Given a finite extension $K$ of $\mathbb{Q}_p$, is there some conjectural statement characterizing which finite-dimensional $p$-adic representations of the absolute Galois group of $K$ are (Tate twists of) subquotients of the cohomology groups of $K$-varieties?
$\begingroup$
$\endgroup$
3
-
3$\begingroup$ Varieties give rise to de Rham and thus potentially semi-stable representations, but not every such representation comes from geometry: for example, the eigenvalues of Frobenius should be Weil numbers. I don't know any conjectural description. $\endgroup$– François BrunaultMay 19, 2019 at 7:06
-
$\begingroup$ The answer to the question in the body of the text is "at present, no" (even for crystalline representations). $\endgroup$– OlivierMay 24, 2019 at 13:24
-
$\begingroup$ @Olivier is there a reference discussing this? What are the obstructions beyond those provided by Weil conjectures? $\endgroup$– user140765May 24, 2019 at 15:56
Add a comment
|