A simple formula for the Picard number of K3 surfaces of BHK type

The Berglund–Hubsch–Krawitz (BHK) mirror symmetry construction applies to certain types of Calabi–Yau varieties that are birational to finite quotients of Fermat varieties. Their definition involves a matrix A and a certain finite abelian group G, and we denote the corresponding Calabi–Yau variety by ZA,G. The transpose matrix AT and the so-called dual groupGT give rise to the BHK mirror variety ZAT,GT. In the case of dimension 2, the surface ZA,G is a K3 surface of BHK type. Let ZA,G be a K3 surface of BHK type, with BHK mirror ZAT,GT. Using work of Shioda, Kelly has shown that the geometric Picard number ρ(ZA,G) of ZA,G may be expressed in terms of a certain subset of the dual group GT. We simplify this formula significantly to show that ρ(ZA,G) depends only upon the degree of the mirror polynomial FAT.