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Kirti Joshi's musings mention "fractional motives". Do you know what are they good for and what the current state of constructions is for them?

Edit: Further cases of "fractional motives" as discussed in the article above (but with other weights) are expected to arise in quantum cohomology, say the experts. I wonder how the idea of such new motives may fit into the "usual" connections between motives, l-adic representations, periods and values of L-functions?

Edit: Conc. L-functions at non-integer values s, someone said that there is a quite old heuristical idea about it "as the dimension of an auxiliary affine space $A^s$ on which you multiply a given scheme over integers". Having either never read about that, or forgotten it: Do you know what it means and where to read more?

Edit: Some links: Yuri Manin had wondered if such things may exist (correct reference to Anderson's article on fractional "arithm. Hodge structures"), M. Marcolli wrote about such things in the context of "dimensional regularization" (and it's connection with log motives and motivic sheaves), Deligne extended representation theory to complex dimensions. It would be interesting to see how such speculations fit to Kedlaya's "fantasy in the key of p"...

Edit: new article by Matilde Marcoli: http://arxiv.org/abs/1310.2261

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    $\begingroup$ This is just vague recollections I have from having conversations about them with him, but I think that the idea is that $\mathbb{Q}(\frac{1}{2})$, for example, should be a motive whose tensor square is $\mathbb{Q}(1)$. So the whole business is trying to come up with what such things should look like. $\endgroup$
    – Cam McLeman
    Commented May 19, 2010 at 16:38
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    $\begingroup$ I should add on that the <i>point</i> of these things seems to be understanding special values of zeta functions at central values. I saw a talk of Ramachandran at one point where he gave an argument as to why these things should be related. $\endgroup$
    – Cam McLeman
    Commented May 19, 2010 at 16:45
  • $\begingroup$ Thanks! Ramachandran's very interesting article is on his website. He mentions that Grothendieck and Deligne had the idea of such motives in the 1960's and refers to articles of Denninger and Manin mentioning them. $\endgroup$
    – Thomas Riepe
    Commented May 29, 2010 at 22:00
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    $\begingroup$ The Tate motive has certain fractional tensor powers in the category of exponential motives, see p. 145 of Exponential Motives by Fresán and Jossen $\endgroup$
    – Julian Rosen
    Commented Oct 15, 2017 at 17:52
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    $\begingroup$ (Can't help it: it's when you don't entirely care, and are just partially motivated...) $\endgroup$
    – paul garrett
    Commented Jul 17, 2019 at 0:21

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