See section 3.3 (and appendix A) of the recent PhD thesis of Brandon Levin (at his IAS webpage) for any reductive group $G$ (with connected fibers) over a Dedekind domain $A$; this fleshes out details sketched in notes of Gaitsgory and Faltings' paper "Algebraic loop groups...". Your case is $A=k[\![t]\!]$ for a field $k$, for which arguments simplify; when $G$ arises over $k\subset k[\![t]\!]$ then additional simplifications occur. A choice of $G\hookrightarrow {\rm{GL}}(V)$ is used to build the ample line bundle. (NB. You cannot "include" $\mathcal{G}r$ in a projective space; it is too big.)

This is a general fact over any field whatsoever, and it is proved as 20.6 in Borel's textbook on linear algebraic groups.

Dear Daniel Litt: Thanks for tracking it down; that is indeed the right paper. It is also worth noting (in view of the question posed) that this paper provides a sense in which mere normalization in higher level (which doesn't have positive-dimensional fibers) is the "wrong" thing to consider, suggesting that perhaps the question posed about normalization is not a "useful" point of view for applications with $g > 1$.

Note that $A_g$ classifies principally polarized abelian schemes (of relative dimension $g$); the polarization aspect comes along for free uniquely when $g=1$ but not otherwise, as you undoubtedly know. I was once told that there is a paper by Chai and Norman (title I do not know) which shows in some sense that there isn't a good version of Drinfeld's idea beyond relative dimension 1. Perhaps email your question to Chai (or Kai-Wen Lan) if you don't receive a satisfactory answer here.

It is perhaps also worth noting that there is a distinct concept called "$A$-pure" for modules over $A$-algebras (under suitable finiteness hypotheses) for a general ring $A$ in the 1971 Inventiones paper of Raynaud and Gruson on flatness criteria (see 3.3.3 of part I of that paper).

Let's first recall what "pure" means: injective after tensoring against any module over the source ring. So a ring map with a section, or more generally one that acquires a section after a faithfully flat base change, is pure. Such maps are often not flat, since there is no condition on the augmentation ideal.

By the way, for a literature reference on the Lie algebra of ${\rm{R}}_{A/k}(G_A)$ for affine algebraic $k$-groups $G$ (or even ${\rm{R}}_{A/k}(\mathcal{G})$ for affine algebraic $A$-groups $\mathcal{G}$), see the more-or-less self-contained section A.7 (esp. Cor. A.7.6) in the book "Pseudo-reductive groups".

Oops, that was silly of me to overlook! Here is an idea over fields $k$ of characteristic 0. Let $H := {\rm{Spec}}(U(\mathfrak{g}_A))$ for a $k$-finite $A$. We seek an isomorphism of affine algebraic $k$-groups $H\simeq{\rm{R}}_{A/k}(G_A)$ extending the "identity" on Lie algebras. This is unique if it exists (since the target is Zariski-connected). Using an inclusion of algebraic groups $G\hookrightarrow{\rm{SL}}_n$, perhaps we can reduce to the case of ${\rm{SL}}_n$ via Lie algebra considerations (at the cost of directly proving $U(\mathfrak{g}_A)$ is a domain with "expected dimension").

For any field $k$, $k$-finite $A$, and affine algebraic $k$-group scheme $G$, ${\rm{Lie}}({\rm{R}}_{A/k}(G_A))$ is the underlying Lie algebra over $k$ of $\mathfrak{g}_A$. But if we regard $\mathfrak{g}_A$ as a Lie algebra over $k$ then $U(\mathfrak{g}_A)$ only "knows" $A$ through its underlying $k$-vector space, so it is $U(\mathfrak{g}^{\oplus n}) = U(\mathfrak{g})^{\otimes n}$ (with $n=\dim_k A$) and thus recovers $G^n$ in your setup. In contrast, the group structure of ${\rm{R}}_{A/k}(G_A)(B)=G(A\otimes_k B)$ involves the $k$-algebra structure of $A$; compare $A=k\times k, k[x]/(x^2)$.

