  We introduce the notion of (nondegenerate) strong-modular fusion algebras.
Here strongly-modular means that the fusion algebra is induced via Verlinde's
formula by a representation of the modular group SL(2,Z) whose kernel contains
a congruence subgroup. Furthermore, nondegenerate means that the conformal
dimensions of possibly underlying rational conformal field theories do not
differ by integers. Our main result is the classification of all
strongly-modular fusion algebras of dimension two, three and four and the
classification of all nondegenerate strongly-modular fusion algebras of
dimension less than 24. We use the classification of the irreducible
representations of the finite groups SL(2,Z_{p^l}) where p is a prime and l a
positive integer. Finally, we give polynomial realizations and fusion graphs
for all simple nondegenerate strongly-modular fusion algebras of dimension less
than 24.
