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Background: There are several known lifts from integral weight modular forms to Siegel forms of genus 2, among them the Saito-Kurokawa lift. Another lift construction that is important for applications combines the Maass lift, which maps Jacobi forms to Siegel forms, with a lift of Skoruppa, which maps forms of integral weight $w$ with respect to ${\rm SL}(2,{\mathbb Z})$ to Jacobi forms of weight $(w-2)$. An example of this construction is the lift of the weight 12 form $\Delta(\tau) = \eta(\tau)^{24}$ to the Igusa form of weight 10. Thinking about motives in the context of Siegel forms of genus 2 leads to the following two questions.

Question 1: Has the lift construction by Maass and Skoruppa been generalized in a systematic way to congruence subgroups, for example $\Gamma_0(N)$?

Question 2: Is there a motivic interpretation of this lift by Maass and Skoruppa?

(Some references: the work of Skoruppa is in his 1992 paper (Math. of Comp. 58); a partial generalization of Skoruppa's lift to congruence subgroups has been considered by Gritsenko and collaborators in several papers. The Maass lift has been extended to restricted levels $N>1$ by Manickam-Ramakrishnan-Vasudevan in a 1993 paper (manuscripta math. 81). Question 1 is asking whether more systematic extensions are known.)

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Does the construction you're talking about send eigenforms to eigenforms? –  Kevin Buzzard Aug 7 '11 at 8:34
    
Yes, it sends cusp eigenforms to such. –  Laie Aug 7 '11 at 21:26
    
I'm not sure what you mean by "more systematic" but maybe <a href="www2.math.ou.edu/~rschmidt/papers/skclass.pdf">this paper</a> by Ralf Schmidt might be what you're interested in. –  ncr Jan 10 at 19:00

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