  For a cardinal kappa and a model M of cardinality kappa let No(M) denote the
number of non-isomorphic models of cardinality kappa which are
L_{infty,kappa}--equivalent to M. In [Sh:133] Shelah established that when
kappa is a weakly compact cardinal and mu<=kappa is a nonzero cardinal, there
exists a model M of cardinality kappa with No(M)=mu. We prove here that if
kappa is a weakly compact cardinal, the question of the possible values of
No(M) for models M of cardinality kappa is equivalent to the question of the
possible numbers of equivalence classes of equivalence relations which are
Sigma^1_1-definable over V_kappa. In math.LO/9911231 we prove that, consistent
wise, the possible numbers of equivalence classes of Sigma^1_1-equivalence
relations can be completely controlled under the singular cardinal hypothesis.
These results settle the problem of the possible values of No(M) for models of
weakly compact cardinality, provided that the singular cardinal hypothesis
holds.
