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Fix $N\ge4$. Let $Y_1(N)$ and $X_1(N)$ be the usual modular curves. I want to view them as schemes over $\mathbb Z$ representing the moduli functors of (usual or generalized) elliptic curves with (Drinfeld) $\Gamma_1(N)$-structures. That they exist in this form is shown in Brian Conrad's paper "Arithmetic moduli of generalized elliptic curves". Let $f\colon E_1(N)\rightarrow Y_1(N)$ be the universal elliptic curve.

Let further $Y_1(N,p)$ and $X_1(N,p)$ denote the moduli spaces for $\Gamma_1(N,p)$ structures used in the definition of Hecke correspondences, and denote the universal elliptic curve over $Y_1(N,p)$ by $E_1(N,p)$. Consider the diagram $$\begin{array}{ccccccccc} & & E_1(N,p) & & \xrightarrow{\phi} & & E_1(N,p)/C & & \\ & \swarrow && \searrow & & \swarrow & & \searrow &\\ E_1(N) & & & & Y_1(N,p) & & & & E_1(N)\\ & \searrow & & \swarrow{\pi_1} & & \searrow{\pi_2} & & \swarrow & \\ & & Y_1(N) & & & & Y_1(N) & & \end{array} $$ where $\pi_1$,$\pi_2$ are the usual degeneracy maps, $\phi$ is the universal $p$-isogeny, and the two squares are cartesian.

This diagram can be used to describe Hecke operators on (Eichler-Shimura type) cohomology groups. More precisely, if we denote analytifications by a superscript "an", then the Hecke operator $T_p$ on the cohomology group $H^1(Y_1(N)^{\mathrm{an}},\operatorname{Sym}^k R^1f_*\mathbb Z)$ (or similar groups) equals $$T_p=\operatorname{tr}_{\pi_1}\circ\phi^*\circ\pi_2^*. $$ This is written e. g. in Delignes paper "Formes modulaires et représentations l-adiques" (Prop. 3.18 (i)). One has a similar relation also in étale cohomology.

In the above-mentioned article "Arithmetic moduli of generalized elliptic curves" (§4.5), Brian Conrad does a similar thing on modular forms themselves, so he considers the cohomology group $H^0(X_1(N),\omega^{\otimes k})$. However, in his case he gets $$pT_p=\operatorname{tr}_{\pi_1}\circ\phi^*\circ\pi_2^*. $$

I am surprised that one has to divide by $p$ in the second case to get the classical Hecke operator from the composition $\operatorname{tr}_{\pi_1}\circ\phi^*\circ\pi_2^*$, while in the first case one doesn't. What is the reason behind this?

Any other references on this topic are welcome.

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This question is somehow a "characteristic 0" question, so let me treat $Y = Y_1(N)$ and $X = X_1(N)$ as $\mathbf{Q}$-varieties rather than doing anything complicated with integral models.

There's an isomorphism of sheaves on $X$, the Kodaira-Spencer map, $$\omega^2 \to \Omega^1_{X / \mathbf{Q}}(C)$$ where $C = X - Y$ is the divisor of cusps. So we have an isomorphism $$ H^0(X, \omega^k) = H^0(X, \omega^{k-2} \otimes \Omega^1(C)).$$ However, this isomorphism is not Hecke equivariant for the obvious actions of Hecke operators on both sides: you pick up a factor of $p$ in the Hecke operator $T_p$. Concretely, a section of $\omega^k$ looks like $f(\tau) (\mathrm{d}z)^k$, and a section of $\omega^{k-2} \otimes \Omega^1$ looks like $f(\tau) (\mathrm{d}z)^{k-2} \mathrm{d}\tau$, where $\tau$ is a coordinate on the upper half-plane, and $z$ is a coordinate on the universal elliptic curve; the action of $T_p$ involves pullback via a map which looks locally like $(\tau, z) \mapsto (\tau/p, z)$, and hence there's an extra factor of $p$ coming from $\mathrm{d}\tau$ which $\omega^k$ doesn't see.

So this is why you have to modify the definition of $T_p$ on $H^0(X, \omega^k)$: it is in order to get it to match with the natural action on $H^0(X, \omega^{k-2} \otimes \Omega^1)$. The latter is somehow the "correct" action, since $H^0(X, \omega^{k-2} \otimes \Omega^1(C))$ is canonically a subspace of $H^1_{\mathrm{dR}}(Y_1(N), \operatorname{Sym}^{k-2} R^1f_*\mathbf{Q})$ and will compare well with etale, Betti, etc cohomology.

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  • $\begingroup$ thanks for your answer, which looks very convincing (I haven't yet checked the details, but I think I will be able to work them out). Anyway, this is an interesting subtlety I was not aware of. $\endgroup$ – Michael Fütterer Jul 8 '16 at 22:14

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