On the Mahler Measure of 1+X+1/X+Y +1/Y

Abstract

We prove a conjectured formula relating the Mahler measure of the Laurent polynomial 1 + X + X −1 + Y + Y −1 to the L-series of a conductor 15 elliptic curve.

Regulator of modular units and Mahler measures

“…and the path of integration γ in (12) corresponds to the range of τ between the two cusps −1/5 and 1/5 of Γ 0 (15). Therefore, Theorem 1 results in 2), which is precisely Boyd's conjecture from [3] first proven in [15].…”

Feynman integrals, L-series and Kloosterman moments

“…where the L-series comes from the modular form η 1 η 3 η 5 η 15 , with weight 2 and level 15. This was proven by Mathew Rogers and Wadim Zudilin [42]. David Boyd conjectured [9] and Anton Mellit proved [37] that…”

Further explorations of Boyd's conjectures and a conductor 21 elliptic curve

“…Remark 3. The expression (19) for m(P a,c ) when a = 1 as well as formulas (17) and (18) show that Proposition 2 is indeed a weaker version of Theorem 2: the rational p is specified in the latter to be 3/4. The integrality of 8p guaranteed by Proposition 2 gives, in fact, a practical recipe to compute the number by providing a simple numerical approximation to p; however the latter has to be done for each particular choice of a, c -the proposition does not guarantee that the integer p is independent of the parameters.…”

“…The work [16] has already incorporated some methods for attacking the conjectural evaluations of Mahler measures via L-values and proving several such cases when the related elliptic curves P (x, y) = 0 have complex multiplication. Some further development of the techniques in the works of Brunault, Mellit, Rogers and others [7,11,13,14,18,19,20] has allowed to establish several new conjectural instances when the elliptic curves are parameterized by modular units (that is, modular functions whose zeroes and poles are only at cusps). The final news is an elegant general formula of Brunault [8] that allows one to deal with parametrization by Siegel units; this creates an efficient way to proving any particular Mahler measure evaluation on a case-by-case study, and the work [8] includes numerous illustrations to the principle.…”