All Questions
7 questions
16
votes
1
answer
1k
views
Some basic questions on crystalline cohomology
Let $X_0$ be a smooth projective variety over $\mathbf{F}_q$ and ${X}$ its base change to an algebraic closure $k$ of $\mathbf{F}_q$.
Crystalline cohomology $H^*_{\rm cris}(X) := H^*((X/W(k))_{\rm ...
user134756
15
votes
2
answers
2k
views
Are there known cases of the Mumford–Tate conjecture that do not use Abelian varieties?
(For a formulation of the Mumford–Tate conjecture, see below.)
The question
As far as I know, all non-trivial known cases of the Mumford–Tate conjecture more or less depend on the Mumford–Tate ...
- 5,504
13
votes
1
answer
591
views
Is there a yoga of effectivity for motives and their realizations?
Inside the 'usual' categories of motives (Chow, Voevodsky ones) there are the categories of effective motives. Similarly, there are effective Hodge structures; they seem to be closely related with ...
- 16.9k
5
votes
1
answer
460
views
Nearby cycles and extension by zero
Let $f: X\to \text{Spec}(R)$ be a proper and smooth morphism, with $R$ a strictly henselian dvr. Call $s = \overline{s}$ the closed point and $\eta$ the geometric point of $\text{Spec}(R)$.
Call $i_s ...
- 181
3
votes
1
answer
235
views
Intrinsicness of Hodge-theoretic properties of Galois representations in a general reductive group
In the paper "The conjectural connections between automorphic representations and Galois representations" by Buzzard and Gee, it is said
"We say that
$\rho$ is crystalline/de Rham/Hodge–Tate if ...
user140765
3
votes
0
answers
220
views
Artin $\ell$-adic comparison and Galois action
Let $X_0$ be a smooth projective variety defined over a number field $k$.
Let $\sigma : k\to\mathbf{C}$ be one of the finitely many field embeddings of $k$ into the complex numbers, and call $X := (...
user97068
2
votes
1
answer
399
views
Classes of hyperplane sections in cohomology
Let $X$ be a smooth projective variety over the algebraic closure of a finite field with Galois group $G$.
Is it true that the vector space $H^{2k}(X,\mathbf{Q}_{\ell}(k))^G$ has always positive ...
user131191