Weil group

Weil group. In algebraic number theory, the Weil group is a fundamental construction that refines the structure of the absolute Galois group of a local field or global field. Introduced by André Weil in the context of class field theory, it serves as a key technical device for formulating the Artin reciprocity law in a unified manner. The group encapsulates the arithmetic of the field by incorporating data from its maximal unramified extension, providing a more tractable object than the full Galois group for classifying abelian extensions.

Definition and construction

For a local field \( K \) such as the p-adic numbers \( \mathbb{Q}_p \), the Weil group \( W_K \) is constructed as a dense subgroup of the absolute Galois group \( G_K = \text{Gal}(\bar{K}/K) \), where \( \bar{K} \) denotes a separable closure. Its definition hinges on the choice of a Frobenius element in the Galois group of the maximal unramified extension \( K^{\text{ur}}/K \). Specifically, one takes the preimage of the cyclic subgroup generated by this Frobenius under the natural restriction map from \( G_K \) to \( \text{Gal}(K^{\text{ur}}/K) \), which is isomorphic to the profinite completion of the integers \( \hat{\mathbb{Z}} \). This yields a topological group extension \( 1 \to I_K \to W_K \to \mathbb{Z} \to 1 \), where \( I_K \) is the inertia subgroup and the quotient is the infinite cyclic group. The topology on \( W_K \) is not the subspace topology from \( G_K \); instead, it is given such that the inertia subgroup remains open. For a global field \( F \), such as an algebraic number field like \( \mathbb{Q} \), the Weil group \( W_F \) is defined via the restricted direct product of the local Weil groups \( W_{F_v} \) over all places \( v \), subject to certain cohomological conditions linking the local and global structures.

Properties

The Weil group possesses several distinctive properties that make it central to modern number theory. It is a locally compact topological group, unlike the profinite absolute Galois group, and its abelianization \( W_K^{\text{ab}} \) is canonically isomorphic to the multiplicative group \( K^\times \) via the local Artin map in the local case. This isomorphism is a reformulation of the main theorem of local class field theory. For global fields, the abelianization of the Weil group relates to the idele class group \( C_F \), encapsulating the global Artin reciprocity law. Furthermore, the Weil group fits into the framework of a class formation, providing a natural setting for Tate's cohomology and the definition of fundamental classes. Its cohomological dimension is finite, and it satisfies a version of Poitou–Tate duality. The representation theory of the Weil group, particularly its continuous finite-dimensional complex representations, is intimately connected to the study of Artin L-functions and automorphic forms.

Examples

The simplest non-trivial example arises from the local field \( \mathbb{Q}_p \). Here, the Weil group \( W_{\mathbb{Q}_p} \) is an extension of \( \mathbb{Z} \) by the inertia subgroup \( I_{\mathbb{Q}_p} \), which is itself a profinite group. A continuous homomorphism from \( W_{\mathbb{Q}_p} \) to \( \mathbb{C}^\times \) corresponds to a character of \( \mathbb{Q}_p^\times \), classifying one-dimensional representations. For the real number field \( \mathbb{R} \), the Weil group \( W_{\mathbb{R}} \) is isomorphic to \( \mathbb{C}^\times \cup j\mathbb{C}^\times \), a non-abelian group of order two extending the complex numbers, which classifies representations via the Langlands correspondence for \( \text{GL}(1) \). In the global setting, for \( F = \mathbb{Q} \), the Weil group \( W_\mathbb{Q} \) is an intricate object whose structure is related to the decomposition groups at each prime, including the Archimedean place represented by the complex conjugation element.

Relation to class field theory

The Weil group provides a unifying language for both local class field theory and global class field theory. The canonical isomorphism \( W_K^{\text{ab}} \cong K^\times \) for a local field \( K \) is the essence of the local reciprocity law, originally formulated by Emil Artin. Globally, the continuous homomorphism from the Weil group \( W_F \) to the idele class group \( C_F \) with dense image realizes the global Artin map, whose kernel is the connected component of the identity. This reformulation bypasses some technicalities of the classical approach using ideles and enables a more functorial treatment. The theory of Langlands correspondence for \( \text{GL}(n) \) generalizes this by relating n-dimensional representations of the Weil group to automorphic representations, with the case \( n=1 \) recovering class field theory. The work of Robert Langlands and Pierre Deligne further established the Weil–Deligne group, which incorporates monodromy operators for handling the inertia subgroup in the study of l-adic representations.

Generalizations

Several profound generalizations of the Weil group have been developed to address deeper arithmetic questions. The most significant is the Weil–Deligne group, introduced by Pierre Deligne, which augments the Weil group with a nilpotent operator to better classify l-adic representations of Galois groups, particularly those arising from étale cohomology of algebraic varieties. This group is central to the formulation of the local Langlands correspondence for reductive groups over local fields. Another direction is the concept of the absolute Weil group of a scheme, which generalizes the construction to a geometric context. In the theory of motives, conjectural objects like the motivic Galois group are expected to play a role analogous to the Weil group. Furthermore, the Langlands group or the conjectural automorphic Galois group is a proposed extension of the Weil group that would fully encapsulate the non-abelian aspects of the Langlands program, linking representations to automorphic forms on arbitrary reductive groups.