number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...

and algebraic geometry, the Tate twist, 'The Tate Twist', https://ncatlab.org/nlab/show/Tate+twist named after John Tate, is an operation on

Galois module 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 ...

s. For example, if ''K'' is a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

, ''G_K'' is its

absolute Galois group In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' t ...

, and ρ : ''G_K'' → Aut_{Q_''p''}(''V'') is a representation of ''G_K'' on a finite-dimensional

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

''V'' over the field Q_''p'' of ''p''-adic numbers, then the Tate twist of ''V'', denoted ''V''(1), is the representation on the

tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...

''V''⊗Q_''p''(1), where Q_''p''(1) is the ''p''-adic cyclotomic character (i.e. 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 the group of

roots of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...

in the separable closure ''K^s'' of ''K''). More generally, if ''m'' is a positive integer, the ''m''th Tate twist of ''V'', denoted ''V''(''m''), is the tensor product of ''V'' with the ''m''-fold tensor product of Q_''p''(1). Denoting by Q_''p''(−1) the

dual representation In mathematics, if is a group and is a linear representation of it on the vector space , then the dual representation is defined over the dual vector space as follows: : is the transpose of , that is, = for all . The dual representation ...

of Q_''p''(1), the ''-m''th Tate twist of ''V'' can be defined as :

V\otimes\mathbf_p(-1)^.

References

{{DEFAULTSORT:Tate Twist Number theory Algebraic geometry