**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.

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