The Picard group of motivic A(1)

Abstract

We show that the Picard group $Pic(A(1))$ of the stable category of modules over $\mathbb{C}$-motivic $A(1)$ is isomorphic to ${\mathbb{Z}}^4$. By comparison, the Picard group of classical $A(1)$ is ${\mathbb{Z}}^2\oplus \mathbb{Z}/2$. One extra copy of $\mathbb{Z}$ arises from the motivic bigrading. The joker is a well-known exotic element of order $2$ in the Picard group of classical $A(1)$. The $\mathbb{C}$-motivic joker has infinite order.