In
number theory and
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 Tate twist,
['The Tate Twist', https://ncatlab.org/nlab/show/Tate+twist ] named after
John Tate John Tate may refer to:
* John Tate (mathematician) (1925–2019), American mathematician
* John Torrence Tate Sr. (1889–1950), American physicist
* John Tate (Australian politician) (1895–1977)
* John Tate (actor) (1915–1979), Australian act ...
, is an operation on
Galois modules.
For example, if ''K'' is a
field, ''G
K'' is its
absolute Galois group, and ρ : ''G
K'' → Aut
Q''p''(''V'') is a
representation
Representation may refer to:
Law and politics
*Representation (politics), political activities undertaken by elected representatives, as well as other theories
** Representative democracy, type of democracy in which elected officials represent a ...
of ''G
K'' on a finite-dimensional
vector space ''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 ''V''⊗Q
''p''(1), where Q
''p''(1) is the
''p''-adic cyclotomic character (i.e. the
Tate module of the group of
roots of unity in the separable closure ''K
s'' of ''K''). More generally, if ''m'' is a
positive integer
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
, 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 of Q
''p''(1), the ''-m''th Tate twist of ''V'' can be defined as
:
References
{{DEFAULTSORT:Tate Twist
Number theory
Algebraic geometry