I'm a beginner to the theory of modular forms trying to understand a certain construction from the point of view of elliptic curves. Let $f(q) = \sum a_n q^n$ be a formal power series. Define $\theta f = q f'(q) = \sum n a_nq^n$. If $f(q)$ is the $q$-expansion of a modular form, it is not true that $\theta f$ is a modular form. However, it is "almost" a modular form: over the complex numbers, it preserves the ring obtained by adding the weight two Eisenstein series $E_2$ to the ring of modular forms $\mathbb{C}[E_4, E_6]$.
If one works $p$-adically, then $E_2$ is the $q$-expansion of a weight two $p$-adic modular form (as I learned from Serre's article "Formes modulaires et fonctions zeta $p$-adiques"), and in fact, by Theorem 5, sec. 2 of the paper, $\theta$ becomes an operator carrying $p$-adic modular forms of weight $k$ to $p$-adic modular forms of weight $k+2$.
The ring of all $p$-adic modular forms has an interpretation of Katz as functions on the following moduli problem: given a $p$-adically complete ring $R$, consider the collection of all elliptic curves over $R$ together with an isomorphism of their formal group with $\widehat{\mathbb{G_m}}$. (There is an action of $\mathbb{Z}_p^{\times}$ on this moduli problem, and the decomposition of functions on it into characters of $\mathbb{Z}_p^{\times}$ is the weight decomposition of $p$-adic modular forms.) Is there a way to interpret $\theta$ in terms of this moduli problem?
