All Questions
19
questions
4
votes
1
answer
405
views
Étale group schemes and specialization
If $A$ is an abelian variety over a finite field $\mathbf{F}_q$, then $A(\mathbf{F}_q)$ (resp. $A(\overline{\mathbf{F}}_q)$) is a finite (resp. infinite torsion) group, but $A(\mathbf{F}_q(t))$ is a ...
user501361
2
votes
0
answers
116
views
Splitting of prime and order of reduction of point of infinite order in an abelian variety
I have already asked this question on stackexchange without much luck. I apologize if the question is too trivial to be asked here.
Let $A$ be an abelian variety defined over a number field $K$, $P \...
- 411
12
votes
2
answers
628
views
What is the correct notion of representation for abelian varieties?
Zeroth question - am I right that in the "ordinary" sense an abelian variety does not possess any representations at all?
More precisely, a representation of an algebraic group $G$ (over an ...
- 17.8k
14
votes
1
answer
1k
views
If it quacks like an abelian variety over a finite field
Consider smooth projective varieties over a finite field. If a curve "looks like" an elliptic curve (i.e. has genus $1$) then it can be made into an elliptic curve.
Is there something ...
- 117
4
votes
4
answers
838
views
Which schemes are divisors of an abelian variety?
Let $X$ be a smooth, projective ireducible scheme over an algebraically closed field $k$. I'm trying to understand when there exists an abelian variety $A$ such that $X$ is isomorphic to a prime ...
- 4,286
3
votes
1
answer
211
views
What is the geometric quotient of the abelian threefold?
Consider a finite field $\mathbb{F}_p$ such that $p \equiv 1 \ (\mathrm{mod} \ 3)$ and its element $\zeta \neq 1$, $\zeta^3 = 1$.
Also, let $E\!: y^2 = x^3 + b$ be an elliptic curve of $j$-invariant ...
- 1,813
3
votes
0
answers
176
views
Subquotients of abelian varieties with good reduction
Let $B$ be an abelian variety over a DVR with good reduction, and let $A$
be a subquotient of $B$. Then $A$ has good reduction.
I know a proof of this statement using Neron-Ogg-Shafarevich. Is ...
- 39
2
votes
0
answers
253
views
Vector extension for p-divisible group
Background:
I am trying to understand a proof of Messing's book at Page 120. My goal it to understand the universal vector extention.
Reference:
Messing, The crystals associated to Barsotti-Tate ...
- 397
3
votes
0
answers
77
views
Under what conditions are superspecial abelian surfaces isomorphic over a finite field?
Let $E_1$, $E_2$, $E_3$, $E_4$ be supersingular elliptic curves over a finite field $\mathbb{F}_{p^2}$, where $p$ is an odd prime. There is a well known theorem stating that over the algebraic closure ...
- 1,813
2
votes
2
answers
326
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 ...
- 1,813
5
votes
1
answer
331
views
Property of bundles with connections on abelian variety doesn't hold for additive or multiplicative group?
This question is a followup to two of my previous questions, see here and here.
Let $A$ be an abelian variety over a field $k$ of characteristic $0$. How do I prove, without using transcendental ...
user61522
5
votes
1
answer
388
views
Is there a covering of Prym variety?
$\mathstrut$Hi, guys!
Let $C$, $C^\prime$ be projective smooth irreducible algebraic curves over an algebraically closed field $k$ ($\mathrm{char}(k) \neq 2$), $\phi : C$ $\to$ $C^\prime$ a two-...
- 1,813
11
votes
0
answers
487
views
Can an abelian variety/Q have no non-trivial points over Q_sol?
Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable
extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial?
This follows from the conjecture that the maximal (pro-)solvable ...
- 11.2k
3
votes
1
answer
643
views
Exactness on rational points of algebraic groups
Let $k$ be a finite extension of the p-adic number field $Q_p$ and G be a connected algebraic (not affine) group over $k$. It is well-known (see e.g. [1] Proposition 3.1) that G decomposes as
$1\...
- 1,672
1
vote
0
answers
117
views
Need information about particular kind of quotients of semisimple algebraic groups by free abelian discrete subgroups
Let me start with the simplest version of the question since already there I don't know anything.
For a complex number $q$, consider the quotient space $X_q:=\mathrm{SL}_2(\mathbb C)\left/\left\{\...
- 17.8k