  The Leopoldt conjecture is concerned with the image of the global units in
the local units at the primes dividing p. In the definition of the global units
the infinite place is distinguished. Exchanging p and infinity in the
formulation one gets a new conjecture. It predicts that certain vectors should
be linearly independent over the reals whose components are arguments of
conjugates of Weil numbers. Using Baker's result on linear forms in logarithms
we prove part of this new conjecture in certain abelian situations.
