On the Mahler measure of a family of genus 2 curves (original) (raw)

We establish a general identity between the Mahler measures m(Q k (x, y)) and m(P k (x, y)) of two polynomial families, where Q k (x, y) = 0 and P k (x, y) = 0 are generically hyperelliptic and elliptic curves, respectively. 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 Rodriguez-Villegas [9], k = 1, 10, −5 (conductor 14) by Mellit [8], and k = 4, −2 (conductor 20) by Rogers and the second author . 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 .