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The standard conjectures imply directly that the category of motives over the finite field $\mathbb{F}_q$ is a polarizable (hence semisimple) Tannakian category. Using only that, we have the following result.

PROPOSITION: Let $X$ be a motive of weight $m$ over $\mathbb{F}_q$, and let $\alpha\mapsto\alpha^t$ be the involution of $\mathrm{End}(X)$ defined by a Weil form $\varphi$. The following statements hold for the Frobenius endomorphism $\pi =\pi_X$ of $X$:

(a) $\pi\cdot\pi^t=q^m$; hence $\mathbb{Q}[\pi ]$ is stable under the involution $\alpha\mapsto\alpha^t$;

(b) $\mathbb{Q}[\pi ]\subset\mathrm{End}(X)$ is a product of fields;

(c) for every homomorphism $\rho\:\mathbb{Q}[\pi ]\rightarrow \mathbb{C}$, $\rho (\pi^t)=\iota (\rho\pi )$, and $|\rho\pi |=q^{m/2}$. ($\iota$ is complex conjugation)

PROOF: (a) By definition, $\varphi$ is a morphism $X\otimes X\to T^{\otimes (-m)}$ ($T$ is the Tate object). It is invariant under $\pi$, and so $$\varphi (\pi x,\pi y)=\pi (\varphi (x,y))=q^m\varphi (x,y)=\varphi (x,q^my).$$ But $\varphi (\pi x,\pi y)=\varphi (x,\pi^t\pi y)$, and because $\varphi$ is nondegenerate, this implies that $\pi^t\cdot\pi =q^m$. Therefore $\mathbb{Q}[\pi ]$ is stable under $\alpha\mapsto\alpha^t$, and we obtain (a).

(b) Let $R$ be a commutative subalgebra of $\mathrm{End}(X)$ stable under $\alpha\mapsto\alpha^t$, and let $r$ be a nonzero element of $R$. Then $s=rr^t\neq 0$ because $\mathrm{Tr}(rr^t)>0$. As $s^t=s$, $\mathrm{Tr}(s^2)=\mathrm{Tr}(ss^t)>0$, and so $s^2\neq 0$. Similarly $s^4\neq 0$, and so on, which implies that $s$ is not nilpotent, and so neither is $r$. Thus $R$ is a finite-dimensional commutative $\mathbb{Q}$-algebra without nonzero nilpotents, and the only such algebras are products of fields.

(c) In an abuse of notation, we set $\mathbb{R}[\pi ]=\mathbb{R}\otimes_{ \mathbb{Q}}\mathbb{Q}[\pi ]$. As in (b), this is a product of fields stable under $\alpha\mapsto\alpha^t$. This involution permutes the maximal ideals of $\mathbb{R}[\pi ]$ and, correspondingly, the factors of $\mathbb{R}[\pi ]$. If the permutation were not the identity, then $\alpha\mapsto\alpha^ t$ would not be a positive involution. Therefore each factor of $\mathbb{R}[\pi ]$ is stable under the involution. The only involution of $\mathbb{R}$ is the identity map (= complex conjugation), and the only positive involution of $\mathbb{C}$ is complex conjugation. Therefore we obtain the first statement of (c), and the second then follows from (a).

This (conjectural) proof of the Riemann hypothesis for motives is very close to Weil's original proof for abelian varieties (Weil 1940).

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The standard conjectures imply directly that the category of motives over the finite field $\mathbb{F}_q$ is a polarizable (hence semisimple) Tannakian category. Using only that, we have the following result.

PROPOSITION: Let $X$ be a motive of weight $m$ over $\mathbb{F}_q$, and let $\alpha\mapsto\alpha^t$ be the involution of $\mathrm{End}(X)$ defined by a Weil form $\varphi$. The following statements hold for the Frobenius endomorphism $\pi =\pi_X$ of $X$:

(a) $\pi\cdot\pi^t=q^m$; hence $\mathbb{Q}[\pi ]$ is stable under the involution $\alpha\mapsto\alpha^t$;

(b) $\mathbb{Q}[\pi ]\subset\mathrm{End}(X)$ is a product of fields;

(c) for every homomorphism $\rho\:\mathbb{Q}[\pi ]\rightarrow \mathbb{C}$, $\rho (\pi^t)=\iota (\rho\pi )$, and $|\rho\pi |=q^{m/2}$.

PROOF: (a) By definition, $\varphi$ is a morphism $X\otimes X\to T^{\otimes (-m)}$ ($T$ is the Tate object). It is invariant under $\pi$, and so $$\varphi (\pi x,\pi y)=\pi (\varphi (x,y))=q^m\varphi (x,y)=\varphi (x,q^my).$$ But $\varphi (\pi x,\pi y)=\varphi (x,\pi^t\pi y)$, and because $\varphi$ is nondegenerate, this implies that $\pi^t\cdot\pi =q^m$. Therefore $\mathbb{Q}[\pi ]$ is stable under $\alpha\mapsto\alpha^t$, and we obtain (a).

(b) Let $R$ be a commutative subalgebra of $\mathrm{End}(X)$ stable under $\alpha\mapsto\alpha^t$, and let $r$ be a nonzero element of $R$. Then $s=rr^t\neq 0$ because $\mathrm{Tr}(rr^t)>0$. As $s^t=s$, $\mathrm{Tr}(s^2)=\mathrm{Tr}(ss^t)>0$, and so $s^2\neq 0$. Similarly $s^4\neq 0$, and so on, which implies that $s$ is not nilpotent, and so neither is $r$. Thus $R$ is a finite-dimensional commutative $\mathbb{Q}$-algebra without nonzero nilpotents, and the only such algebras are products of fields.

(c) In an abuse of notation, we set $\mathbb{R}[\pi ]=\mathbb{R}\otimes_{ \mathbb{Q}}\mathbb{Q}[\pi ]$. As in (b), this is a product of fields stable under $\alpha\mapsto\alpha^t$. This involution permutes the maximal ideals of $\mathbb{R}[\pi ]$ and, correspondingly, the factors of $\mathbb{R}[\pi ]$. If the permutation were not the identity, then $\alpha\mapsto\alpha^ t$ would not be a positive involution. Therefore each factor of $\mathbb{R}[\pi ]$ is stable under the involution. The only involution of $\mathbb{R}$ is the identity map (= complex conjugation), and the only positive involution of $\mathbb{C}$ is complex conjugation. Therefore we obtain the first statement of (c), and the second then follows from (a).

This (conjectural) proof of the Riemann hypothesis for motives is very close to Weil's original proof for abelian varieties (Weil 1940).