this question is related to this one https://mathoverflow.net/questions/24282/geometry-for-andersons-motives/30315#30315, though the previous one doesn't answer exactly my question. 

 Let $\mathbb{C}_{\infty}$ be the function field analog of $\mathbb{C}$ for $\mathbb{F}_q (\theta) = \mathbb{Q}_{\infty}$ and $A = \mathbb{C}_{\infty} [T, \tau]$ the Anderson ring ( i.e.,  $T$ is central and $\tau$ acts as the Frobenius endomorphism).

 An Anderson $T$-motive is simply a left $A$-module $M$ which is free and finitely generated over $\mathbb{C}_{\infty} [\tau]$, and satisfies $$(T - \theta)^n M/\tau M = \{ 0 \}$$ for some  $n > 0$. If, furthermore, $M$ is finitely generated over $\mathbb{C}_{\infty}[T]$ (or equivalently, free of finite rank), then it's called an abelian $T$-motive or $t$-motive.

  First of all, why the word **"motives"** in "Anderson $T$-motives"?

  Are $T$-motives analogous to motives of arithmetic schemes? What's its number field analogue?
 
  Thanks in advance.