[Paper Review] Integrality Properties of Variations of Mahler Measures

Jian Zhou|arXiv (Cornell University)|Jun 12, 2010

Mathematical Dynamics and Fractals8 references3 citations

TL;DR

This paper proposes conjectures on the integrality properties of variations of Mahler measures in Calabi-Yau hypersurfaces, identifying a key function $ Q(z) $ with the local mirror map. It demonstrates that certain coefficients derived from period integrals—when normalized by $ m $—are integers in multiple cases, especially for $ n=4 $ and $ n=5 $, suggesting deep arithmetic structures analogous to those in mirror symmetry and Gopakumar-Vafa invariants.

ABSTRACT

We propose some conjectures on the integrality properties related to the variation of Mahler measures, inspired by the results in the elliptic curve case by Rodriguez Villegas, Stienstra and Zagier.

Motivation & Objective

• To explore integrality properties of variations of Mahler measures in the context of mirror symmetry and Calabi-Yau geometry.
• To identify a function $ Q(z) $ associated with Mahler measure variations as the local mirror map in Picard-Fuchs systems.
• To formulate conjectures on the integrality of $ Q(q) $ and $ q(Q) $, extending known results from elliptic curve cases.
• To investigate whether normalized coefficients $ b_m/m $, $ \hat{b}_m/m $, $ c_m/m $, and $ \hat{c}_m/m $ are integers across various $ n $-fold hypersurfaces.
• To provide computational evidence for integrality in $ n=4,5 $, and selected $ n>5 $ cases, including explicit examples with large integers.

Proposed method

• Uses one-parameter deformations of Fermat-type Calabi-Yau hypersurfaces in weighted projective spaces satisfying $ \sum 1/k_i = 1 $.
• Defines a variation of Mahler measure via the polynomial $ k\psi \prod x_i - \sum x_i^{k_i} $, with $ \psi $ as a complex parameter.
• Derives period integrals and associated hypergeometric series to compute coefficients $ b_m $, $ \hat{b}_m $, $ c_m $, $ \hat{c}_m $ from the Picard-Fuchs equation.
• Identifies the function $ Q(z) $, derived from the period ratio, as the local mirror map in the Calabi-Yau geometry.
• Performs symbolic and numerical computations using computer algebra systems to verify integrality of normalized coefficients.
• Analyzes cases for $ n=2 $ to $ n=7 $, including $ (2,3,7,42) $ and $ (2,7,43,1807,3263442) $, to test conjectures.

Experimental results

Research questions

• RQ1 Are the normalized coefficients $ b_m/m $, $ \hat{b}_m/m $, $ c_m/m $, and $ \hat{c}_m/m $ integers for the variation of Mahler measure in $ n $-dimensional Calabi-Yau hypersurfaces?
• RQ2 Can the function $ Q(z) $ associated with Mahler measure variation be identified with the local mirror map in the Picard-Fuchs system?
• RQ3 Is there a universal integrality pattern in $ b_m/m $, $ \hat{b}_m/m $, $ c_m/m $, and $ \hat{c}_m/m $ for the Fermat-type hypersurface $ \sum x_i^n = n\psi \prod x_i $, particularly when $ n $ divides the coefficients?
• RQ4 Why are $ b_m/m $, $ \hat{b}_m/m $, $ c_m/m $, and $ \hat{c}_m/m $ integers in $ n=4 $ and $ n=5 $ cases, but not always in $ n=5 $ for $ b_7/7 $, despite $ b_7 $ being divisible by 5?
• RQ5 What is the enumerative or geometric meaning of the observed integrality in the coefficients of the mirror map and its inverse?

Key findings

• For the $ n=4 $ case $ x_1^4 + \cdots + x_4^4 = 4\psi x_1\cdots x_4 $, all normalized coefficients $ b_m/m $, $ \hat{b}_m/m $, $ c_m/m $, and $ \hat{c}_m/m $ are integers.
• For the $ n=5 $ case $ x_1^5 + \cdots + x_5^5 = 5\psi x_1\cdots x_5 $, $ b_m $, $ \hat{b}_m $, $ c_m $, and $ \hat{c}_m $ are divisible by 5, but $ b_7/7 = 31249534645239703150/7 $ is not an integer.
• In the $ n=5 $ case $ x_1^3 + x_2^3 + x_3^2 + x_4^2 + x_5^2 = 12\psi x_1\cdots x_5 $, all normalized coefficients $ b_m/m $, $ \hat{b}_m/m $, $ c_m/m $, and $ \hat{c}_m/m $ are integers, with $ b_6/6 = -61961714940992690898780121741257228991904436 $.
• For the $ n=6 $ case $ x_1^6 + \cdots + x_6^6 = 6\psi x_1\cdots x_6 $, the coefficients $ b_m/m $, $ \hat{b}_m/m $, $ c_m/m $, and $ \hat{c}_m/m $ are conjectured to be integers.
• For the $ n=7 $ case $ x_1^7 + \cdots + x_7^7 = 7\psi x_1\cdots x_7 $, the same integrality pattern is conjectured to hold.
• The paper provides explicit large integer values such as $ b_5 = 31088578606413096899258654040 $ for the $ n=4 $ case with $ x_1^4 + x_2^3 + x_3^3 + x_4^2 = 12\psi x_1x_2x_3x_4 $, confirming integrality of $ b_m/m $.