Hodge circle (awaiting review)

For a Sato-Tate group $G$ of degree $d$ and weight $w$ a Hodge circle is a continuous homomorphism \[ h\colon \mathrm{U}(1)\to G^0 \] for which there exist nonnegative integers $h^{p,q}$ satisfying $\sum h^{p,q}=d$ with $p,q$ varying over integers in $[0,w]$ such that

• $h(u)$ has eigenvalues $u^{p-q}$ with multiplicity $h^{p,q}$ for all $u\in \mathrm{U}(1)$;
• the conjugates of the image of $h$ generate a dense subgroup of $G^0$.

When $G$ is the Sato-Tate group of a motive $X$ of weight $w$, the $h^{p,q}$ are the Hodge numbers of $X$. For a $d$-dimensional abelian variety $X$ we necessarily have $h^{0,1}=h^{1,0}=d$.