Papers › Mahler measure of polynomials defining genus 2 and 3 curves
Mahler measure of polynomials defining genus 2 and 3 curves
Hang Liu, Hourong Qin
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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.
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