Further explorations of Boyd's conjectures and a conductor 21 elliptic curve

Abstract

We prove that the (logarithmic) Mahler measure $m(P)$ of $P(x,y)=x+1/x+y+1/y+3$ is equal to the $L$-value $2L^{\prime }(E,0)$ attached to the elliptic curve $E:P(x,y)=0$ of conductor 21. In order to do this we investigate the measure of a more general Laurent polynomial $P_{a,b,c}(x,y)=a(x+1/x)+b(y+1/y)+c$ and show that the wanted quantity $m(P)$ is related to a "half-Mahler" measure of $\tilde{P}(x,y)=P_{\sqrt{7},1,3}(x,y)$. In the finale we use the modular parametrization of the elliptic curve $\tilde{P}(x,y)=0$, again of conductor 21, due to Ramanujan and the Mellit--Brunault formula for the regulator of modular units.

21 pages

References in corpus (2)

• Regulator of modular units and Mahler measures
• Regulators of Siegel units and applications

Cited by in corpus (5)

• 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
• Mahler's measure and elliptic curves with potential complex multiplication
• The Mahler measure of exact polynomials in three variables