# Is there a "universal" cohomology theory for varieties over p-adic fields?

Let $$K$$ be a $$p$$-adic field, $$X$$ a smooth proper algebraic variety over $$K$$, and $$0 \le i \le 2 \dim X$$. For a prime $$\ell \ne p$$ one can consider the $$\ell$$-adic cohomology $$H^i(\overline{X}, \mathbb{Q}_\ell)$$, and massage this in the usual way (via Grothendieck's abstract monodromy theorem) to get a Weil–Deligne representation of $$K$$ with coefficients in $$\mathbb{Q}_\ell$$. For $$p$$-adic étale cohomology, there is a more complicated construction starting from $$H^i(\overline{X}, \mathbb{Q}_p)$$ going via Fontaine's $$D_{\text{pst}}$$ functor. I gather it is conjectured that all of these Weil–Deligne representations are in fact definable over $$\mathbb{Q}$$, and they should have the same character and thus be isomorphic up to semisimplification; and this is known in some cases.

Is it expected that there should be a "universal" cohomology theory taking values in the category of Weil–Deligne representations over $$\mathbb{Q}$$, from which all of the above can be obtained by extending scalars? If so, have there been any attempts to construct such a cohomology theory?

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 $$\mathbb{Q}$$ vector 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, $$H^1(X)$$ and $$H^1(Y)$$ 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.

• Ah, yes, I had forgotten about Serre's nasty example. Of course one can solve the equations in 2x2 matrices over $\overline{\mathbb{Q}}$ but not in any "canonical" fashion, so it seems like there is no hope for a nice universal theory. Sep 3 '12 at 15:12
• With this post on the front page, I would like to take the chance to register my appreciation for a sentence about a theory for $\mathbb F_p$-varieties taking values in $\mathbb Q$-vector spaces that is followed by "this is not my field". Perhaps "these are not my fields"? 😄 Nov 11 at 0:57

I'm seeing this old question just now, and simply wanted to remark that the situation may be slightly better.

Namely, enlarging the category of $$\mathbb Q$$-vector spaces into the larger semisimple $$\mathbb Q$$-linear tensor category $$\operatorname{Rep}(\mathrm{Kt}_{\mathbb Q})$$ defined by Kottwitz (see for example the article Representations of the Kottwitz gerbes of Iakovenko for a definition), one expects that there is a (canonical) cohomology theory valued in Weil–Deligne representations with coefficients in $$\operatorname{Rep}(\mathrm{Kt}_{\mathbb Q})$$. This should recover the other constructions via natural functors to $$\operatorname{Rep}(\mathrm{Kt}_{\mathbb Q_\ell})$$ for all primes $$\ell$$. This is a variant of Conjecture 9.5 in my 2018 ICM paper. (Warning: In that paper, I write $$\mathrm{Kt}_{\mathbb Q}$$ for what I denote $$\operatorname{Rep}(\mathrm{Kt}_{\mathbb Q})$$ in this answer.)

While the definition of $$\operatorname{Rep}(\mathrm{Kt}_{\mathbb Q})$$ is as representations of a gerbe $$\mathrm{Kt}_{\mathbb Q}$$ that is more-or-less explicitly constructed by class field theory — quite similar to the global Weil group —, we still lack a direct linear-algebraic description of $$\operatorname{Rep}(\mathrm{Kt}_{\mathbb Q})$$, unfortunately.

• I edited to link to point to the more-stable arXiv version, as the statement of Conjecture 9.5, at the very least, is identical to your personal copy (not that I expect any differences at all!) Nov 11 at 0:54