HEDGEHOGS IN LEHMER’S PROBLEM

Motivated by a famous question of Lehmer about the Mahler measure, we study and solve its analytic analogue.

Mahler measure of 3D Landau–Ginzburg potentials

We express the Mahler measures of 23 families of Laurent polynomials in terms of Eisenstein–Kronecker series. These Laurent polynomials arise as Landau–Ginzburg potentials on Fano 3-folds, sixteen of which define K ⁢ 3 {K3} hypersurfaces of generic Picard rank 19, and the rest are of generic Picard rank less than 19. We relate the Mahler measure at each rational singular moduli to the value at 3 of the L-function of some weight-3 newform. Moreover, we find ten exotic relations among the Mahler measures of these families.

ON AN INTEGRAL OF -BESSEL FUNCTIONS AND ITS APPLICATION TO MAHLER MEASURE

Cogdell et al. [‘Evaluating the Mahler measure of linear forms via Kronecker limit formulas on complex projective space’, Trans. Amer. Math. Soc. (2021), to appear] developed infinite series representations for the logarithmic Mahler measure of a complex linear form with four or more variables. We establish the case of three variables by bounding an integral with integrand involving the random walk probability density $a\int _0^\infty tJ_0(at) \prod _{m=0}^2 J_0(r_m t)\,dt$ , where $J_0$ is the order-zero Bessel function of the first kind and a and $r_m$ are positive real numbers. To facilitate our proof we develop an alternative description of the integral’s asymptotic behaviour at its known points of divergence. As a computational aid for numerical experiments, an algorithm to calculate these series is presented in the appendix.

Volume function and Mahler measure of exact polynomials

We study a class of two-variable polynomials called exact polynomials which contains $A$ -polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the two-dimensional torus and give a closed formula for the Mahler measure in terms of these extremal values. This formula shows that the Mahler measure of an irreducible and exact polynomial divided by $\pi$ is greater than the amplitude of the volume function. We also prove a K-theoretic criterion for a polynomial to be a factor of an $A$ -polynomial and give a topological interpretation of its Mahler measure.