Home > algebraic geometry, number theory > Adèlic groups and principal bundles

## Adèlic groups and principal bundles

Today we will establish a result which, in some form, goes back to Weil: the “uniformization” by the idèles of the Picard group of a curve. More generally, the moduli of $G$-bundles on a curve can be uniformized by the adèlic points of $G$ (here $G$ is a connected smooth affine algebraic group satisfying some additional hypotheses which which hold, for instance, if the ground field is $\mathbb{C}$ or $G = \text{GL}_n$), so taking the multiplicative group $G = \mathbb{G}_m$ we recover the previous sentence. This fact is remarkable because it connects the geometry of principal bundles on our curve with the arithmetic of its ring of adèles. There is even a “stacky” version of the theorem, which we will not treat here, but can be found in a slightly altered form as Theorem 5.1.1 in Sorger’s Lectures on moduli of principal $G$-bundles over algebraic curves.

First let us define the geometric objects appearing in the theorem, which are of basic interest in algebraic geometry and representation theory. Fix a ground field $k$, let $X$ be a $k$-scheme, and $G$ an affine algebraic group over $k$, meaning an affine group scheme of finite type over $k$.

Definition A (principal) $G$-bundle on $X$ is a scheme $E$ over $X$ equipped with a right action of $G$ which, locally in the étale topology, is isomorphic to the trivial bundle $X \times G \to X$.

Morphisms of principal bundles are simply $G$-equivariant $X$-morphisms, which are automatically isomorphisms by local triviality.

There are two basic constructions we perform with $G$-bundles. If $f : Y \to X$ is a morphism, then we can form the pullback bundle $f^*E = Y \times_X E$, which becomes a $G$-bundle when equipped with the obvious right action. Less obviously, for any scheme $Z$ with a left action of $G$, there is an associated bundle $E_Z = (E \times Z)/G$, where $E \times Z$ has the diagonal action

$(e,z) \cdot g = (e \cdot g,g^{-1} \cdot z)$.

One observes $E_Z$ is a scheme over $X$ which is isomorphic to $Z \times X \to X$ locally in the étale topology.

The latter construction allows us to produce a vector bundle on $X$ from a linear representation of $G$. It is a basic fact that, when applied to the standard representation of $\text{GL}_n$, this establishes an equivalence between $\text{GL}_n$-bundles and rank $n$ vector bundles. When $n = 1$, which is the case we are interested in for the purposes of class field theory, the inverse is easy to describe: just remove the image of the zero section from a line bundle to obtain a $\mathbb{G}_m$-bundle.

If $p \in X(S)$ is an $S$-point of $X$, then by a trivialization of $E$ at $p$ we mean an isomorphism of $p^*E$ with the trivial $G$-bundle on $S$, or equivalently a section of the projection $p^*E \to S$. It is not hard to see that the group of automorphisms of the trivial $G$-bundle on $S$ is $G(S)$, and in particular $G(S)$ acts simply transitively on the set of trivializations of $E$ at $p$, provided that this set is nonempty.

Next, we explain the construction of the adèles. For this let $X$ be a smooth connected curve over $k$, and denote its field of rational functions by $K$. For each closed point $x \in X$, write $\mathcal{O}_x$ for the completed local ring of $X$ at $x$ (we will not use the non-completed local rings) with maximal ideal $\mathfrak{m}_x \subset \mathcal{O}_x$, residue field $k_x = \mathcal{O}_x/\mathfrak{m}_x$, and fraction field $K_x$. Alternatively, $K_x$ can be viewed as the completion of $K$ with respect to the $x$-adic absolute value. Recall that if we choose a local coordinate $\epsilon \in \mathfrak{m}_x \setminus \mathfrak{m}_x^2$ (a number theorist would say “uniformizer” or “prime element”) we get an isomorphism $\mathcal{O}_x \cong k_x[[\epsilon]]$ and hence $K_x \cong k_x((\epsilon))$.

Definition For any finite set $S \subset X$ of closed points, the $S$-adèles are the topological ring

$\mathbb{A}_S = \prod_{x \notin S} \mathcal{O}_x \times \prod_{x \in S} K_x$.

The full ring of adèles is the union $\mathbb{A} = \bigcup_S \mathbb{A}_S$ over all finite $S \subset X$, endowed with the colimit topology. The integral adèles are just $\mathbb{O} = \mathbb{A}_{\varnothing} = \prod_{x \in X} \mathcal{O}_x$.

Although the topology on $\mathbb{A}$ is important for many purposes, especially when $k$ is finite, we will not need it in what follows.

Now we begin to interpret these arithmetic constructions geometrically. Write $D_x = \text{Spec } \mathcal{O}_x$ and $D_x^o = \text{Spec } K_x$, which we view as the disk and punctured disk centered at $x$, respectively. If $G$ is an affine algebraic group over $k$ as before, then $G(D_x^o) = G(K_x)$ is analoguous to a topological loop group, and sometimes this terminology is still employed in algebraic geometry. In our global situation, the group of adèlic points $G(\mathbb{A})$ packages together all the loop groups $G(D_x^o)$ as $x$ varies through the closed points of $X$. It contains $G(K)$ and $G(\mathbb{O})$ as subgroups, which in particular act on $G(\mathbb{A})$ by translations. We declare that by convention $G(K)$ acts on the left and $G(\mathbb{O})$ acts on the right.

