An explicit Mahler measure (Q2745744)

Let \(P(x,y) = (x+1)(y^2+x)+(x^2-6x+1)y\). The author gives a proof of the formula \(m(P(x,y)) = 5d_3\), where \(m\) denotes the logarithmic Mahler measure and \(d_3 = L'(\chi_{-3},-1) = 3^{3/2}L(\chi_{-3},2)/(4\pi)\), \(\chi_{-3}\) being the odd quadratic character of conductor \(3\). This formula was conjectured by the reviewer in [Exp. Math. 7, 37-82 (1998; Zbl 0932.11069)] from experimental evidence. The author uses modular forms associated to a family of elliptic curves as in a paper of \textit{F. Rodriguez Villegas} [in Topics in Number Theory, ed. Ahlgren et al., 17-48 (1999; Zbl 0980.11026)]. By coincidence, the same identity has recently been established independently by two other quite different methods. The reviewer has shown that \(A(L,M)=P(L,M^2)\) is the so-called A-polynomial of the hyperbolic manifold \(m412\) which can be triangulated by \(5\) regular ideal tetrahedra and hence that \(\pi m(A) = \text{vol}(m412) = 5\pi d_3\) [Can. Math. Soc. Notes 34.2, 3-4 \& 26-28 (2002)]. Rodriguez Villegas has given an essentially elementary proof by finding an explicit primitive of the differential form \(\log|x|d\arg y - \log|y|d\arg x\) on the curve \(P(x,y) = 0\) from which the result follows by a direct calculation.