  We compute the lattice 1-site probabilities, on the upper half-plane, of the
four height variables in the two-dimensional Abelian sandpile model. We find
their exact scaling form when the insertion point is far from the boundary, and
when the boundary is either open or closed. Comparing with the predictions of a
logarithmic conformal theory with central charge c=-2, we find a full
compatibility with the following field assignments: the heights 2, 3 and 4
behave like (an unusual realization of) the logarithmic partner of a primary
field with scaling dimension 2, the primary field itself being associated with
the height 1 variable. Finite size corrections are also computed and
successfully compared with numerical simulations. Relying on these field
assignments, we formulate a conjecture for the scaling form of the lattice
2-point correlations of the height variables on the plane, which remain as yet
unknown. The way conformal invariance is realized in this system points to a
local field theory with c=-2 which is different from the triplet theory.
