A characterization of locally finite congruence modular varieties with the number of at most k-generated models being bounded from above by a polynomial in k is given. These are exactly the varieties polynomially equivalent to the varieties of unitary modules over a finite ring of finite representation type.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm170-1-3, author = {Pawe\l\ M. Idziak and Ralph McKenzie}, title = {Varieties with polynomially many models, I}, journal = {Fundamenta Mathematicae}, volume = {167}, year = {2001}, pages = {53-68}, zbl = {1002.08003}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm170-1-3} }
Paweł M. Idziak; Ralph McKenzie. Varieties with polynomially many models, I. Fundamenta Mathematicae, Tome 167 (2001) pp. 53-68. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm170-1-3/