# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [3] doi:10.1090/S0002-9939-97-03928-2 · Zbl 0894.11020 [5] doi:10.2140/ant.2007.1.87 · Zbl 1172.11037 [6] doi:10.2307/2001551 · Zbl 0725.33014 [8] doi:10.1063/1.526675 · Zbl 0587.40007 [11] doi:10.1007/978-1-4612-1624-7 [13] doi:10.1007/978-1-4612-0879-2 [14] doi:10.1007/978-1-4612-0965-2 [15] doi:10.1007/s11139-009-9186-9 · Zbl 1183.33037 [17] doi:10.1007/BF02465410 · Zbl 0780.33013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.