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The following is an adaptation of an argument of Serre, explaining why there shouldn't be a universal cohomology theory for $\mathbb{F}_p$ varieties taking values in $\mathbb{Q}$ vetor spaces. Be warned that this is not my field, so I may be missing something basic.
Let $p$ be a prime which is $3 \bmod 4$, let $X$ be the elliptic curve $y^2 = x^3-x$ over $\mathbb{Q}_p$ and let $Y$ be the base change of $X$ to $\mathbb{Q}_p(i)$. \mathbb{Q}_p(i)$. In any of the cohomology theories you describe,$H^1(X)$is naturally isomorphic to$H^1(Y)$, and both$H^1(Y)$are two dimensional. In etale cohomology,$H^1(X) \cong H^1(Y)$; in$p$-adic cohomologies I believe you usually have $H^1(X) \otimes_{\mathbb{Q}_p} \mathbb{Q}_p(i) \cong H^1(Y)$. I assume in your hypothetical$\mathbb{Q}$-valued theory, you would have$H^1(X) \cong H^1(Y)$. Let$F$be the Frobenius automorphism of$H^1(X)$; let$J$be the automorphism of$H^1(Y)$induced by$(x,y) \mapsto (-x, iy)$. Identifying$H^1(X)$and$H^1(Y)$, these maps should obey the relations $$F^2 = -p \quad J^2 = -1 \quad FJ=-JF$$ These equations are not solvable in$2 \times 2$matrices over$\mathbb{Q}$(or even over$\mathbb{R}$). So any theory would have to be "unnatural" enough that this is not an obstacle. The category of motives is designed to be the recipient of a universal cohomology theory. It gets around this issue by being$\mathbb{Q}$-linear, meaning that$\mathrm{Hom}(U,V)$is a$\mathbb{Q}$-vector space for any motives$U$and$V$, but not having a natural functor to$\mathbb{Q}$-vector spaces, so the motives themselves cannot be thought of as$\mathbb{Q}$-vector spaces. I don't know if there is any way in which motives over$p$-adic fields are better than motives over general fields. 1 The following is an adaptation of an argument of Serre, explaining why there shouldn't be a universal cohomology theory for$\mathbb{F}_p$varieties taking values in$\mathbb{Q}$vetor spaces. Be warned that this is not my field, so I may be missing something basic. Let$p$be a prime which is$3 \bmod 4$, let$X$be the elliptic curve$y^2 = x^3-x$over$\mathbb{Q}_p$and let$Y$be the base change of$X$to$\mathbb{Q}_p(i)$. In any of the cohomology theories you describe,$H^1(X)$is naturally isomorphic to$H^1(Y)$, and both are two dimensional. Let$F$be the Frobenius automorphism of$H^1(X)$; let$J$be the automorphism of$H^1(Y)$induced by$(x,y) \mapsto (-x, iy)$. Identifying$H^1(X)$and$H^1(Y)$, these maps obey the relations $$F^2 = -p \quad J^2 = -1 \quad FJ=-JF$$ These equations are not solvable in$2 \times 2$matrices over$\mathbb{Q}$(or even over$\mathbb{R}$). So any theory would have to be "unnatural" enough that this is not an obstacle. The category of motives is designed to be the recipient of a universal cohomology theory. It gets around this issue by being$\mathbb{Q}$-linear, meaning that$\mathrm{Hom}(U,V)$is a$\mathbb{Q}$-vector space for any motives$U$and$V$, but not having a natural functor to$\mathbb{Q}$-vector spaces, so the motives themselves cannot be thought of as$\mathbb{Q}$-vector spaces. I don't know if there is any way in which motives over$p\$-adic fields are better than motives over general fields.