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What is the idea behind a "motive of a modular form"? (I know what a motive is and what a Weil cohomology is. I want to know how to get the motive [what is the idea of Scholl?], and why this is helpful.) I am aware of Scholl's article http://www.dpmms.cam.ac.uk/~ajs1005/preprints/mf.pdf

What does it help if I know that there is a motive of a modular form?

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Giving a motive underlying a modular form is a strengthening of the association of a compatible families of $\ell$-adic representations attached to the modular forms. It means that their is an underlying geometric context from which these $\ell$-adic representations arise by taking etale cohomology. –  Emerton Sep 7 '11 at 18:44
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One application is given already in Scholl's article: that the characteristic polynomial of the crystalline Frobenius coincides with the one for the $\ell$-adic Frobenii. –  Dan Petersen Sep 7 '11 at 19:04
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Dear norondion, The other realizations give other data attached to the modular form: the crystalline/$p$-adic de Rham realizations give an admissible filtered $\varphi$-module, which (thanks to the $p$-adic comparison theorem) we know is attached to the $p$-adic Galois representation locally at $p$ via the mechanisms of $p$-adic Hodge theory. The Betti/rational de Rham realizations give the periods of the modular form. The Hodge realization gives the Gamma factors in the functional equation for the $L$-function of the modular form. Regards, Matthew –  Emerton Sep 7 '11 at 19:09
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As for how Scholl gets his motive, he carefully analyzes Deligne's consruction of Galois representations, so that he can cut out the motive from the cohomology of (a particular desingularization of) the appropriate symmetric power of the universal elliptic curve using the Hecke operators (these are the most important correspondences involved, since they pick out the piece of cohomology corresponding to the Hecke eigenvalues coming from the particular newform under consideration) and some auxiliary correspondences which make sure that cohomology outside the middle degree and/or coming ... –  Emerton Sep 7 '11 at 19:14
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The analog of the Birch and Swinnerton-Dyer conjecture (namely, the Bloch-Kato conjecture) for modular forms is naturally stated in terms of the various realizations of the motive attached to the form. –  Ramsey Sep 7 '11 at 19:15
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One reason why modular motives are of interest is physics oriented, which may not be what you're looking for. In string theory modular forms arise naturally via the propagation of the 1-dimensional string itself, because the 2-dimensional worldsheet that is swept out by the string supports a conformal field theory. On the other hand, the string worldsheet is thought to be embedded in a spacetime manifold. The question then becomes wether the compact dimensions of this spacetime manifold can be constructed by considering the motives associated to the string theoretic modular forms. This strategy has been shown to work in some examples. At least in those cases one can view the modular motives as providing a string theoretic explanation of the extra dimensions, given e.g. by Calabi-Yau varieties.

As regards to the work of Deligne and Scholl, their construction is concerned with motives associated to the Kuga-Sato varieties. In the application just mentioned the focus is more on Calabi-Yau varieties, and the goal is to construct motives arising from such spaces.

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+1 for a point of view I wasn't aware of. –  James D. Taylor Sep 7 '11 at 19:32
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