Mesure de Mahler d'hypersurfaces K3

New 5 F 4 hypergeometric transformations, three-variable Mahler measures, and formulas for 1/π

“…, z n )) = We will define g 1 (u) and g 2 (u) in terms of the following three-variable Mahler measures + (x + x −1 )(z + z −1 ) + (y + y −1 )(z + z −1 )). (1.2) We can recover Bertin's original notation by observing that g 1 (u) = m(P u ), and after substituting (xz, y/z, z/x) → (x, y, z) in (1.2) we see that g 2 (u + 4) = m(Q u ) [6]. In Sect.…”

“…Recall that Bertin proved q-series expansions for a pair of three-variable Mahler measures in [6]. As usual the Mahler measure of an n-variable polynomial, P (z 1 , .…”

Feynman integrals and critical modular $L$-values

“…For example, Bertin [4,5] proved that, for some integral values of t which are corresponding to singular K3 surfaces, the Mahler measures of…”

An algebraic integration for Mahler measure

“…The first case would lead to instances of Beilinson's conjectures and produces special values of L-functions of surfaces. Examples in this direction can be found in Bertin's work [Be05]. Bertin relates the Mahler measure of some K3 surfaces to Eisenstein-Kronecker series in a similar way as Rodriguez-Villegas does for two-variable cases [R-V99].…”