There is a counterexample of Serre showing that there is no Weil cohomology theory with coefficients in $\mathbf{Q}, \mathbf{Q}_p, \mathbf{R}$ over $\mathbf{F}_{p^2}$ (a supersingular elliptic curve). So what happens if we tensor motivic cohomology with $\mathbf{Q}$ or crystalline cohomology with $\mathbf{Q}_p$?
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