Oh, OK, so you're taking the Lie algebra $\mathfrak{g}_A$ over $A$ and viewing it instead just as a Lie algebra over $\mathbf{C}$ for the formation of the universal enveloping algebra (so it isn't an $A$-algebra, etc.). Then your question makes sense, though for $\mathbf{C}$-finite $A$ do you know if the Weil restriction ${\rm{R}}_{A/\mathbf{C}}(G_A)$ has coordinate ring equal to that enveloping algebra? (I would be surprised if this is true, but maybe there is a simple trick.)

Dear Chuck: Your $\epsilon$ defines a homomorphism from $U:=U(\mathfrak{g}_A)$ into $A$, not into $\mathbf{C}$; what does $\epsilon$ do to the natural image of $A$ in $U$? Likewise, your $\Delta$ is a map from the $A$-algebra $U$ into $U\otimes_A U$, not into $U \otimes_{\mathbf{C}}U$; if you think $\Delta$ is a map into the latter then where do $aX$ and $a$ go under this map for $a \in A$? For representability when $A=\mathbf{C}[t]$, if $G=\mathbf{G}_a$ then this amounts to a "universal polynomial" in $t$; considering nilpotent coefficients rules it out (EGA IV$_3$, 8.14.2 makes it rigorous).

A (co-commutatie) Hopf algebra $R$ over $\mathbf{C}$ is equipped with an "identity section" $R \rightarrow \mathbf{C}$ as $\mathbf{C}$-algebras and a co-multiplication $R \rightarrow R \otimes_{\mathbf{C}} R$ as $\mathbf{C}$-algebras. For the Hopf algebra $U(\mathfrak{g}_A)$ over $A$, what such data do you have in mind? Also, your proposed functor fails to be representable for $A=\mathbf{C}[t]$ and $G$ any linear algebraic $\mathbf{C}$-group with positive dimension. (Weil restriction through an algebra map not of finite dimension arises for "affine Grassmannians", which are not representable.)

At a very elementary level, there are special algebraicity properties for GL$_1$ over CM fields, due to a theorem of Artin and Weil. See the recent PhD thesis of Stefan Patrikis for an extension of these algebraicity considerations to more general reductive groups.

The degeneracy map $X_0(N)\rightarrow X_0(p)$ over $\mathbf{Q}$ induces $J_0(N)\twoheadrightarrow J_0(p)$ over $\mathbf{Q}$ by Albanese functoriality. Toric reduction for $J_0(p)$ at $p$ is a consequence of the determination of the minimal regular proper model of $X_0(p)$ over $\mathbf{Z}_{(p)}$, due to the relationship (proved by Raynaud) between the Neron model of a Jacobian and the relative Picard scheme of the minimal regular proper model of a curve over a dvr, combined with the description of the special fiber of that relative Pic. See Ch. 9 of "Neron Models", especially 9.2/8 and 9.7.

Suppose $X_0(N)$ has genus $>0$ ($N=11,14,15,17$ or $N>18$ except $N=25$). If $p|N$ with $p>7$ but $p\ne 13$ then $J_0(N)$ has the quotient $J_0(p)$ with toric reduction at $p$, so CM over $\overline{\mathbf{Q}}$ is ruled out by Kestutis' argument. Thus, you just have to handle $N$ whose prime factors are in $\{2,3,5,7,13\}$ with $N$ not in the genus-0 list. For $N = 27, 32, 49$, $X_0(N)$ is an elliptic curve with geometric CM by $\mathbf{Q}(\zeta_3)$, $\mathbf{Q}(i)$, $\mathbf{Q}(\sqrt{-7})$ respectively. I am sure that others on MO with more computational expertise can address the rest.

OK, good to see what I was overlooking.

The proof in your link is also flawed; why should $v$ there have anything to do with infinitesimal automorphisms of $X$ if it does not arise from the tangent space to the automorphism scheme (but just that of the closure of its image)?

How do you know that the map ${\rm{Aut}}_{X/k} \rightarrow {\rm{PGL}}(\Gamma(X,\omega^{\otimes n})^{\vee})$ has closed image (e.g., why is it finite type?)? It is certainly not sufficient that it has trivial kernel. One can make plenty of maps from a locally finite type $k$-group to a finite type $k$-group with trivial kernel.