Abstract
In this work, we verify all the conjectural formulas for the Mahler measure of the Laurent polynomial
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
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.
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The authors are grateful to the anonymous referee for his/her very useful comments, suggestions, and corrections.
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Correspondence to Dongxi Ye. The second author Dongxi Ye is supported by the Natural Science Foundation of China (Grant No. 11901586), the Natural Science Foundation of Guangdong Province (Grant No. 2019A1515011323) and the Sun Yat-sen University Research Grant for Youth Scholars (Grant No. 19lgpy244)
