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First of all, I need to declare my extreme ignorance on the topic of modular forms, so, please, does not assume that I know Deligne's construction in details.

In Motives for modular forms, Scholl constructs motives associated to modular forms . Deligne's also constructed $\ell$-adic representations attached to modular forms of weight $k \geq 2$. For $k=2$, Shimura constructed an abelian variety $A_f$ and then obtained a $\ell$-adic representation from it's Tate module. It's known that Scholl construction coincides with Shimura construction, i.e., the motive attached to a modular form of weight $k = 2$, namely $M_f$ coincides with $M(A_f)$.

My question is if Scholl construction is compatible with Deligne's construction in the sense that these motives are the underlying geometric objects of Deligne's constructions. More explicitly, I would like to know if from these motives $M_f$ I can create an $\ell$-adic representation with values in some object of cohomological nature arising from $M_f$ (like motivic cohomology) such that this representation is the one constructed by Deligne.

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More explicitly, I would like to know if from these motives $M_{f}$ I can create an $\ell$-adic representation with values in some object of cohomological nature arising from $M_{f}$ (like motivic cohomology) such that this representation is the one constructed by Deligne.

The answer is: of course, and in some sense almost by definition. When we say a Galois representation $\rho_{f}$ is attached to an eigencuspform $f$, this means that the $L$-function of $\rho_f$ and of $f$ are equal and, likewise, when we say that a motive $M(f)$ is attached to $f$, this means that the motive and the modular form have the same $L$-function (where the $L$-function of a motive $M$ is the $L$-function of the $p$-adic étale representation attached to the $p$-adic étale realization of $M$ at a prime $p$ of good reduction of $M$). Hence, $\rho_f$ and $M(f)_{\operatorname{et},p}$ have the same $L$-function by definition and so (as $\rho_{f}$ is semi-simple) the $\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)$-representations $\rho_{f}$ and $M(f)$ are isomorphic.

But let me be more precise, because we actually know much more in this case. Let $k≥2$ be an integer and let $M=(KS_k,\Pi)$ be the (Chow) motive constructed by Scholl, where $KS_{k}$ is the Kuga-Sato variety (that is to say the canonical desingularization of the $k-2$-fold fiber product of the universal generalized elliptic curve $\pi:\bar{E}\longrightarrow X(N)$ over the modular curve $X(N)$) and where $\Pi$ is the projector arising from the action of the wreath product of $((\mathbb Z/2\mathbb Z)^2\rtimes\{\pm 1\})$ with $\mathfrak S_{k-2}$ on $\bar{E}^{k-2}$ (and hence on $KS_k$). Let $f\in S_{k}(\Gamma(N))$ be a normalized eigencuspform. Let $M(f)$ be the (Grothendieck) motive cut out of $M$ by the $f$-component of the action of the Hecke algebra. Finally, denote by $H^{1}_{\operatorname{et}}(X(N)\times_{\mathbb Q}\bar{\mathbb Q},\operatorname{Sym}^{k-2}R^1\pi_*\mathbb Q_{p})$ the Galois representation constructed by Deligne and by $V(f)$ its quotient corresponding to $f$.

Then there is a canonical Galois equivariant isomorphism \begin{equation} M_{\operatorname{et},p}:=H^{k-1}(KS_k(\mathbb C),\mathbb Q)(\Pi)\otimes_{\mathbb Q}\mathbb Q_{p}\simeq H^{1}_{\operatorname{et}}(X(N)\times_{\mathbb Q}\bar{\mathbb Q},\operatorname{Sym}^{k-2}R^1\pi_*\mathbb Q_{p}) \end{equation} inducing a Galois equivariant isomorphism $M(f)\simeq V(f)$. So there is a geometric description of Deligne's Galois representation for $k\geq 2$ just as you suspected.

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  • $\begingroup$ Thanks for the answer. Sorry, but how do you attach a representation to the p-adic étale realization of $M$? $\endgroup$ – user40276 Feb 25 '15 at 19:01
  • $\begingroup$ I mean is this just the usual action on étale cohomology? $\endgroup$ – user40276 Feb 25 '15 at 19:05
  • $\begingroup$ Furthermore, where can I find a proof of the Galois equivariant isomorphism you cited? $\endgroup$ – user40276 Feb 25 '15 at 19:06
  • $\begingroup$ @user40276 1,2) Yes 3) In Scholl's article. $\endgroup$ – Olivier Feb 25 '15 at 19:52
  • $\begingroup$ Sorry for bothering you again. But Scholl construction is over a full $N$ level structure or just a $\Gamma_0 (N)$-structure? Furthermore the second cohomology shouldn't be something in parabolic cohomology (according to Deligne's construction)? $\endgroup$ – user40276 Feb 26 '15 at 12:25

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