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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:

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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. – Cam McLeman May 19 '10 at 16:38
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. – Cam McLeman May 19 '10 at 16:45
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. – Thomas Riepe May 29 '10 at 22:00
I cannot say this is related directly, though it has special values as with Cam McLeman's second comment, but Shimura's paper for Hecke Grossencharacters and periods for abelian varieties can handle more cases than future authors, as he allows $\infty$-types to have half-integral values I think. Harris notes this in Remark 1.7 (p647) of "L-functions of 2x2 unitary matrices" I do not know if there is a motivic explanation of these as in Schappacher's book. – Junkie Apr 9 '11 at 6:39
I have a specific criticism of Ramachandran's example (page 1), if I am not mistaken. "The work of Harris/Kudla [7] relates the central critical value (at $s=1/2$) to periods, cf. [22, (48), p.459]." For (48) in Sarnak's survey, the $s=1/2$ central value is renormalized from a triple product with $s\rightarrow (k_1+k_2+k_3-3)+1-s$ in the natural motivic way, so the central value is of an integer when all $k$ are even. So Watson's formula is in the integral motive framework I think. Harris/Kudla do the same (p607): " that $k_1+k_2+k_3-3$ is odd." – Junkie Apr 9 '11 at 7:06

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