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Functional equations for Mahler measures of genus-one. (English) Zbl 1172.11037

Let \(m(P)\) be the (logarithmic) Mahler measure and let \(L(E,s)\) be the \(L\)-function of the elliptic curve \(E\). In this paper, the authors study the values of \(m(k+x+1/x+y+1/y)\) for various values of \(k\). In particular, they prove that \[ m(2+x+1/x+y+1/y)= L'(E_{3\sqrt{2}},0) \] and \[ m(8+x+1/x+y+1/y)=L'(E_{3\sqrt{2}},0). \] Those identities were conjectured by Boyd in 1998. Using some modular equations they also prove the identity \[ \begin{split} m(4/k^{2}+x+1/x+y+1/y)\\=m(2k+2/k+x+1/x+y+1/y)+m(2i(k+1/k)+x+1/x+y+1/y)\end{split} \] for \(|k|<1\) and establish some new transformations for hypergeometric functions.

MSC:

11R09 Polynomials (irreducibility, etc.)
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
33C05 Classical hypergeometric functions, \({}_2F_1\)
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