AbstractWe prove the conjectural relations between Mahler measures andL-values of elliptic curves of conductors 20 and 24. We also present new hypergeometric expressions forL-values of elliptic curves of conductors 27 and 36. Furthermore, we prove a new functional equation for the Mahler measure of the polynomial family (1+X) (1+Y)(X+Y)−αXY,α∈ℝ.

We define the notion of a spanning tree generating function (STGF) a n z n , which gives the spanning tree constant when evaluated at z = 1, and gives the lattice Green function (LGF) when differentiated. By making use of known results for logarithmic Mahler measures of certain Laurent polynomials, and proving new results, we express the STGFs as hypergeometric functions for all regular two and three dimensional lattices (and one higher-dimensional lattice). This gives closed form expressions for the spanning tree constants for all such lattices, which were previously largely unknown in all but one three-dimensional case. We show for all lattices that these can also be represented as Dirichlet L-series. Making the connection between spanning tree generating functions and lattice Green functions produces integral identities and hypergeometric connections, some of which appear to be new. 1

In this paper we will establish functional equations for Mahler measures of families of genus-one two-variable polynomials. These families were previously studied by Beauville, and their Mahler measures were considered by Boyd, Rodriguez-Villegas, Bertin, Zagier, and Stienstra. Our functional equations allow us to prove identities between Mahler measures that were conjectured by Boyd. As a corollary, we also establish some new transformations for hypergeometric functions.

New relations are established between families of three-variable Mahler measures. Those identities are then expressed as transformations for the 5 F 4 hypergeometric function. We use these results to obtain two explicit 5 F 4 evaluations, and several new formulas for 1/π.

We compute the critical L-values of some weight 3, 4, or 5 modular forms, by transforming them into integrals of the complete elliptic integral K . In doing so, we prove closed-form formulas for some moments of K 3 . Many of our L-values can be expressed in terms of Gamma functions, and we also obtain new lattice sum evaluations.