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.
Reviewer: Artūras Dubickas (Vilnius)
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, \({}_2F_1\) |