Lalín, Matilde N. On a conjecture by Boyd. (English) Zbl 1201.11098 Int. J. Number Theory 6, No. 3, 705-711 (2010). Let \(P_k(x,y) = x + 1/x + y + 1/y +k\), and let \(m(k) = m(P_k)\), \(m\) denoting the two dimensional logarithmic Mahler measure. Let \(E_k\) denote the (usually) elliptic curve defined by \(P_k(x,y) = 0\). In [Exp. Math. 7, No. 1, 37–82 (1998; Zbl 0932.11069)], the reviewer described computations to support the conjecture that \(m(k) = r_k L'(E_k,0)\) for integer \(k\), where the \(r_k\) are certain rational numbers. If \(E_{k_1}\) and \(E_{k_2}\) are isogenous then their \(L\)-functions coincide and then the conjectural identities would imply that \(r_{k_2} m(k_1) = r_{k_1} m(k_2)\). One of these conjectured identities, \(m(8) = 4m(2)\) was proved by the author and M. D. Rogers in [Algebra Number Theory 1, No. 1, 87–117 (2007; Zbl 1172.11037)].In this paper, the author establishes a second such identity, \(m(5) = 6m(1)\). The proof is accomplished by calculations in \(K_2(\mathbb C(E_k))\) for the two curves in question, establishing relationships between the regulators of the two curves which give enough linear relationships to prove the desired identity, as well as to give another proof of \(m(8) = 4m(2)\) and \(m(16) = 11m(1)\). Reviewer: David W. Boyd (Vancouver) Cited in 2 ReviewsCited in 10 Documents MSC: 11R09 Polynomials (irreducibility, etc.) 19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) Keywords:Mahler measure; elliptic curve; elliptic dilogarithm; regulator Citations:Zbl 0932.11069; Zbl 1172.11037 PDFBibTeX XMLCite \textit{M. N. Lalín}, Int. J. Number Theory 6, No. 3, 705--711 (2010; Zbl 1201.11098) Full Text: DOI arXiv References: [1] Bertin M. J., J. Reine Angew. Math. 569 pp 175– [2] Bloch S. J., CRM Monograph Series 11, in: Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves (2000) · Zbl 0958.19001 [3] DOI: 10.1080/10586458.1998.10504357 · Zbl 0932.11069 · doi:10.1080/10586458.1998.10504357 [4] DOI: 10.1017/CBO9781139172530 · doi:10.1017/CBO9781139172530 [5] DOI: 10.1090/S0894-0347-97-00228-2 · Zbl 0913.11027 · doi:10.1090/S0894-0347-97-00228-2 [6] Kurokawa N., Comment. Math. Univ. St. Pauli 54 pp 121– [7] DOI: 10.2140/ant.2007.1.87 · Zbl 1172.11037 · doi:10.2140/ant.2007.1.87 [8] F. Rodriguez-Villegas, Topics in Number Theory (University Park, PA, 1997), Math. Appl. 467 (Kluwer Acad. Publ., Dordrecht, 1999) pp. 17–48. [9] F. Rodriguez-Villegas, Number Theory for the Millennium, III (Urbana, IL, 2000) (A. K. Peters, Natick, MA, 2002) pp. 223–229. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.