Determinants of Mahler Measures and Special Values of L -Functions

We consider Mahler measures of two well-studied families of bivariate polynomials, namely Pt=x+x−1+y+y−1+t and Qt=x3+y3+1−t3xy, where t is a complex parameter. In the cases when the zero loci of these polynomials define CM elliptic curves over number fields, we derive general formulas for their Mahler measures in terms of L-values of cusp forms. For each family, we also classify all possible values of t in number fields of degree not exceeding 4 for which the corresponding elliptic curves have complex multiplication. Finally, for all such values of t in totally real number fields of degree n=2 and n=4, corresponding to elliptic curves Ft (resp. Ct), we prove that determinants of n×n matrices whose entries are Mahler measures corresponding to their Galois conjugates are non-zero rational multiples of L(n)(Ft,0) (resp. L(n)(Ct,0)).