this question is related to this one Geometry for Anderson's motives?, 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.

this question is related to this one Geometry for Anderson's motives?, 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.