All Questions
7 questions
11
votes
1
answer
1k
views
Pointless groups
This question now has two sequels, Pointless groups II (to which @R.vanDobbendeBruyn gave a counterexample for an infinite, imperfect field) and Pointless groups III, both using revised wording ...
- 12.9k
2
votes
0
answers
154
views
Reference request - obtaining finite simple groups from algebraic groups
I'm looking for references for the following statements, which I believe are true:
Let $G$ be a simply connected simple linear algebraic group over a finite field $k$ of cardinality $q\ge 4$. Let $Z\...
- 7,527
5
votes
1
answer
203
views
Orbit counting polynomials over finite fields
Let $X$ be an affine variety defined over $\mathbb{Z}$ and let $G$ be an algebraic group defined over $\mathbb{Z}$. Let $q$ be a power of a prime number. We write $\mathbb{F}_q$ for the field with $q$ ...
- 5,039
1
vote
1
answer
154
views
Lifting Lang-Steinberg to DVR's in Characteristic 0
Let $A$ be a compact DVR in characteristic $0$, uniformizer $\pi$ and residue field $k$. Let $A\subset B$ be a complete DVR with the same uniformizer $\pi$ and algebraicly closed residue field $F$. ...
- 273
2
votes
2
answers
327
views
How to prove that $A$ is supersingular iff the Picard number $\rho(A)$ is equal to the second $l$-adic Betti number $b_2(A) = 6$?
Let $A$ be an abelian surface over algebraically closed field $k$ of characteristic $p > 2$. How to prove that $A$ is supersingular (in other words, there is an isogeny between $A$ and $E^2$, where ...
- 1,833
6
votes
1
answer
560
views
Over a finite field, does a torsor under the component group of G lift to a torsor under G?
Let $k$ be a finite field and $G$ a finite type smooth $k$-group scheme. Let $G^0$ and $\Gamma$ be the connected component of identity and the component group of $G$, so there is an exact sequence $1 \...
- 1,103
7
votes
2
answers
2k
views
The Lang isogeny
Let $G$ be a connected commutative algebraic group over $\mathbb{F}_q$. If $\text{Fr}_q : G \to G$ denotes the $q$-Frobenius morphism, we define the Lang isogeny $L_q$ to be the endomorphism of $G$ ...
- 3,605