If $\#A$ is the counting version of some corresponding decision problem $A$, when, if ever, can we determine solely on the basis of the complexity of the underlying decision problem that $\#A$ is #P-complete knowing that $\#A$ is ModkP-complete (J.-Y. Cai and L. A. Hemachandra. On the power of parity polynomial time, Proceedings of the Symposium on Theoretical Aspects of Computer Science (STACS), Lecture Notes in Computer Science 349, pp. 229-240, Springer, 1989.) for some $k>1$?When, if ever, can we determine solely on the basis of the complexity of the underlying decision problem that $\#A$ is ModkP-complete for some $k>1$ provided that we know it is #P-complete, this being something that does not hold in the general case if $A$ is not NP-hard?
#P-completeness and ModkP-completeness
cc.complexity theory;counting complexity