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Anderson's $t$-motives satisfy most of what is expected of a reasonable category of mixed motives, except of course that everything is in positive characteristic. For instance, it is a linear category with a tensor product, we have a well behaved weight filtration, and there are realisation functors with good properties. (After all, the name "motive" was not chosen at random.)

However, a classical motive is something that one wants to associate with a scheme or variety, thinking of it as its universal cohomology, and now it bugs me that I have no geometric objects at hand to which I could associate $t$-motives. I don't even know whether these, if any, should be varieties or something else.

To make this a halfway real question: Let $C$ be a smooth proper curve over $\mathbb{F}_p(t)$ of genus $g$. Is there a natural, functorial way of associating with $C$ a pure $t$-motive $M(C)$ of rank $g$, in such a way of course that the cohomology of $C$ is related, via a functor morphism, to the realisations of $M(C)$? So I'm asking here for some kind of Jacobian construction.

But as I said, I don't even know whether varieties over $\mathbb{F}_p(t)$ are the right geometric objects to look at.

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One object with a very geometric flavor associated with Anderson's $t$-motives are $\tau$-sheaves, which are sheaves on the curve with a certain Frobenius structure. For Drinfeld modules (so Anderson $t$-motives of dimension 1), these are Drinfeld shtukas, which were then generalized to higher dimensions by several authors. There is a nice treatment of them in the recent book by Böckle and Pink, Cohomological Theory of Crystals over Function Fields. These are probably what you are looking for.

Pink has also defined Hodge structures for $t$-motives in the paper "Hodge structures over function fields," which is available on his web page: http://www.math.ethz.ch/~pink/.

To make this a halfway real question: Let C be a smooth proper curve over Fp(t) of genus g. Is there a natural, functorial way of associating with C a pure t-motive M(C) of rank g, in such a way of course that the cohomology of C is related, via a functor morphism, to the realisations of M(C)? So I'm asking here for some kind of Jacobian construction.

What you are proposing will have trouble working in some generality. For example, the characteristic polynomials of Frobenius acting on the Tate module of the Jacobian of the curve will have coefficients in ℤ, while those acting on the Tate module of the $t$-motive itself will have coefficients in the function field.

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