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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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I think you should state their theorem here in addition to giving a link. –  Felipe Voloch May 7 '12 at 12:54
Toy model (i.e. GL_1 and "infinitesimal") for Narasimhan-Seshadri theorem is Hodge theory. As far as I understand this step is a major problem to generalize from C to function fields and moreover to alg.numbers. I guess if this would be clear, then proper exponentiating and non-abelianization would be not so difficult as the first step... See also: mathoverflow.net/questions/90928/… mathoverflow.net/questions/85943/… –  Alexander Chervov May 7 '12 at 15:41
Let us look on the "abelian" NS-theorem: the moduli space of line bundles (which we know to be the Jacobian I.e. torus of real dimension 2g) is identified as topological manifold (not as algebraic) with unitary 1-dimensional irreps of fundamental group (which is also 2g-dim torus, obviously ). Is there an analogue for this in alg.num setup? It smells like class field theory... J –  Alexander Chervov May 7 '12 at 19:25
Let us go a little further analysing "abelian" version. Consider "infinitesimal" versions (i.e. tangent spaces on the both sides). Consider "holomorphic side". Moduli space of line bundles is H^1(O^*), so the tangent space is H^1(O), by Serre's duality it is H^0(K) - holomorphic differentials. Consider the "unitary side": the moduli space of unitary 1-dimensional representations of \pi_1. It is the same as H^1(\pi_1, S^1), so the tangent space is H^1(\pi_1, R), the same as H^1(X,R), where X is our Riemann surface. Continued... –  Alexander Chervov May 8 '12 at 7:34
You might look at the articles of Deninger and Werner: wwwmath.uni-muenster.de/u/deninger/about/publikat/cd49.ps –  Jason Starr May 8 '12 at 12:52
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1 Answer

Theorem of Narasimhan and Seshadri is a special case of what Carlos Simpson calls nonabelian Hodge theory developed by Hitchin and Simpson. This theory was generalized to the characteristic $p$ case in the paper of Ogus and Vologodsky, Nonabelian Hodge Theory in Characteristic p. Hope this helps.

Update: Below, on Alexander's request, is a brief explanation of relation between nonabelian Hodge theory and NS theorem. Consider vector bundles with vanishing 1st and 2nd Chern classes (I will call this condition ($\star$)), then the story in higher dimensions is exactly the same as for complex curves. The detailed explanation is in the pages 12-19 of Simpson's 1992 paper [S1992]. From there, it follows that flat unitary connections correspond exactly to vanishing Higgs fields (subject to ($\star$)). Briefly, every semistable Higgs bundle $E=(V, \bar\partial, \theta)$ has a hermitian YM metric $K$. Define the connection $D_K$ (as in [S1992], page 13), then then $D_K$ is flat (subject to ($\star$), page 17 of [S1992]).

If Higgs field $\theta$ vanishes then $D_K=\partial_K+ \bar\partial$ and, hence, by definition of $\partial_K$, connection $D_K$ preserves the metric $K$. Thus, our bundle reduces to a flat unitary bundle. Conversely, if bundle is flat unitary (with unitary metric denoted $K$) then the associated (multivalued) map $\Phi_K$ defined on page 16 of [S1992], is constant, so it has zero derivative. But its derivative is $0=d\Phi_K=\theta+ \bar\theta$ (here $\theta$ is the Higgs field determined by $K$). Since $\theta, \bar\theta$ have different types, the only way we can have $\theta+ \bar\theta=0$ is that $\theta=0$.

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G. Faltings "A p-adic Simpson correspondence" imperium.lenin.ru/~kaledin/math/falt.pdf , see also mathoverflow.net/questions/21100/… , see also Deninger-Werner e.g. arxiv.org/abs/math/0403516 (as Jason Starr commented above). Small survey of surrounding works: sfb45.de/graduate-school/Mainz.pdf –  Alexander Chervov May 9 '12 at 5:57
I am not great expert, but "NS is SPECIAL CASE of Simpson correspondence" seems to me too strong claim, although they are certainly related... Simposon's correspondence is for manifolds of arbitrary dimensions and takes irreps of pi_1 to GL to pairs vector bunle "F" + Higgs Field "Phi". While in NS you have 1-dimensional manifolds and instead of GL you have U(n) and there is NO Higgs field Phi. However the for 1-dimensional manifolds NS and Simposon (which was done by Hitchin before Simpson's N-dim case) are COMPATIBLE ... continued ... –  Alexander Chervov May 9 '12 at 6:04
Continued ... In the sense that if we take Simposon's map with zero Phi we will get U(n) irrep of \pi_1. (If I remember correctly), however it seems to me it will require additional work to prove this COMPATIBILITY starting from Hitchin-Simpson... –  Alexander Chervov May 9 '12 at 6:07
@Alexander: Nonabelian Hodge theory (in the context of Riemann surfaces) contains Narasimhan-Seshadri theorem as a special case: $\Phi=0$ iff the representation of the fundamental group has pre-compact image, in the case of interest, conjugate to a unitary representation, which is exactly NS theorem. You are right, however, in the sense that this was not done in the characteristic $p$ case. –  Misha May 9 '12 at 13:54
@Misha yes, it is the same that I wrote in the last comment, however it seems to this will require some addtional work, comparing with Simpson correspondence, am I wrong ? By the way what will happen in higher dims if $ \Phi =0$ ? –  Alexander Chervov May 9 '12 at 19:57
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