Definition Let $E$ be a $G$-bundle on $X$ and $x \in X$ a closed point. A (full) level structure on $E$ at $x$ is a trivialization of $E$ at the canonical morphism $D_x \to X$.

The reason for the parenthetical terminology “full” is that if $N \subset X$ is a finite subscheme (for example, an $n^{\text{th}}$-order neighborhood of $x$, of which $D_x$ is the colimit as $n$ varies) then a trivialization of $E$ at $N$ is called a structure of level $N$. We will not need the latter notion in this post.

Finally we come to the promised result. The proof is actually quite instructive, which is one reason we chose to state it in this generality. Recall our standing assumption that $X$ is a smooth connected curve over $k$.

Theorem (Uniformization) Let $G$ be a smooth affine algebraic group over $k$. Then there is a canonical bijection from $G(\mathbb{A})$ to the set of isomorphism classes of $G$-bundles on $X$ equipped with a generic trivialization (i.e. a trivialization at the generic point $\eta : \text{Spec } K \to X$) and full level structures at all closed points $x \in X$. Moreover, this bijection is equivariant for $G(K)$ and $G(\mathbb{O})$, which act on generic trivializations and full level structures respectively.

Proof. If we are given a $G$-bundle $E$ equipped with a generic level structure and full level structures at all closed points of $X$, we can construct an adèlic point of $G$ as follows. First observe that the given generic trivialization $\zeta : \text{Spec } K \times G \ \tilde{\to} \ \eta^*E$ “spreads out” to a trivialization $U \times G \ \tilde{\to} \ E|_U$ over a nonempty open set $U \subset X$ (this is where we use the hypothesis that $G$ is of finite type over $k$). Now the desired adèlic point is a Cech cocycle representative for $E$ with respect to the fpqc cover of $X$ consisting of all the formal disks $D_x$ together with $U$. More precisely, restricting the full level structures and the generic trivialization gives two trivializations of $E$ on every punctured disk $D_x^o$, and the difference of these two trivializations is an automorphism of the trivial $G$-bundle on $D_x^o$, or equivalently a point $g_x \in G(D_x^o)$ of the loop group (we should really be careful about the order we take this difference, but let’s avoid writing lots of formulas). Notice that $g_x \in G(D_x)$ for all $x \in U$, so as $x$ varies the $g_x$ give a well-defined element of $G(\mathbb{A})$.

Conversely, given an adèlic point $g = (g_x) \in G(\mathbb{A})$, we continue to think of it as a Cech cocycle and construct the desired $G$-bundle by “gluing,” or rather fpqc descent. Let $S \subset X$ be the finite set of closed points such that $g_x \notin G(D_x)$ and write $U = X \setminus S$, so that $U$ and the $D_x$ for $x \in S$ form an fpqc cover of $X$. Then the $g_x \in G(D_x^o)$ for $x \in S$ can be viewed as transition maps along which we glue the trivial bundles on the $D_x$ to the trivial bundle on $U$. The resulting $G$-bundle $E$ is a priori only locally trivial in the fpqc topology, but then the étale local triviality follows by smoothness of $G$. Of course $E$ comes equipped with a trivialization over $U$, hence a generic trivialization, as well as full level structures at each $x \in S$. Now restricting the trivialization of $E$ over $U$ to $D_x$ for each $x \in U$ gives a full level structure there, but the level structure we really want is obtained by composing this one with $g_x$, thought of as an automorphism of the trivial bundle on $D_x$.

One can inspect the constructions to see that they are mutually inverse and equivariant for $G(K)$ and $G(\mathbb{O})$.

$\Box$

Our final result, which is the punchline of this post, is an immediate consequence of the theorem.

Corollary If every $G$-bundle on $X$ admits a generic trivialization, then the bijection from the theorem descends to a $G(\mathbb{O})$-equivariant bijection from $G(K) \backslash G(\mathbb{A})$ to the set of isomorphism classes of $G$-bundles on $X$ equipped with full level structures at every closed point $x \in X$. If, in addition, every $G$-bundle on $X$ admits a trivialization at $\text{Spec } k(x) \to X$ for each closed point $x \in X$, then we obtain a bijection from $G(K) \backslash G(\mathbb{A})/G(\mathbb{O})$ to the set of isomorphism classes of $G$-bundles on $X$.

Proof. The only part that is not obvious is that if a $G$-bundle $E$ on $X$ trivializes over $\text{Spec } k(x)$, then it trivializes over $D_x$. But a trivialization is the same as a section, so by smoothness of $G$ this follows from Hensel’s lemma.

$\Box$

We conclude by pointing out when $G = \mathbb{G}_m$ is the multiplicative group, both sides of the final bijection have a group structure, and we actually obtain an isomorphism of groups $\mathbb{A}^{\times}/K^{\times}\mathbb{O}^{\times} \ \tilde{\to} \ \text{Pic}(X)$. The only additional content in this assertion is that tensoring line bundles corresponds to multiplying representative $\mathbb{G}_m$-valued Cech cocycles.

Next time, we’ll get into some specifics of the situation where $k$ is finite, and in particular explain how the hypotheses of the corollary then hold quite often. We will probably not use the results in this post very much moving forward, but they are important for establishing the connection between our approach and harmonic analysis on adèlic groups.