# Deligne-Scholl's motives for modular forms: Hecke operators vs. transposed Hecke operators

EDIT: I moved my original question down to the bottom. The question here at the top is related, at least I suppose that the same phenomenon is behind both of them.

In the article "Valeurs de fonctions L et périods d'intégrales" (p. 333/334), Deligne sketches how to define the motive $M(f)$ attached to a newform $f=\sum a_nq^n$. The construction involves taking the eigenspace (inside a certain cohomology) where the Hecke operators act with the eigenvalues given by the Fourier coefficients of $f$, i.e. the common kernel of all $T_p-a_p$. He then uses the differential form $\omega_f:=f(z)\,2\pi i\mathrm dz$ to calculate periods (I'm assuming that the weight is $2$ here, for simplicity).

Deligne claims that the transposes ${}^tT_n$ of the Hecke operators act on $\omega_f$ by the eigenvalues $a_n$, so that $\omega_f$ represents a class in the de Rham realization of the motive $M(f^*)$ attached to the dual cusp form.

Why is this true? Let ${}^N_kW$ be the motive defined by Scholl having $M(f)$ as a submotive. If we view $\omega_f$ as a class in $\operatorname{fil}^1{}^N_kW_{\mathrm{dR}}\otimes\mathbb C\cong S_k(X(N),\mathbb C)$, then the complex comparison isomorphism to ${}^N_kW_{\mathrm B}\otimes\mathbb C$ (which is the Eichler-Shimura isomorphism) sends $\omega_f$ to the usual cocycle given by integrating $\omega_f$ along geodesics on the upper half plane, which corresponds to $f$ (and not $f^*$!). Since the Eichler-Shimura isomorphism is Hecke equivariant, $\omega_f$ should lie in the eigenspace for $f$ and not $f^*$, i.e. it should be just $f$ under the identification $\operatorname{fil}^1{}^N_kW_{\mathrm{dR}}\otimes\mathbb C\cong S_k(X(N),\mathbb C)$.

EDIT: This was my original question.

I have a question concerning a detail in Tony Scholl's construction of motives attached to modular forms.

To my understanding, the construction of Deligne's Galois representation attached to modular forms works as follows. Let $Y_1(N)$ be the modular curve over $\mathbb Q$ classifying elliptic curves with a point of exact order $N$, and let $f\colon E_1(N)\rightarrow Y_1(N)$ be the universal elliptic curve over it. Then one looks at the étale cohomology group $${}_k^NW_\ell:=H^1_{\mathrm p}(Y_1(N),\operatorname{Sym}^{k-2}R^1f_*\mathbb Q_\ell).$$ From the Hecke correspondences on $Y_1(N)$ one gets a Hecke action on this $\mathbb Q_\ell$-vector space, and using the (Eichler-Shimura) congruence relation and the "corrected scalar product", or "twisted Poincaré duality" (involving the Atkin-Lehner involution) one can show that $$\det(1-F_pX,{}_k^NW_\ell)=1-T_p+p^{k-1}\langle p \rangle X^2$$ where $F_p$ is a geometric Frobenius at $p$. Hence to get Deligne's Galois representation attached to a fixed newform $f$ of weight $k$ and level $N$ with Fourier coefficients in a number field $K$, one just tensors ${}_k^NW_\ell$ over the Hecke algebra $T$ with the morphism $T\rightarrow K_\lambda$ induced by $f$, where $\lambda$ is a prime of $K$ lying above $\ell$ (or equivalently, one projects to the eigenspace of ${}_k^nW_\ell\otimes_{\mathbb Q_\ell}K_\lambda$ where the Hecke operators act by the eigenvalues given by the Fourier coefficients of $f$).

Now in the article "Motives for modular forms" Scholl defines a motive over $\mathbb Q$ called ${}_k^NW$ whose $\ell$-adic étale realization is precisely ${}^N_kW_\ell$. He then shows that the Hecke correspondences induce endomorphisms of this motive and that on the Betti realization these endomorphisms are the same as the ones defined by Deligne (Prop. 4.1.1). Hence also on the $\ell$-adic realization these are the same as Deligne's.

But then Scholl projects to the eigenspace where the transposes ${}^tT_p$ of the Hecke operators act by the eigenvalues given by $f$ to get his motive attached to $f$. He claims (4.2.2) that on the $\ell$-adic realization we get precisely Deligne's Galois representation attached to $f$. To me this seems to contradict the above, it should rather give the representation attached to the "dual" modular form $f^*$. Also Scholl writes that on the space of cusp forms (which is the intermediate step $\operatorname{fil}^1{}^N_kW_{\mathrm{dR}}=\operatorname{fil}^{k-1}{}^N_kW_{\mathrm{dR}}$ in the Hodge filtration on the de Rham realization of ${}^N_kW$) the endomorphisms of ${}^N_kW$ given by the Hecke correspondences induce the tranposes of the usual Hecke operators. But I do not see why this statement is true. If on the Betti realization they are the same as Deligne's, then by the Eichler-Shimura isomorphism one should see that the are the usual ones on the space of cusp forms... (?)

• The motive and Deligne representation indeed coincides. See this question I asked some time ago mathoverflow.net/questions/198453/… – user40276 Aug 10 '16 at 21:44
• Well, I think this is the whole point of Scholl's construction – to find a motive underlying Deligne's representation. My question is why his construction – with the normalization of Hecke eigenspaces as above – does this, and unfortunately in the question you linked this point is not explained... – Michael Fütterer Aug 11 '16 at 12:00

You can define Deligne's representation $V_f$ associated to $f$ either a subspace of the \'etale cohomology group $W_{\ell}$ or as a quotient. If you choose to define it as a subspace, then $V_f$ is indeed the subspace on which the usual Hecke operators $T_p$ for $p\nmid N \ell$ acts as multiplication by $a_p(f)$, as you were expecting.
If you prefer to see $V_f$ as a quotient, note that there is a Galois-equivariant perfect pairing $$H^1(Y,Sym^{k-2}(R^1f_*\mathbb{Q}_{\ell})) \times H^1(Y,Sym^{k-2}(R^1f_*\mathbb{Q}_{\ell})^\vee) \rightarrow \mathbb{Q}_{\ell}(1)$$ where $(1)$ denotes the Tate twist by the cyclotomic character.
This means that in order to obtain the same $V_f$ as above, it must arise as a quotient of the dual of $H^1(Y,Sym^{k-2}(R^1f_*\mathbb{Q}_{\ell}))$, which the above pairing shows that may be identified with $H^1(Y,Sym^{k-2}(R^1f_*\mathbb{Q}_{\ell})^\vee)(1)$. Under this isomorphism, the Hecke operator $T_p$ on the left-hand side is identified with the adjoint Hecke operator $T_p^*$ on the right-hand side.
This explains why the definition of $V_f$ you encounter in Deligne coincides with the one we described above as a subspace.