Mahler Measure of Families of Polynomials Defining Genus 2 and 3 Curves

Abstract

In this article, we study the Mahler measures of more than 500 families of reciprocal polynomials defining genus 2 and genus 3 curves. We numerically find relations between the Mahler measures of these polynomials with special values of L-functions. We also numerically discover more than 100 identities between Mahler measures involving different families of polynomials defining genus 2 and genus 3 curves. Furthermore, we study the Mahler measures of several families of nonreciprocal polynomials defining genus 2 curves and numerically find relations between the Mahler measures of these families and special values of L-functions of elliptic curves. We also find identities between the Mahler measures of these nonreciprocal families and tempered polynomials defining genus 1 curves. We will explain these relations by considering the pushforward and pullback of certain elements in K₂ of curves defined by these polynomials and applying Beilinson’s conjecture on K₂ of curves. We show that there are two and three explicit linearly independent elements in K₂ of certain families of genus 2 and genus 3 curves, respectively.

Keywords:

• Mahler measure
• special value of L-function
• Beilinson’s conjecture
• genus 2 and 3 curves
• computation

2010 Mathematics Subject Classification:

• 11R06
• 19F27
• 11Y99

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Acknowledgments

The authors thank to David Boyd, Francois Brunault, Xuejun Guo, Qingzhong Ji, Maltilde Lalin, Riccardo Pengo, and Wadim Zudilin for very helpful conversations and/or correspondence, as well as the referees for very useful comments and suggestions.

Additional information

Funding

The first author was supported by the National Natural Science Foundation of China (Grant nos. 11726606, and 11801345) and General Program Class A of Shenzhen Stable Support Plan. The second author was supported by the National Natural Science Foundation of China (Grant nos. 11726605, 11571163 and 11631009).