ROBERTO DVORNICICH and UMBERTO ZANNIER

On the Hasse principle for the division of points in a commutative algebraic group

Pages: 67-81
Received: 7 November 2004   
Mathematics Subject Classification (2000): 14G05 - 11R34 - 14G25 - 14G05

Abstract: Let A be a commutative algebraic group defined over a number field k. We consider the following question: Let r be a positive integer and let P ∈ A (k). Suppose that for all but a finite number of primes v of k we have P = r D_v, for some D ∈ A (k_v). Can one conclude that there exists D ∈ A (k) such that P = r D? A complete answer for the case of the multiplicative group G_m is classical. We study other instances, mainly concerning elliptic curves and algebraic tori, obtaining results in both directions: namely, we have families of examples for which the answer is positive and families of examples for which the answer is negative.