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

TL;DR

This paper proves a specific Mahler measure of a polynomial equals an elliptic curve's L-value, using modular parametrization and regulator formulas, advancing understanding of Boyd's conjectures for conductor 21 elliptic curves.

Contribution

It establishes a new explicit link between Mahler measures and L-values for a conductor 21 elliptic curve using modular techniques.

Findings

Proved that the Mahler measure m(P) equals 2L'(E,0) for a specific polynomial.

Connected Mahler measures to half-Mahler measures of related polynomials.

Utilized Ramanujan's modular parametrization and Mellit–Brunault formula in the proof.

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 $2L'(E,0)$ 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.