algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, the Mumford–Tate group (or Hodge group) ''MT''(''F'') constructed from a

Hodge structure In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structure ...

''F'' is a certain

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...

''G''. When ''F'' is given by a

rational representation In mathematics, in the representation theory of algebraic groups, a linear representation of an algebraic group is said to be rational if, viewed as a map from the group to the general linear group, it is a rational map In mathematics, in particu ...

of an

algebraic torus In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by \mathbf G_, \mathbb_m, or \mathbb, is a type of commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher ...

, the definition of ''G'' is as the

Zariski closure In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is no ...

of the image in the representation of the

circle group In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \ ...

, over the

rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...

. introduced Mumford–Tate groups over the complex numbers under the name of Hodge groups. introduced the ''p''-adic analogue of Mumford's construction for

Hodge–Tate module In mathematics, a Hodge–Tate module is an analogue of a Hodge structure over p-adic fields. introduced and named Hodge–Tate structures using the results of on p-divisible groups. Definition Suppose that ''G'' is the absolute Galois group o ...

s, using the work of on p-divisible groups, and named them Mumford–Tate groups.

Formulation

The algebraic torus ''T'' used to describe Hodge structures has a concrete matrix representation, as the 2×2 invertible matrices of the shape that is given by the action of ''a''+''bi'' on the basis of the complex numbers C over R: :

\begin a & b \\ -b & a \end.

The circle group inside this group of matrices is the

unitary group In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is an ...

''U''(1). Hodge structures arising in geometry, for example on the

cohomology group In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...

s of

Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnold ...

s, have a

lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...

consisting of the integral cohomology classes. Not quite so much is needed for the definition of the Mumford–Tate group, but it does assume that the vector space ''V'' underlying the Hodge structure has a given rational structure, i.e. is given over the rational numbers ''Q''. For the purposes of the theory the complex vector space ''V''_''C'', obtained by extending the scalars of ''V'' from ''Q'' to ''C'', is used. The weight ''k'' of the Hodge structure describes the action of the diagonal matrices of ''T'', and ''V'' is supposed therefore to be homogeneous of weight ''k'', under that action. Under the action of the full group ''V''_''C'' breaks up into subspaces ''V''^''pq'', complex conjugate in pairs under switching ''p'' and ''q''. Thinking of the matrix in terms of the complex number λ it represents, ''V''^''pq'' has the action of λ by the ''p''th power and of the complex conjugate of λ by the ''q''th power. Here necessarily :''p'' + ''q'' = ''k''. In more abstract terms, the torus ''T'' underlying the matrix group is the Weil restriction of the

multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred to ...

''GL''(1), from the complex field to the real field, an algebraic torus whose character group consists of the two homomorphisms to ''GL''(1), interchanged by complex conjugation. Once formulated in this fashion, the rational representation ρ of ''T'' on ''V'' setting up the Hodge structure ''F'' determines the image ρ(''U''(1)) in ''GL''(''V''_''C''); and ''MT''(''F'') is by definition the smallest algebraic group defined over ''Q'' containing this image.

Mumford–Tate conjecture

The original context for the formulation of the group in question was the question of the

Galois representation In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring ...

on the

Tate module In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group ''A''. Often, this construction is made in the following situation: ''G'' is a commutative group scheme over a field ''K'', ...

of an

abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...

''A''. Conjecturally, the image of such a Galois representation, which is an l-adic Lie group for a given prime number ''l'', is determined by the corresponding Mumford–Tate group ''G'' (coming from the Hodge structure on ''H''¹(''A'')), to the extent that knowledge of ''G'' determines the

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

of the Galois image. This conjecture is known only in particular cases. Through generalisations of this conjecture, the Mumford–Tate group has been connected to the

motivic Galois group In algebraic geometry, motives (or sometimes motifs, following French usage) is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham coho ...

, and, for example, the general issue of extending the Sato–Tate conjecture (now a theorem).

Period conjecture

A related conjecture on abelian varieties states that the

period matrix In mathematics, in the field of algebraic geometry, the period mapping relates families of Kähler manifolds to families of Hodge structures. Ehresmann's theorem Let be a holomorphic submersive morphism. For a point ''b'' of ''B'', we deno ...

of ''A'' over number field has

transcendence degree In abstract algebra, the transcendence degree of a field extension ''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of ...

, in the sense of the field generated by its entries, predicted by the dimension of its Mumford–Tate group, as in the previous section. Work of

Pierre Deligne Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...

has shown that the dimension bounds the transcendence degree; so that the Mumford–Tate group catches sufficiently many algebraic relations between the periods. This is a special case of the full Grothendieck period conjecture.https://arxiv.org/abs/0805.2569v1, p. 7.

Notes

References

* * *

External links

Lecture slides (PDF)
by

Phillip Griffiths Phillip Augustus Griffiths IV (born October 18, 1938) is an American mathematician, known for his work in the field of geometry, and in particular for the complex manifold approach to algebraic geometry. He was a major developer in particul ...

''Mumford-Tate groups, families of Calabi-Yau varieties and analogue André-Oort problems I'', preprint (PDF)
{{DEFAULTSORT:Mumford-Tate group Hodge theory Algebraic groups