number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

and

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, the Tate twist, 'The Tate Twist', https://ncatlab.org/nlab/show/Tate+twist named after John Tate, is an operation on Galois modules. For example, if ''K'' is a field, ''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'' ...

, 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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

''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 V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...

''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 mathematics, a root of unity 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 number theory, the theory of group char ...

in the separable closure ''K^s'' of ''K''). More generally, if ''m'' is a

positive integer In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...

, 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 :

V\otimes\mathbf_p(-1)^{\otimes m}.

References

Number theory Algebraic geometry