What are "fractional motives"? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:37:36Z http://mathoverflow.net/feeds/question/25216 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25216/what-are-fractional-motives What are "fractional motives"? Thomas Riepe 2010-05-19T09:35:29Z 2011-01-01T10:47:17Z <p>Kirti Joshi's <a href="http://arxiv.org/abs/1005.3008" rel="nofollow" title="arxiv">musings</a> mention "fractional motives". Do you know what are they good for and what the current state of constructions is for them? </p> <p>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?</p> <p>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?</p> <p>Edit: Some links: Yuri Manin had <a href="http://arxiv.org/abs/math/0502016" rel="nofollow" title="arxiv">wondered</a> if such things may exist (correct reference to <a href="http://archive.numdam.org/ARCHIVE/CM/CM_1986__57_2/CM_1986__57_2_153_0/CM_1986__57_2_153_0.pdf" rel="nofollow" title="numdam">Anderson's article</a> on fractional "arithm. Hodge structures"), M. Marcolli <a href="http://arxiv.org/abs/0804.4824" rel="nofollow" title="arxiv">wrote</a> about such things in the context of "dimensional regularization" (and it's connection with log motives and motivic sheaves), Deligne <a href="http://www.math.ias.edu/files/deligne/Symetrique.pdf" rel="nofollow" title="pdf">extended</a> representation theory to complex dimensions. It would be interesting to see how such speculations fit to Kedlaya's <a href="http://math.mit.edu/~kedlaya/papers/nagoya2010.pdf" rel="nofollow" title="slides">"fantasy in the key of p"</a>... </p>