New period representations for $L(E,3)$

Contact: Pieter Moree (moree @ mpim-bonn.mpg.de)

Special values of $L$-functions at integers are expected to be periods in the sense of Kontsevich–Zagier. More precisely, the Beilinson conjectures provide expressions for them in terms of so-called regulator integrals. For $L$-values of modular forms, the regulator integrals considered by Beilinson are double modular values in the sense of Manin and Brown. We consider in this talk a regulator integral of Goncharov type. We show it is equal to a linear combination of triple modular values, and evaluate it as the $L$-value of a weight 2 modular form at $s=3$. As an application, we prove a conjecture of Boyd and Rodriguez Villegas relating the Mahler measure of the polynomial $(1+x)(1+y)+z$ and $L(E,3)$, where $E$ is an elliptic curve of conductor $15$. This is (in part) joint work with Wadim Zudilin.