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From \(L\)-series of elliptic curves to Mahler measures. (English) Zbl 1260.11062
The authors prove unconditionally some relations between Mahler measures of families of polynomials and \(L\)-values of elliptic curves. In particular, they prove that \[ m(8+X+1/X+Y+1/Y)=4m(2+X+1/X+Y+1/Y)=\frac{24}{\pi^2} L(E_{24},2) \] and \[ m((1+X)(1+Y)(X+Y)-4XY)=\frac{3}{4}m(X^3+Y^3+1-\sqrt[3]{32} XY)=\frac{10}{\pi^2} L(E_{20},2), \] where \(E_{20}\) and \(E_{24}\) are elliptic curves of conductors \(20\) and \(24\), respectively. These formulas have been earlier found by Boyd (without proof). The proofs are quite technical although authors call them elementary. In Theorem 6 they derive the functional equation \[ g(4p(1+p))+g(4(1+p)p^{-2})=2g(2(1+p)^2 p^{-1}) \] for each \(p\) in the range \((\sqrt{3}-1)/2 \leq p \leq 1\) and \(g(p):=m((1+X)(1+Y)(X+Y)-pXY)\). The authors also establish some formulas for \(L(E_{27},2)\) and \(L(E_{36},2)\) in terms of hypergeometric functions and derive some other (known and unknown) results using elliptic and hypergeometric reduction.

MSC:
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
33C20 Generalized hypergeometric series, \({}_pF_q\)
11F03 Modular and automorphic functions
11G05 Elliptic curves over global fields
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
33C75 Elliptic integrals as hypergeometric functions
33E05 Elliptic functions and integrals
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