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.