Samart, Detchat (2014-08). Mahler Measures of Hypergeometric Families of Calabi-Yau Varieties. Doctoral Dissertation.
The logarithmic Mahler measure of a nonzero n-variable Laurent polynomial P ? C [X_1^(?1),...,X_n^(?1) ], denoted by m(P), is defined to be the arithmetic mean of log|P| over the n-dimensional torus. It has been proved or conjectured that the logarithmic Mahler measures of some classes of polynomials have connections with special values of L-functions. However, the precise interpretation of m(P) in terms of L-values is not clearly known, so it has become a new trend of research in arithmetic geometry and number theory to understand this phenomenon. In this dissertation, we study Mahler measures of certain families of Laurent polynomials of two, three, and four variables, whose zero loci define elliptic curves, K3 surfaces, and Calabi-Yau threefolds, respectively. On the one hand, it is known that these Mahler measures can be expressed in terms of hypergeometric series and logarithms. On the other hand, we derive explicitly that some of them can be written as linear combinations of special values of Dirichlet and modular L-functions, which potentially carry some arithmetic information of the corresponding algebraic varieties. Our results extend those of Boyd, Bertin, Lal?n, Rodriguez Villegas, Rogers, and many others. We also prove that Mahler measures of those associated to families of K3 surfaces are related to the elliptic trilogarithm defined by Zagier. This can be seen as a higher dimensional analogue of relationship between Mahler measures of bivariate polynomials and the elliptic dilogarithm known previously by work of Guillera, Lal?n, and Rogers.