Grothendieck showed the Riemann Hypothesis for finite fields would follow from the so-called "Standard Conjectures".
Among other things, these conjectures would imply:
Every variety \(X\) has \(h(X) \cong X_0 \oplus \cdots \oplus X_n\) where the motive \(X_k\) has "weight" \(k\), meaning it contributes terms of the form \(\alpha^n\) with \(|\alpha| = p^{k/2}\) to the count of points of \(X\) over \(\mathbb{F}_{p^n}\).
The category \(\mathsf{Mot}\) is "abelian": it has well-behaved kernels, cokernels, subobjects and quotient objects.
The category \(\mathsf{Mot}\) is "semisimple":
every motive is a finite direct sum of motives that have only two subobjects, 0 and that motive itself.
