From $L$-series of elliptic curves to Mahler measures

Abstract

We prove the conjectural relations between Mahler measures and $L$-values of elliptic curves of conductors 20 and 24. We also present new hypergeometric expressions for $L$-values of CM elliptic curves of conductors 27 and 36. Furthermore, we prove a new functional equation for the Mahler measure of the polynomial family $(1+X)(1+Y)(X+Y)-\alpha XY$, $\alpha \in \mathbb{R}$.

33 pages

Cited by in corpus (14)

• On the Mahler measure of $1+X+1/X+Y+1/Y$
• Mahler measure of polynomials defining genus 2 and 3 curves
• Regulator of modular units and Mahler measures
• Regulators of Siegel units and applications
• Mahler measures as linear combinations of $L$-values of multiple modular forms
• On the Mahler measure of a family of genus 2 curves
• Further explorations of Boyd's conjectures and a conductor 21 elliptic curve
• Regulator proofs for Boyd's identities on genus 2 curves
• Régulateurs modulaires explicites via la méthode de Rogers-Zudilin
• Mahler measures of a family of non-tempered polynomials and Boyd's conjectures
• Mahler measure of a non-reciprocal family of elliptic curves
• A functional identity for Mahler measures of non-tempered polynomials
• $L$-values for conductor $32$
• The Mahler measure of exact polynomials in three variables