Let $\mathcal{E}$ be the universal elliptic curve over the moduli stack $\mathcal{M}$ of elliptic curves. As $\mathcal{E}$ is an abelian group scheme over $\mathcal{M}$, we obtain a product-preserving functor $\mathrm{Lat} \to \mathrm{Sch}_{/\mathcal{M}}$ from the category of finitely generated abelian groups to that of relative scheme over $\mathcal{M}$, sending $L$ to $\mathrm{Hom}(L, \mathcal{E})$, i.e. $\mathbb{Z}^n$ to $\mathcal{E}^{\times_{\mathcal{M}}n}$.
As discussed in Deligne and Rapoport's classic paper, we can compactify the moduli stack of elliptic curves to get $\overline{\mathcal{M}}$ and we have a generalized elliptic curve $\overline{\mathcal{E}}$. As this is no longer a group scheme (I believe), we cannot have a product-preserving functor $\mathrm{Lat} \to \mathrm{Sch}_{/\overline{\mathcal{M}}}$, sending $\mathbb{Z}$ to $\overline{\mathcal{E}}$, as this would precisely give the abelian group structure. Nevertheless, there might be a non-product preserving functor.
Is there a functor $F\colon \mathrm{Lat} \to \mathrm{Sch}_{/\overline{\mathcal{M}}}$ such that $F(L)$ is proper over $\overline{\mathcal{M}}$ and has (naturally) $\mathrm{Hom}(L, \mathcal{E})$ as an open dense substack?
