# Frobenius for modular curves.

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?