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Geometric class field theory

Today, after a long absence from the blogosphere, I’m starting a series of posts on geometric class field theory. My goal is to make the presentation so geometrical that it is easily comprehensible to readers with backgrounds in algebraic geometry but not number theory. Of course, the story is enriched by the analogy with number fields, and I will frequently draw attention to this analogy, but it will be unnecessary for both the statements and the proofs of the main results.

The main character is a smooth, projective, and (geometrically) connected curve X over a field k, which we will generally assume is either a finite field \mathbb{F}_q or the complex numbers \mathbb{C}. Very broadly speaking, the goal is to understand all “covers” of X, by which we mean finite separable maps Y \to X where Y is another curve over k, but this is far too ambitious for us. We will focus our attention on abelian covers, which are the connected Galois covers whose automorphism group is abelian (recall that a connected cover is called Galois if its automorphism group acts transitively on the geometric generic fiber, or equivalently has cardinality equal to the degree of the cover). Then there is a correspondence involving moduli of line bundles on X, as we will explain at length. When k = \mathbb{F}_q, abelian covers correspond to finite-index subgroups of the Picard group of X (with level structure in the ramified case).

This is very much like the number theorist’s goal of understanding (abelian) extensions of a number field. Indeed, we are doing the same for the field of rational functions on X. The case k = \mathbb{F}_q was developed classically along the same lines as arithmetic class field theory, and to my knowledge it was Deligne who first gave a purely geometric proof in the sixties.

Here is a more precise outline of the plan. Our short-term goal will be to prove the main theorem of class field theory in the unramified setting. After that, we will move on to local class field theory, which we will approach using a geometric version of Lubin-Tate theory. The natural next step is to return to and prove the the general ramified case of global class field theory. Along the way we will explain how, in the case k = \mathbb{F}_q, the basic correspondence can be realized using moduli of shtukas on X, and how this relates to Drinfeld modules and explicit class field theory. Finally, in the distant future we might say some words about the higher rank case, which is the geometric Langlands correspondence for \text{GL}_n, and especially Drinfeld’s proof of the case n = 2 in positive characteristic.

So that this post is not entirely devoid of content, let’s go ahead and state the main theorem of unramified global geometric class field theory when k = \mathbb{F}_q (the case k = \mathbb{C} is slightly harder to formulate, but we’ll get to it). Next time we’ll give (some) definitions and explain how our statement relates to more classical formulations, and probably move on to the proof two posts from now.

We will denote by \pi_1(X) the étale fundamental group of X based at the geometric generic point and \pi_1(X)^{\text{ab}} its abelianization (as a profinite group). The structure morphism X \to \text{Spec } \mathbb{F}_q induces a homomorphism \pi_1(X) \to \widehat{\mathbb{Z}}, and we write W_X (respectively W_X^{\text{ab}}) for the Weil group of X, i.e. the preimage of \mathbb{Z} in \pi_1(X) (respectively \pi_1(X)^{\text{ab}}). It is not hard to see that the Weil group is a dense discrete subgroup of \pi_1(X). Any closed point x : \text{Spec } \mathbb{F}_{q^d} \to X induces a map \mathbb{Z} \to W_X, well-defined up to conjugation, and the image of 1 is a conjugacy class in W_X called the (arithmetic) Frobenius at x, which we denote by \text{Fr}_x. In particular, \text{Fr}_x maps to a single element of W_X^{\text{ab}}, which we also denote by \text{Fr}_x.

The other object which appears in the theorem is the Picard group \text{Pic}_X(\mathbb{F}_q) of isomorphism classes of line bundles on X under tensor product. As the notation suggests, the Picard group is the group of rational points of the Picard group scheme \text{Pic}_X, which will be relevant later. For now, just observe that \text{Pic}_X(\mathbb{F}_q) is generated by the line bundles \mathcal{O}(x) as x varies through the closed points of X. Now we can state the theorem.

Theorem (Unramified global class field theory) There is a unique map \text{Pic}_X(\mathbb{F}_q) \to \pi_1(X)^{\text{ab}} which sends \mathcal{O}(x) \mapsto \text{Fr}_x for each closed point x \in X. This map induces an isomorphism \text{Pic}_X(\mathbb{F}_q) \cong W_X^{\text{ab}}.

Note that the isomorphism \text{Pic}_X(\mathbb{F}_q) \cong W_X^{\text{ab}} intertwines the degree map \text{Pic}_X(\mathbb{F}_q) \to \mathbb{Z} with the natural map W_X^{\text{ab}} \to \mathbb{Z}. This is because if x \in X is a degree d point, then \mathcal{O}(x) is a degree d line bundle and \text{Fr}_x induces the automorphism a \mapsto a^{q^d} on \overline{\mathbb{F}}_q.

The uniqueness in the theorem is obvious, since the line bundles \mathcal{O}(x) generate the Picard group. But the existence of this map is already a highly nontrivial statement: this says that if \sum n_ix_i is a principal divisor on X, then \prod \text{Fr}_{x_i}^{n_i} is trivial in \pi_1(X)^{\text{ab}}. This is an example of a reciprocity law in the sense of arithmetic class field theory.

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