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

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 (Dri …