# Global functions on generalized Jacobians

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?

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I don't know the answer, but I'll point out that this is the sort of situation where you may get different answers in the analytic and algebraic categories. Note that the top two answers to mathoverflow.net/questions/68421 feature $\mathbb{G}_a$ bundles over elliptic curves. –  David Speyer Jul 12 '11 at 17:53
Woops! I'll add explicitly that I want algebraic functions. –  Moosbrugger Jul 12 '11 at 18:08

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.