From \(L\)-series of elliptic curves to Mahler measures. (English) Zbl 1260.11062

Rogers, Mathew; Zudilin, Wadim

Compos. Math. 148, No. 2, 385-414 (2012).

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.

Reviewer: Artūras Dubickas (Vilnius)

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

Keywords:

Mahler measure; \(L\)-value of elliptic curve; hypergeometric series; lattice sum; elliptic dilogarithm

References:

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