  The main message of the paper is that for Gorenstein singularities, whose
(real) link is rational homology sphere, the Artin--Laufer program can be
continued. Here we give the complete answer in the case of elliptic
singularities. The main result of the paper says that in the case of an
elliptic Gorenstein singularity whose link is rational homology sphere, the
geometric genus is a topological invariant. Actually, it is exactly the length
of the elliptic sequence in the minimal resolution (or, equivalently, in S.
S.-T. Yau's terminology: these singularities are maximally elliptic). In the
paper we characterize the singularities with this property, and we compute
their Hilbert-Samuel function from their resolution graph (generalizing some
results of Laufer and Yau). The obstruction for a normal surface singularity to
be maximally elliptic can be connected with the torsion part of some Picard
groups, this is the new idea of the paper.
