Gross and Zagier prove the following fantastic result in their paper "Singular Moduli":

Let $R$ be a discrete valuation ring over $\mathbb Z_p$ with uniformizer $\pi$ such that $k = R/\pi$ is algebraically closed and normalize the valuation so that $v(\pi) = 1$.

Now, let $E_1,E_2$ be two distinct (ie, non isomorphic) elliptic curves over $R$ with complex multiplication with $j$-invariants $j_i = j(E_i)$. These are algebraic integers in $R$ and we can talk about their reduction mod $\pi$. Define $$i(n) = \frac{|\operatorname{Isom}_{R/\pi^n}(E_1,E_2)|}{2}.$$

Then Gross-Zagier show that: $$v(j_1-j_2) = \sum_{n\geq 1}i(n).$$

Unfortunately, their proof is a very explicit case by case analysis (depending on both $n,p$) of both sides of the equation.

This proof is very unsatisfying to me because of how uniform the statement of the result is (it doesn't depend on p or the Elliptic curves, it doesn't even distinguish between the curves with extra automorphisms and those without).

However, I suspect that there should be a very nice uniform proof using maybe a moduli space (stack..?) of Elliptic curves. For instance, we easily see that $v(j_1-j_2) \geq 1 \iff i(1) \geq 1$ because over an algebraically closed field the $j$-invariant determines the isomorphism class.

Does anyone know such a uniform proof?

(The paper I am referring to is here, the theorem is 2.3 on page 196.)