Let $C$ be the Fermat curve of degree $d$, defined by the equation $x^d+y^d=z^d$ in $\mathbb{P}^2$. The first cohomology group $H^1(C, \mathbb{Q})$ carries a pure Hodge structure, so it has an associated Mumford-Tate group. I could not find any reference for its computation. I guess this is an exercise, but I am having trouble doing it :(
The idea is that the simple factors of the Jacobian are abelian varieties with complex multiplication by the cyclotomic field $E=\mathbb{Q}(\mu_d)$, so their Mumford-Tate groups should be tori, which are quotients of the Tannaka group of the category of Hodge structures with CM by $E$. This group is the torus with group of characters the CM types of $E$, which is just the Serre torus of $E$, right? My impression is also that, under some assumptions ($d$ prime?) $H^1(C, \mathbb{Q})$ generates that category, so we already get the Serre torus by only looking at this object.
Does anybody know a reference or can help me?
