On conjectures of Samart

Abstract

In this work, we verify all the conjectural formulas for the Mahler measure of the Laurent polynomial $$\begin{aligned} \left( X+\frac{1}{X}\right) ^{2}\left( Y+\frac{1}{Y}\right) ^{2}(1+Z)^{3}Z^{-2}-s \end{aligned}$$ parametrized by s posed by Samart using properties of spherical theta functions, and show that when s is induced by a CM point, these Mahler measures are all expressible in terms of special values of modular L-functions. In addition, we derive all new Samart-type formulas attached to a family of particular s as byproducts of this work. We remark that our method may also be used to verify all Samart’s remaining conjectural formulas associated to the Laurent polynomials $$\begin{aligned} \left( X+\frac{1}{X}\right) \left( Y+\frac{1}{Y}\right) \left( Z+\frac{1}{Z}\right) +s^{1/2}\quad \text{ and }\quad X^{4}+Y^{4}+Z^{4}+1+s^{1/4}XYZ, \end{aligned}$$ validating his hypothesis that $$n_{2}(s)$$ must be a linear combination of modular L-values at the s induced by the modularity of the associated K3 surface. At the end, we also affirm a conjecture of Samart on elliptic trilogarithms related to $$n_{2}(s)$$ by showing that the value of the elliptic trilogarithm associated to an elliptic curve E induced by an imaginary quadratic point at some 4-torsion point of E can be written as a linear combination of special values of Dirichlet L-series and modular L-functions.