Are Anderson $T$-motives motives for the function field analogy?

Question

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.

Answer 1

Disclaimer: this is not an answer, and I am not an expert in the subject. I hope that the pointers I can give below are helpful nonetheless.

Some idea on motivic aspects of $t$-motives can be found on the website for a Banff workshop from 2009, see here.

There are several papers (including Anderson's original paper or this paper of Bornhofen and Hartl) which say that Anderson's pure $t$-motives are function field analogues of abelian varieties. In this direction, I would think that the category of Anderson's pure $t$-motives is a function field analogue of the category of Chow motives of abelian varieties; similarly, the full category of Anderson's $t$-motives should correspond to the category of mixed motives generated by motives of abelian varieties (although this does not exist at the moment). Conjecturally (cf. this MO-question) the category of motives is generated by motives of abelian varieties. [Edit: This is expected over finite fields, thanks to Mikhail Bondarko and guest for pointing this out: if hom=num then the category of motives over the finite field is generated by abelian varieties and is the category described in Jannsen's "Motives, numerical equivalence and semi-simplicity".] If this is the right track, then Anderson's $t$-motives are the function field analogue of something that at the moment is expected to exist, but whose existence depends on serious conjectures...

In another direction, Anderson's $t$-motives seem to be analogues of motives for automorphic forms. Historically, they appeared after Carlitz modules were used to find function field analogues of cyclotomic extensions, and Drinfeld modules were used to find the function field analogues of the extensions of number fields arising from elliptic curves (see e.g. Rosen's book on number theory in function fields or Drinfeld's paper on elliptic modules). So one use of $t$-motives is the construction of Galois extensions of $\mathbb{F}_q(T)$. Furthermore, there are papers talking about uniformization of $t$-motives, analogues to lattices etc. which also supports the belief that $t$-motives should be function field analogues of motives associated to summands of locally symmetric varieties $\Gamma\backslash G/K$ (and hence to the motives that appear in the Langlands program).

I guess the above two directions can actually be explained in a bigger picture, but you'll have to ask an expert in $t$-motives or Langlands program about this...

Answer 2

This is just a small supplement to Wendt's (as usual) excellent response.

Drinfel'd modules (or elliptic modules as Drinfel'd called them) are the function field analogues of elliptic curves. Note that elliptic curves make sense also over function fields. So elliptic modules are exotic analogues of elliptic curves. Similarly, Anderson t-motives are the function field analogues of the Tannakian category of (mixed) motives generated by abelian varieties. The Tannakian word is because the category of t-motives has a tensor product and duals, abelian varieties because the category contains elliptic modules.

But even this is not quite accurate. Because only Drinfeld modules of rank two correspond to elliptic curves, and those of rank one correspond to Tate motive (or its twists), and (it is tempting to think) those of rank >2 as corresponding to non-commutative tori (these are not connected to motives, as far as I know). So Anderson t-motives is the function field analogue of the Tannakian category of mixed motives generated by abelian varieties, Artin-Tate motives, and ... Here motives would mean under homological/numerical equivalence. There is no underlying geometry for Anderson's t-motives (varieties whose Chow groups etc could give t-motives).

@Wendt: it is only over finite fields that abelian varieties generate the category of motives. this does not hold over number fields.