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.



Why is there a factor $p$ in the definition of $T_p$ via Hecke correspondences on modular curves?
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. 


