A refined counter-example to the support conjecture for abelian\n varieties

2005/02/13 by Michael Larsen, Larsen, Michael, René Schoof +1
Computer Science · Mathematics · #11G10 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.math/0502261

openalex publication_date 2005/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

If A/K is an abelian variety over a number field and P and Q are rational\npoints, the original support conjecture asserted that if the order of Q (mod p)\ndivides the order of P (mod p) for almost all primes p of K, then Q is obtained\nfrom P by applying an endomorphism of A. This is now known to be untrue. In\nthis note we prove that it is not even true modulo the torsion of A.\n

Related

• The support problem for abelian varieties
• The constant of the support problem for abelian varieties
• Reduction of homomorphisms mod p and algebraicity
• The multilinear support problem for products of abelian varieties and tori
• Torsion of Abelian varieties over solvable extensions of number fields