Let $X/\mathbb{C}$ be a smooth projective curve of genus $g>0$ (here $\mathbb{C}$ is any algebraically closed field, say of characteristic $0$). Let $S$ be a finite set of closed points of X and let $\operatorname{Pic}^{level,S}(X)$ be the (commutative group) scheme (of infinite type) parametrizing line bundles on $X$ with trivialization in a formal neighborhood of $S$. Is it true that every (global) function on $\operatorname{Pic}^{level,S}(X)$ is constant on connected components?


Here's an essentially equivalent problem, which is perhaps a little more concrete. Let $x\in X$ be a closed point and let $\operatorname{Jac}^{n\cdot x}(X)$ be the (commutative group) scheme of finite type parametrizing degree $0$ line bundles on $X$ with trivialization up to $n$th order at $x$ (a generalized Jacobian in the sense of Serre et al). For $n\geq 1$, $\operatorname{Jac}^{(n+1)\cdot x}(X)$ is a $\mathbb{G}\_a$-bundle over $\operatorname{Jac}^{n\cdot x}(X)$. Is this bundle non-trivial?



Global functions on generalized Jacobians
The assumption that $g>0$ is justified by the fact that the answer is clearly no in genus zero: in fact, if $C=\mathbb{P}^1$ and $S=\{\infty\}$, then the connected component of $\mathrm{Pic}^{\mathrm{level},\infty}(\mathbb{P}^1)$ is the affine group scheme $U$ with $U(\mathbb{C})=(1+u\,\mathbb{C}[[u]],\times)$ (where $u$ is a local coordinate at $\infty$). In particular it has lots of global functions. 


Now, for arbitrary $C$, choose a finite morphism $f:C\to \mathbb{P}^1$ which is étale above $\infty$, and take $S=f^{-1}(\infty)$. It is then easy to define a norm map $f\_\*:P:=\mathrm{Pic}^{\mathrm{level},S}(C)\to \mathrm{Pic}^{\mathrm{level},\infty}(\mathbb{P}^1)$ such that $f\_{\*}\,f^{\*}=$ multiplication by $\deg(f)$. In particular the induced morphism $P^\circ\to U$ is surjective, hence by composition $P^\circ$ has nonconstant global functions.


