## 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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