Mahler Measure, Eisenstein Series and Dimers

Mahler Measure Variations, Eisenstein Series and Instanton Expansions

This paper points at an intriguing inverse function relation between Eisenstein series connected with ``Modular Mahler Measures'' and instanton numbers for ``Non-Critical Strings''. In a companion paper Mahler measures are related to dimer models.

Mesure de Mahler d'hypersurfaces K3

We express the logarithmic Mahler measure of two families of polynomials defining K3 hypersurfaces of generic Picard number 19 in terms of Eisenstein-Kronecker series. For some singular K3 surfaces (i.e of Picard number 20) we write this measure as Hecke L-series of weight 3 with "Gr\" ossencharacters".

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, $16$ of which define $K3$ hypersurfaces of generic Picard rank $19$, and the rest are of generic Picard rank $< 19$. 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 $10$ exotic relations among the Mahler measures of these families.

Mahler Measuring the Genetic Code of Amoebae

Amoebae from tropical geometry and the Mahler measure from number theory play important roles in quiver gauge theories and dimer models. Their dependencies on the coefficients of the Newton polynomial closely resemble each other, and they are connected via the Ronkin function. Genetic symbolic regression methods are employed to extract the numerical relationships between the 2d and 3d amoebae components and the Mahler measure. We find that the volume of the bounded complement of a d-dimensional amoeba is related to the gas phase contribution to the Mahler measure by a degree-d polynomial, with…

Mahler Measure and the Vol-Det Conjecture

The Vol-Det Conjecture relates the volume and the determinant of a hyperbolic alternating link in $S^3$. We use exact computations of Mahler measures of two-variable polynomials to prove the Vol-Det Conjecture for many infinite families of alternating links. We conjecture a new lower bound for the Mahler measure of certain two-variable polynomials in terms of volumes of hyperbolic regular ideal bipyramids. Associating each polynomial to a toroidal link using the toroidal dimer model, we show that every polynomial which satisfies this conjecture with a strict inequality gives rise to many infin…

马勒测度与实二次域上椭圆曲线的 $L$-值

Mahler measures and $L$-values of elliptic curves over real quadratic fields

本文证明 Villegas 公式中次数可由 CM 点类数界定，由此推导 28 个新恒等式及 5 个将实二次域上 CM 椭圆曲线 $L$-值表为 $m(k)$ 行列式的公式，推广了 Guo 等人的工作。