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David Loeffler
David Loeffler
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Suppose $X$ is a variety over $\mathbb{Q}$ and it has a Galois twist $X_1$$X'$, i.e. $X$ is isomorphic to $X_1$$X'$ over $\overline{\mathbb{Q}}$. The Galois twistset of isomorphism classes of twists of $X$ is classified by $H^1(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}), \text{Aut}(X_{\overline{\mathbb{Q}}}))$. 

If $X_1$$X'$ corresponds to a cycleclass $c_1$ of $H^1(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}), \text{Aut}(X_{\overline{\mathbb{Q}}}))$$c \in H^1(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}), \text{Aut}(X_{\overline{\mathbb{Q}}}))$, what is the relation between the Galois representation obtained fromrepresentations $H^q_{et}(X,\mathbb{Q}_{\ell})$$H^q_{et}(X_{\overline{\mathbb{Q}}},\mathbb{Q}_{\ell})$ and $H^q_{et}(X_1,\mathbb{Q}_{\ell})$$H^q_{et}(X'_{\overline{\mathbb{Q}}},\mathbb{Q}_{\ell})$?

Suppose $X$ is a variety over $\mathbb{Q}$ and it has a Galois twist $X_1$, i.e. $X$ is isomorphic to $X_1$ over $\overline{\mathbb{Q}}$. The Galois twist of $X$ is classified by $H^1(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}), \text{Aut}(X_{\overline{\mathbb{Q}}}))$. If $X_1$ corresponds to a cycle $c_1$ of $H^1(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}), \text{Aut}(X_{\overline{\mathbb{Q}}}))$, what is the relation between the Galois representation obtained from $H^q_{et}(X,\mathbb{Q}_{\ell})$ and $H^q_{et}(X_1,\mathbb{Q}_{\ell})$?

Suppose $X$ is a variety over $\mathbb{Q}$ and it has a Galois twist $X'$, i.e. $X$ is isomorphic to $X'$ over $\overline{\mathbb{Q}}$. The set of isomorphism classes of twists of $X$ is classified by $H^1(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}), \text{Aut}(X_{\overline{\mathbb{Q}}}))$. 

If $X'$ corresponds to a class $c \in H^1(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}), \text{Aut}(X_{\overline{\mathbb{Q}}}))$, what is the relation between the Galois representations $H^q_{et}(X_{\overline{\mathbb{Q}}},\mathbb{Q}_{\ell})$ and $H^q_{et}(X'_{\overline{\mathbb{Q}}},\mathbb{Q}_{\ell})$?

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Wenzhe
Wenzhe
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Galois twist of a variety

Suppose $X$ is a variety over $\mathbb{Q}$ and it has a Galois twist $X_1$, i.e. $X$ is isomorphic to $X_1$ over $\overline{\mathbb{Q}}$. The Galois twist of $X$ is classified by $H^1(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}), \text{Aut}(X_{\overline{\mathbb{Q}}}))$. If $X_1$ corresponds to a cycle $c_1$ of $H^1(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}), \text{Aut}(X_{\overline{\mathbb{Q}}}))$, what is the relation between the Galois representation obtained from $H^q_{et}(X,\mathbb{Q}_{\ell})$ and $H^q_{et}(X_1,\mathbb{Q}_{\ell})$?