Serre group

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Serre group S is a 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. Because it is a projective limit of finite-dimensional tori, S is abelian. It was introduced by Serre in 1968 and is a subgroup of the Taniyama group. There are two related objects called the Serre group: the connected component of the identity (the version usually meant by “the Serre group”) and a larger, disconnected version. This article focuses on the connected Serre group.

Serre groups can also be defined for algebraic number fields; the Serre group over Q is the inverse limit of the Serre groups over all number fields. Equivalently, S is the projective limit of the Serre groups attached to the tori coming from finite Galois extensions L of Q. For each such L, the associated torus L* has dimension [L:Q], and its rational characters form a finite free Z-module with an action of Gal(L/Q). The Serre torus SL is a quotient of L*, so it is described by its module X*(SL) of rational characters.

X*(SL) can be realized as integral functions on Gal(L/Q) that are compatible with complex conjugation and carry the natural Galois action. The full Serre group S is described in the same way by its character module X*(S). Concretely, X*(S) can be viewed as locally constant integer-valued functions on the full Galois group Gal(Q̄/Q) that satisfy the complex-conjugation relation.