Functional equations for Mahler measures of genus-one curves

Abstract

In this paper we will establish functional equations for Mahler measures of families of genus-one two-variable polynomials. These families were previously studied by Beauville, and their Mahler measures were considered by Boyd, Rodriguez-Villegas, Bertin, Zagier, and Stienstra. Our functional equations allow us to prove identities between Mahler measures that were conjectured by Boyd. As a corollary, we also establish some new transformations for hypergeometric functions.

Equations for Mahler measure and isogenies

“…Following the same techniques, similar results were given by Bertin [2,3], Touafek and Kerada [23], Lalín and Rogers [13], Lalín [12], and Guillera and Rogers [10]. Some of these proofs rely on functional equations involving m(α) or the other functions.…”

Divisor divisibility sequences on tori

“…The equation P T (X, Y ) = 0, which defines a family of elliptic curves over Q(T ), has been much studied. 6 For t ∈ Z, the Mahler measure M(P t ) is conjecturally related to the value of L ′ (E t , 0), and a number of deep relations between various M(P t ) values have been proven, for example M(P 8 ) = M(P 2 ) 4 and M(P 5 ) = M(P 1 ) 6 ; see [12,13]. It is thus natural to ask whether W n (P 8 ) and W n (P 2 ) 4 , or W n (P 5 ) and W n (P 1 ) 6 , are similarly related.…”

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

“…Because of this, we cannot establish a connection between y and y ± for the entire set of points in E a,c . As defined in equations (13), S ± and T ± have y-coordinate with absolute value 1, which in principle can be interpreted as either y + or y − . Combining the equations in Proposition 2 shows that y = y + over [S − , S + ] while y = y − over [T − , T + ].…”