In number theory and algebraic geometry, the **Tate twist**,^{[1]} named after John Tate, is an operation on Galois modules.

For example, if *K* is a field, *G _{K}* is its absolute Galois group, and ρ :

*G*→ Aut

_{K}_{Qp}(

*V*) is a representation of

*G*on a finite-dimensional vector space

_{K}*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*of

^{s}*K*). More generally, if

*m*is a positive integer, the

**, denoted**

*m*th Tate twist of*V**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

\( {\displaystyle V\otimes \mathbf {Q} _{p}(-1)^{\otimes m}.} \)

References

'The Tate Twist', in Lecture Notes in Mathematics', Vol 1604, 1995, Springer, Berlin p.98-102

Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

Hellenica World - Scientific Library

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