Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 etale cohomology, there is a more complicated construction starting from $H^i(\overline{X}, \mathbb{Q}_p)$ going via Fontaine's $D_{\mathrm{pst}}$ 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?

share|improve this question

1 Answer 1

up vote 10 down vote accepted

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, $H^1(X)$ 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.

share|improve this answer
    
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. –  David Loeffler Sep 3 '12 at 15:12

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.