We establish an explicit formula for the number of ideals of codimension (colength) of the algebra of Laurent polynomials in two variables over a finite field of cardinality . This number is a palindromic polynomial of degree in . Moreover, , where is another palindromic polynomial; the latter is a -analogue of the sum of divisors of , which happens to be the number of subgroups of of index .
"Counting the Ideals of Given Codimension of the Algebra of Laurent Polynomials in Two Variables." Michigan Math. J. 67 (4) 715 - 741, November 2018. https://doi.org/10.1307/mmj/1529114453