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REDUCTIONS OF POINTS ON ALGEBRAIC GROUPS

Part of: Arithmetic algebraic geometry Algebraic groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  14 November 2019

Davide Lombardo Open the ORCID record for Davide Lombardo [Opens in a new window]  and
Antonella Perucca
Davide Lombardo
Affiliation:
University of Pisa, Largo Bruno Pontecorvo 5, 56127Pisa, Italy ([email protected])
Antonella Perucca
Affiliation:
University of Luxembourg. 6, avenue de la Fonte, 4364Esch-sur-Alzette, Luxembourg ([email protected])
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Abstract

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.

Keywords

reductionKummer theoryalgebraic groupsemiabelian varietyelliptic curveGalois representations

MSC classification

Primary: 11F80: Galois representations
Secondary: 14L10: Group varieties 11G05: Elliptic curves over global fields 11G10: Abelian varieties of dimension $> 1$
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 5 , September 2021 , pp. 1637 - 1669
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Copyright
© Cambridge University Press 2019

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