Mixing
Lefschetz fixed point formula for Frobenius and automorphisms of finite order
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In Proposition 3.3 of the paper "Reductive groups over finite fields" by Deligne and Lusztig, the authors mention the fact that if $X$ is a quasi-projective scheme over $overline{F_p}$, $sigma$ is an automorphism of finite order acting on it, both $X$ and $sigma$ can be defined over $F_q$, Let $F$ be the corresponding geometric Frobenius, then for $ngeq 1$, the composite $F^{n}circsigma$ is the Forbenius map relative to some new way of lowering the field of definition of $X$ from $overline{F_p}$ to $F_{q^n}$ and the Lefschetz fixed point formula for Frobenius shows that $Tr((F^{n}sigma)^{*},H_{c}^{*}(X,Q_l))$ is the number of fixed points of $F^{n}sigma$.
I think this means that there exists some twists of $X$ over $F_{q^n}$ such that when we restrict $F^{n}sigma$ onto it we get $F^n$. I would like to know whether my understanding is correct and why such twists exist. Any comments or references will be appreciated.
algebraic-geometry
asked Jan 1 at 22:16
Sailun ZhanSailun Zhan
62
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add a comment |
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In Proposition 3.3 of the paper "Reductive groups over finite fields" by Deligne and Lusztig, the authors mention the fact that if $X$ is a quasi-projective scheme over $overline{F_p}$, $sigma$ is an automorphism of finite order acting on it, both $X$ and $sigma$ can be defined over $F_q$, Let $F$ be the corresponding geometric Frobenius, then for $ngeq 1$, the composite $F^{n}circsigma$ is the Forbenius map relative to some new way of lowering the field of definition of $X$ from $overline{F_p}$ to $F_{q^n}$ and the Lefschetz fixed point formula for Frobenius shows that $Tr((F^{n}sigma)^{*},H_{c}^{*}(X,Q_l))$ is the number of fixed points of $F^{n}sigma$.
I think this means that there exists some twists of $X$ over $F_{q^n}$ such that when we restrict $F^{n}sigma$ onto it we get $F^n$. I would like to know whether my understanding is correct and why such twists exist. Any comments or references will be appreciated.
algebraic-geometry
asked Jan 1 at 22:16
Sailun ZhanSailun Zhan
62
$endgroup$
add a comment |
$begingroup$
In Proposition 3.3 of the paper "Reductive groups over finite fields" by Deligne and Lusztig, the authors mention the fact that if $X$ is a quasi-projective scheme over $overline{F_p}$, $sigma$ is an automorphism of finite order acting on it, both $X$ and $sigma$ can be defined over $F_q$, Let $F$ be the corresponding geometric Frobenius, then for $ngeq 1$, the composite $F^{n}circsigma$ is the Forbenius map relative to some new way of lowering the field of definition of $X$ from $overline{F_p}$ to $F_{q^n}$ and the Lefschetz fixed point formula for Frobenius shows that $Tr((F^{n}sigma)^{*},H_{c}^{*}(X,Q_l))$ is the number of fixed points of $F^{n}sigma$.
I think this means that there exists some twists of $X$ over $F_{q^n}$ such that when we restrict $F^{n}sigma$ onto it we get $F^n$. I would like to know whether my understanding is correct and why such twists exist. Any comments or references will be appreciated.
algebraic-geometry
asked Jan 1 at 22:16
Sailun ZhanSailun Zhan
62
$endgroup$
In Proposition 3.3 of the paper "Reductive groups over finite fields" by Deligne and Lusztig, the authors mention the fact that if $X$ is a quasi-projective scheme over $overline{F_p}$, $sigma$ is an automorphism of finite order acting on it, both $X$ and $sigma$ can be defined over $F_q$, Let $F$ be the corresponding geometric Frobenius, then for $ngeq 1$, the composite $F^{n}circsigma$ is the Forbenius map relative to some new way of lowering the field of definition of $X$ from $overline{F_p}$ to $F_{q^n}$ and the Lefschetz fixed point formula for Frobenius shows that $Tr((F^{n}sigma)^{*},H_{c}^{*}(X,Q_l))$ is the number of fixed points of $F^{n}sigma$.
I think this means that there exists some twists of $X$ over $F_{q^n}$ such that when we restrict $F^{n}sigma$ onto it we get $F^n$. I would like to know whether my understanding is correct and why such twists exist. Any comments or references will be appreciated.
algebraic-geometry
algebraic-geometry
asked Jan 1 at 22:16
Sailun ZhanSailun Zhan
62
asked Jan 1 at 22:16
Sailun ZhanSailun Zhan
62
asked Jan 1 at 22:16
Sailun ZhanSailun Zhan
62
asked Jan 1 at 22:16
Sailun ZhanSailun Zhan
62
asked Jan 1 at 22:16
Sailun ZhanSailun Zhan
62
62
add a comment |
add a comment |
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