by Jeffrey D. Achter], we can express the density as an 
$\ell$-adic integral without requiring any assumption. We also prove that the density is always a rational number whose denominator (up to powers of 
$\ell$) is uniformly bounded in a very strong sense. For elliptic curves, we describe a strategy for computing the density which covers every possible case.
