Tate twist explained

In number theory and algebraic geometry, the Tate twist,^{[1] [2]} 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_K → Aut_Qp(V) is a representation 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, the mth 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 -mth Tate twist of V can be defined as

V ⊗ Q ^{⊗ m} _{p( -1 )} .

References

1. 'The Tate Twist', in Lecture Notes in Mathematics', Vol 1604, 1995, Springer, Berlin p.98-102
2. 'The Tate Twist', https://ncatlab.org/nlab/show/Tate+twist