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The question is kind of contained in the title but let me add a few words.

If $f$ is a cusp form of weight $k$ for $SL(2, \mathbb{Z})$ then Scholl constructed a Grothendieck motive $M(f)$. In this case $k$ is an even natural number.

What happens for modular forms of half weight? Is there some motivic construction attached to them? I would be particularly interested in the case of the $\eta$ function.

Thanks for your help.

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  • $\begingroup$ I'm far from an expert but it seems very ambitious to ask for a motive, I don't think it's even clear what the correct notion of an attached Galois representation or $L$-function is in this context. What properties do you want from this motive? Try googling "l-function attached to modular form of half integral weight" and you'll find lots of places to start reading. $\endgroup$
    – Dan Petersen
    Sep 15, 2013 at 1:22
  • $\begingroup$ I'd agree with Dan that this is essentially the same question as "do L-functions exist for half-integral weight modular forms". This was asked before, and got a very nice answer from Marty Weissman: [mathoverflow.net/questions/113811/… $\endgroup$
    – David Loeffler
    Sep 15, 2013 at 8:16

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