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Let $k$ be a field with $char(k)\neq p$. Take a representation of the absolute Galois group $G=Gal(k_{alg}/k)$ and a continuous representation $V$ of $G$, with coefficients in, say, $\mathbb{Z}_p$.

Now if $\mathbb{Z}_p(1)$ the Tate module of $p$th-power roots of unity in $k_{alg}$, one usually denotes by $V(1)$ the representation $V\otimes_{\mathbb{Z}_p}\mathbb{Z}_p(1)$ and set $V(-1)=Hom_{\mathbb{Z}_p}(\mathbb{Z}_p(1),V)$.

I get these definitions but i would like some intuition about the reason we consider these twists. What are the main reasons we consider them? What are the main basic applications of these? (i don't mean here deep geometric conjectures but mainly the reasons they were introduced in the first place).

user654981
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  • Do you know étale cohomology ? Or the Tate module of an abelian variety ? (I am not talking about deep geometric conjectures there, there is nothing deep and nothing conjectural in fact) – Roland Dec 02 '19 at 20:33
  • I know only a bit of it and it would be interesting to have some relations betweens these of course. Yet did Tate twists first appear in this setting? It is funny – user654981 Dec 03 '19 at 00:05
  • Have you seen https://math.stackexchange.com/questions/2923709/about-the-definition-of-l-adic-tate-twist? – Mathmo123 Dec 03 '19 at 09:50
  • See https://math.stackexchange.com/questions/57750/what-is-the-intuition-behind-the-concept-of-tate-twists – Pol van Hoften Jan 03 '20 at 15:07

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