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Actually, I am a bit confused about the notation and what is called Frobenius for modular curves. Let $N\geq 4$ be an integer, let $R$ be an $\mathbb{F}_p$-algebra, where $p$ is a prime not dividing $N$, and let $E\rightarrow X$ to be the universal elliptic curve over the modular curve $X$, which solves the moduli problem of representing elliptic curves over $R$-algebras with $N$-level structure. It is well known that, since we are working in characteristic $p$, there exists for every elliptic curve $E$ over an $R$-algebra $S$, the Frobenius morphism relative to $S$. Moreover, this defines a morphism of the modular curve, which is usually defined on the universal elliptic curve again by sending $E$ to the quotient over the kernel of its relative Frobenius, which can be done since we know that Frobenius is an isogeny. Moreover, since $X$ lives in characteristic $p$, it admits his own relative Frobenius over $R$, which is a morphism $X\rightarrow X^{(p)}$. Is there any relation between the two? I mean, is it true that the relative Frobenius defined universally over the modular curve coincides with the relative Frobenius defined in the usual way?

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