On conjectures of Samart

Abstract: In this work, we verify all the conjectural formulas for the Mahler measure of the Laurent polynomial \(\left(X+\frac{1}{X}\right)^{2}\left(Y+\frac{1}{Y}\right)^{2}(1+Z)^{3}Z^{-2}-s\) 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 \(\left(X+\frac{1}{X}\right)\left(Y+\frac{1}{Y}\right)\left(Z+\frac{1}{Z}\right)+s^{1/2}\quad\mbox{and}\quad X^{4}+Y^{4}+Z^{4}+1+s^{1/4}XYZ,\) 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 $K$3 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.