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Further explorations of Boyd’s conjectures and a conductor 21 elliptic curve. (English) Zbl 1337.11072
Let \(P_{a,c}(x,y)=a(x+1/x)+y+1/y+c\). Writing \(yP_{a,c}(x,y)=(y-y_{+}(x))(y-y_{-}(x))\), one can define \(m^{\pm}(P_{a,c})=m(y-y_{\pm})\), so that the logarithmic Mahler measure \(m(P_{a,c})\) is equal to \(m^{+}(P_{a,c})+m^{-}(P_{a,c})\). For any real \(k\) satisfying \(0<k<4\) and \(a=\sqrt{(4+k)/(4-k)}\), \(c=k/\sqrt{4-k}\) the authors show that \(m(P_{1,k})=m^{-}(P_{a,c})-3m^{+}(P_{a,c})\) (Theorem 2). They also prove that \(m^{-}(P_{\sqrt{7},3})=\frac{1}{2}L'(f_{21},0)+\frac{3}{8} \log 7\) (Theorem 3). As a consequence they recover Boyd’s conjecture for \(k=3\) by showing that \[ m(x+1/x+y+1/y+3)=2L'(E,0), \] where \(E\) is the elliptic curve \(x+1/x+y+1/y+3=0\) (Corollary 1).

MSC:
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G05 Elliptic curves over global fields
11G16 Elliptic and modular units
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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