I will sketch the proof of a conjecture of Boyd and Rodriguez Villegas relating the Mahler measure of the polynomial $(1+x)(1+y)+z$ and the L-value $L(E,3)$ associated to an elliptic curve E of conductor 15. The proof uses the theory of multiple modular values (MMV) initiated by Manin and Brown. We show that the MMV associated to Eisenstein series satisfy differential relations with respect to the elliptic parameters. We obtain in this way relations between MMV of lengths 2 and 3, giving a connection between the Beilinson regulator and the Goncharov regulator for modular curves. We conjecture that these relations have a motivic origin and extend in higher weight.
Vladimir Rubtsov, Ilia Gaiur
