FromL-series of elliptic curves to Mahler measures

Abstract

<jats:title>Abstract</jats:title><jats:p>We prove the conjectural relations between Mahler measures and<jats:italic>L</jats:italic>-values of elliptic curves of conductors 20 and 24. We also present new hypergeometric expressions for<jats:italic>L</jats:italic>-values of elliptic curves of conductors 27 and 36. Furthermore, we prove a new functional equation for the Mahler measure of the polynomial family (1+<jats:italic>X</jats:italic>) (1+<jats:italic>Y</jats:italic>)(<jats:italic>X</jats:italic>+<jats:italic>Y</jats:italic>)−<jats:italic>αXY</jats:italic>,<jats:italic>α</jats:italic>∈ℝ.</jats:p>

Equations for Mahler measure and isogenies

“…Parallel to this, Rodriguez-Villegas [15] made a more detailed analysis of Boyd's results in the light of Beilinson's conjectures. Since then, results in the direction of equations (1.1) have been proven for m(α), g(α), n(α), and r(α) by Rodriguez-Villegas [15], Brunault [6,7], Mellit [14], and Rogers and Zudilin [18,19]. For example, Rogers and Zudilin proved in [18] that…”

Арифметические Гипергеометрические Ряды

“…Использование гипергеометрического аппарата играет центральную роль при получении в [125] доказательства равенства (10), а также при доказательстве некоторых других гипотез из списка Бойда в [124], которые оставались откры-тыми в течение многих лет. Из (10) и (11) следует, что…”

On the Mahler measure of a family of genus 2 curves

“…We believe that Theorem 1 and its proof below provides us with at least a partial understanding of the magic behind the L-series evaluations of Mahler measures. The results in [8,9,10] and the theorem also lead to different proofs of the formulae for m(Q 2 ) and m(Q −1 ), as well as to four more equalities conjectured by Boyd in [6].…”

“…The required evaluations follow from the formulae for m(P 0 ), m(P −2 ), m(P 1 ), m(P 4 ), m(P 6 ) and m(P 10 ) obtained in [8,9,10].…”

“…One of the exemplar families include evaluations of m(P k (x, y)) for P k (x, y) = (x + 1)y 2 + (x 2 + kx + 1)y + (x 2 + x), k ∈ Z, which can be characterised by m(P k )/L ′ (E, 0) ∈ Q × , where E : P k (x, y) = 0 is (generically) an elliptic curve and L ′ (E, 0) is the derivative of its L-function L(E, s) at s = 0. Very few of these are proven so far: k = 0, 6 (conductor 36) by RodriguezVillegas [9], k = 1, 10, −5 (conductor 14) by Mellit [8], and k = 4, −2 (conductor 20) by Rogers and the second author [10]. Note that these particular cases (as well as all other proven cases of elliptic type, for both CM and non-CM curves) accidentally fall under application of the Mellit-Brunault formula [12].…”