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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
DLMF
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