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On a conjecture by Boyd. (English) Zbl 1201.11098

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 \(m(16) = 11m(1)\).

MSC:

11R09 Polynomials (irreducibility, etc.)
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
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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
[4] DOI: 10.1017/CBO9781139172530
[5] DOI: 10.1090/S0894-0347-97-00228-2 · Zbl 0913.11027
[6] Kurokawa N., Comment. Math. Univ. St. Pauli 54 pp 121–
[7] DOI: 10.2140/ant.2007.1.87 · Zbl 1172.11037
[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.
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