mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Serre group ''S'' is the pro-algebraic group whose representations correspond to CM-motives over the algebraic closure of the rationals, or to polarizable rational Hodge structures with abelian Mumford–Tate groups. It is a

projective limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can ...

of finite dimensional tori, so in particular is abelian. It was introduced by . It is a subgroup of the

Taniyama group In mathematics, the Taniyama group is a group that is an extension of the absolute Galois group of the rationals by the Serre group. It was introduced by using an observation by Deligne, and named after Yutaka Taniyama. It was intended to be the g ...

. There are two different but related groups called the Serre group, one the connected component of the identity in the other. This article is mainly about the connected group, usually called the Serre group but sometimes called the connected Serre group. In addition one can define Serre groups of

algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...

s, and the Serre group is the inverse limit of the Serre groups of number fields.

Definition

The Serre group is the projective limit of the Serre groups of ''S''_''L'' of finite Galois extensions of the rationals, and each of these groups ''S''_''L'' is a torus, so is determined by its module of characters, a finite free Z-module with an action of the finite Galois group Gal(''L''/Q). If ''L''* is the algebraic group with ''L''*(''A'') the units of ''A''⊗''L'', then ''L''* is a torus with the same dimension as ''L'', and its characters can be identified with integral functions on Gal(''L''/Q). The Serre group ''S''_''L'' is a quotient of this torus ''L''*, so can be described explicitly in terms of the module ''X''*(''S''_''L'') of rational characters. This module of rational characters can be identified with the integral functions λ on Gal(''L''/Q) such that :(σ−1)(ι+1)λ = (ι+1)(σ−1)λ = 0 for all σ in Gal(''L''/Q), where ι is complex conjugation. It is acted on by the Galois group. The full Serre group ''S'' can be described similarly in terms of its module ''X''*(''S'') of rational characters. This module of rational characters can be identified with the locally constant integral functions λ on Gal(/Q) such that :(σ−1)(ι+1)λ = (ι+1)(σ−1)λ = 0 for all σ in Gal(/Q), where ι is complex conjugation.

References

* *{{citation, mr=0263823 , last=Serre, first= Jean-Pierre , title=Abelian l-adic representations and elliptic curves. , series=McGill University lecture notes , publisher= W. A. Benjamin, Inc. , place=New York-Amsterdam, year= 1968 Algebraic groups Langlands program