In his famous 1940 letter from prison in Rouen to his sister Simone, André Weil talks about the analogy between number fields and functions fields (in one variable) over finite fields, and the analogy between these functions fields and functions fields over $\mathbf{C}$ (or equivalently compact connected curves over $\mathbf{C}$). This letter is reproduced in his *Scientific papers* and has been recently translated into English (*Notices of the AMS* **52**(3) 2005).

**Question** What is the number field analogue of the Narasimhan-Seshadri theorem (*Stable and unitary vector bundles on a compact Riemann surface*. Ann. of Math. (2) **82** 1965 540–567) ?

**Addendum** (in response to Felipe's comment) The original paper of Narasimhan and Seshadri is available on JSTOR. An excerpt from their introduction : D. Mumford has defined the notion of a stable vector bundle on a compact
Riemann surface $X$ and proved that the set of equivalence classes of stable
bundles (of fixed rank and degree) has a natural structure of a non-singular,
quasi-projective, algebraic variety [13]. We prove in this paper that, if $X$ has
genus $\ge2$, the stable vector bundles are precisely the holomorphic vector bundles
on $X$ which arise from certain *irreducible unitary* representations of suitably defined fuchsian groups acting on the unit disc and having $X$ as quotient
(Theorem 2, $\S12$).
[...]
A particular case of our result is that a holomorphic vector bundle of
degree zero on $X$ is stable if and only if it arises from an irreducible unitary
representation of the fundamental group of $X$. As a consequence one sees that
a holomorphic vector bundle on $X$ arises from a unitary representation of the
fundamental group of $X$ if and only if each of its indecomposable components
is of degree zero and stable.

Their main result is summarised by Atiyah (MR0170350) and Le Potier (Séminaire Bourbaki Exposé 737) as follows :

**Atiyah:** Let $X$ be a compact Riemann surface. If $W$ is a (holomorphic) vector bundle of rank $n$ over $X$ we define $d(W)$ to be the degree of the associated line bundle $\bigwedge^n W$. A bundle $W$ is stable, in the sense of Mumford, if $(\mathrm{rank}W)d(V)<(\mathrm{rank}V)d(W)$ for all proper sub-bundles $V$ of $W$. According to Mumford [Proc. Internat. Congr. Mathematicians (Stockholm, 1962), pp. 526--530, Inst. Mittag-Leffler, Djursholm, 1963], the set of isomorphism classes of stable bundles of rank $n$ and degree $q$ over $X$ has a natural structure of an algebraic variety. In this paper the authors give a complete characterization of stable bundles in terms of unitary representations of a certain discrete group (provided genus $X$ $≥2$).

Their main theorem runs as follows. Given integers n and q, with $-n< q \le0$, we can choose (i) a discrete group $\pi$ acting on a simply connected Riemann surface $Y$ with $Y/\pi=X$ and with the map $p:Y\to X$ being ramified over only one point $x_0\in X$; (ii) a representation $\tau:\pi_{y_0}→\mathrm{GL}(n,\mathbf{C})$ of the isotropy group of $\pi$ at a point $y_0\in p^{−1}(x_0)$ by scalars such that the following holds. A vector bundle over $X$ of rank $n$ and degree $q$ is stable if and only if the corresponding sheaf is isomorphic to a sheaf of the form $p_∗^\pi(\mathbf{V})$, where $\mathbf{V}$ denotes the $\pi$-sheaf of holomorphic mappings $Y\to V$, $V$ is an irreducible unitary representation of $\pi$ coinciding with $\tau$ when restricted to $\pi_{y_0}$, $p_∗$ is the direct image functor and $p_∗^\pi$ denotes the subsheaf invariant under $\pi$. Moreover, two such stable bundles are isomorphic on $X$ if and only if the corresponding unitary representations of $\pi$ are equivalent.

It should be observed that the inequality $-n< q \le0$ presents no essential restriction since it can always be realized by tensoring with a line bundle $L$ and, on the other hand, the definition of stable bundle shows that $W$ is stable if and only if $W\otimes L$ is stable.

**Le Potier:** En 1965 Narasimhan et Seshadri établissaient une correspondence bijective entre l’ensemble des classes d’équivalence de représentations unitaires irréductibles du groupe fondamentale $\pi$ d’une surface de Riemann compacte $X$, et l’ensemble des classes d’isomorphisme de fibrés vectoriels stables de degré $0$ sur $X$ : ils associent à une representation $\rho:\pi\to\mathbf{U}(r)$ le fibré vectoriel holomorphe $E_\rho$ défini par
$$
E_\rho=\tilde X\times_\pi\mathbf{C}^r
$$
où $\tilde X$ est le revêtement universel de $X$, et où le produit ci-dessus est le quotient de $\tilde X\times\mathbf{C}^r$ par l’action de $\pi$ définie par
$(\gamma,(x,v))\mapsto(x\gamma^{-1},\gamma v)$ pour $\gamma\in\pi$ et $(x,v)\in \tilde X\times\mathbf{C}^r$.

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