A Mahler measure of a \(K3\) surface expressed as a Dirichlet \(L\)-series. (English) Zbl 1273.11152

Bertin, Marie José

Can. Math. Bull. 55, No. 1, 26-37 (2012).

Let \(m(P)\) be the logarithmic Mahler measure of a Laurent polynomial \(P\). Set \[ Q_k:=X+\frac{1}{X}+ Y+\frac{1}{Y}+Z+\frac{1}{Z}+XY+\frac{1}{XY}+ZY+\frac{1}{ZY}+XYZ+\frac{1}{XYZ}-k. \] Using a quite complicated formula for \(m(Q_k)\) the author derives a simple formula for \(m(Q_{-3})\) in terms of Dirichlet \(L\)-series, namely, \(m(Q_{-3})=8d_3/3\). This formula was guessed numerically by Boyd.

Reviewer: Artūras Dubickas (Vilnius)

MSC:

11R09	Polynomials (irreducibility, etc.)
11M41	Other Dirichlet series and zeta functions
14J28	\(K3\) surfaces and Enriques surfaces

Keywords:

modular Mahler measure; Eisenstein-Kronecker series; \(L\)-series of \(K3\)-surfaces; \(l\)-adic representations; Livné criterion; Rankin-Cohen brackets