The (logarithmic) Mahler measure of a Laurent polynomial \(P \in \mathbb{C}[x_1^{\pm 1},\ldots, x_n^{\pm 1}]\) is defined by the integral \[ m(P) = \frac{1}{(2\pi i)^n} \int_{T^n} \log |P(x_1,\ldots,x_n)| \frac{dx_1}{x_1} \ldots \frac{dx_n}{x_n}, \] where \(T^n:|x_1|=\cdots=|x_n|=1\) is the unit torus in \((\mathbb{C}^\times)^n\).
In this article, the authors investigate the Mahler measure of the polynomial \[ P_k(x,y) = x+\frac{1}{x}+y+\frac{1}{y}+k \] in the cases \(k = 4 \pm 4 \sqrt{2}\). The Mahler measure of this family of polynomials has already been studied by Boyd and Rodriguez Villegas, among others. Based on numerical evidence, D. W. Boyd [Exp. Math. 7, No. 1, 37–82 (1998; Zbl 0932.11069)] has conjectured that for \(k \in \mathbb{Z} \setminus \{0,\pm 4\}\), we have \[ m(P_k) \stackrel{?}{=} c_k L'(E_k, 0),\tag{1} \] where \(c_k \in \mathbb{Q}^\times\) and \(L(E_k,s)\) is the \(L\)-function of the elliptic curve \(E_k\) defined by the equation \(P_k(x,y)=0\). This conjecture has been proved only for finitely many values of \(k\). On the other hand, F. Rodriguez Villegas [Math. Appl., Dordr. 467, 17–48 (1999; Zbl 0980.11026)] has studied the Mahler measure \(m(P_k)\) when the elliptic curve \(E_k\) has complex multiplication. He proved (1) in the cases \(k \in \{2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}, 4i\}\). In these cases, although \(k \not\in \mathbb{Q}\), the elliptic curve \(E_k\) has a Weierstrass equation defined over \(\mathbb{Q}\). For a comprehensive list of the proven cases, see Table 1 in [D. Samart, Integral Transforms Spec. Funct. 32, No. 1, 78–89 (2021; Zbl 1471.11264)].
In the case under study, namely \(k = 4 \pm 4 \sqrt{2}\), the elliptic curve \(E_k\) has complex multiplication, but is not isomorphic to the base change of an elliptic curve over \(\mathbb{Q}\). The authors prove that \begin{align*} m(P_{4 + 4 \sqrt{2}}) & = \mathrm{Re}(L'(g_{64},0)) + \mathrm{Im}(L'(g_{64},0)), \\ m(P_{4 - 4 \sqrt{2}}) & = \mathrm{Re}(L'(g_{64},0)) - \mathrm{Im}(L'(g_{64},0)), \end{align*} where \(g_{64}\) is a newform of weight \(2\) on \(\Gamma_1(64)\) with character \((\frac{8}{\bullet})\), having Fourier coefficients in \(\mathbb{Q}(i)\). The proof follows the method of Rodriguez Villegas, by expressing the Mahler measure \(m(P_k)\) as an Eisenstein-Kronecker series.
A particular feature here is that the elliptic curve \(E=E_{4+4\sqrt{2}}\) is isogenous to its Galois conjugate \(E'=E_{4-4\sqrt{2}}\), so that \(E\) is a \(\mathbb{Q}\)-curve. The \(L\)-function of \(E\) decomposes as the product \(L(g_{64},s) L(\overline{g}_{64},s)\), where \(\overline{g}_{64}\) is the complex conjugate of \(g_{64}\).
Furthermore, the authors prove a weak version of Beilinson’s conjecture for the \(L\)-value \(L''(E,0)\), by constructing two linearly independent elements in the integral subspace of \(K_2(E) \otimes \mathbb{Q}\) such that the associated Beilinson regulator is proportional to \(L''(E,0)\). The proof proceeds by relating the above Mahler measures to regulator integrals on \(E\) and \(E'\).