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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 $m((1+X)(1+Y)(X+Y)-4XY)=\frac{3}{4}m(X^3+Y^3+1-\sqrt{32} XY)=\frac{10}{\pi^2} L(E_{20},2),$ 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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References:
  doi:10.1090/S0002-9939-97-03928-2 · Zbl 0894.11020  doi:10.2140/ant.2007.1.87 · Zbl 1172.11037  doi:10.2307/2001551 · Zbl 0725.33014  doi:10.1063/1.526675 · Zbl 0587.40007  doi:10.1007/978-1-4612-1624-7  doi:10.1007/978-1-4612-0879-2  doi:10.1007/978-1-4612-0965-2  doi:10.1007/s11139-009-9186-9 · Zbl 1183.33037  doi:10.1007/BF02465410 · Zbl 0780.33013
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