6
votes
1answer
197 views
Prym varieties as Jacobian varieties
A generic abelian variety of dimension 2 or 3 is a jacobian of a curve.
Is there a canonical way to determine a curve whose jacobian is a prym variety of a unramified double cover …
0
votes
1answer
53 views
Ramification in Division field of Abelian Varieties
This might be a very simple question, and that might be the reason that I could not find any reference on this.
My question is
Let $A$ be an abelian variety defined over a numb …
0
votes
1answer
58 views
Is the stabilizer of an irreducible subvariety of an abelian variety irreducible ?
Let $A$ be a (semi-)abelian variety over an algebraically closed field $K$, and $X$ be a closed irreducible subvariety. Can $X$ have a non-trivial finite stabilizer ? By stabilizer …
8
votes
0answers
106 views
Infinitely many curves with isogenous Jacobians
Let $g\geq 4$. Are there infinitely many compact genus $g$ Riemann surfaces with (mutually) isogenous Jacobians?
Does the situation change in positive characteristic?
2
votes
2answers
195 views
Is the moduli space of ppAVs smooth?
Let $A_g$ be the moduli space of principally polarised abelian varieties of dimension $g$ over the complex numbers. (EDIT: I mean the coarse moduli space.) Is this smooth?
Since …
2
votes
0answers
100 views
Automorphisms of Generic Abelian Varieties
Automorphism groups of elliptic curves are very well understood. Of course, every elliptic curve has the automorphism $[-1]$ of order $2$. If we are over a (algebraically closed) f …
5
votes
1answer
145 views
Local Norm Mapping for Abelian Varieties
Let $A/K$ be an abelian variety defined over a nonarchimedean local field $K$ of characteristic $0$ and let $L$ be a finite extension of $K$. Consider the norm map $$A(L)\xrightarr …
3
votes
1answer
180 views
Weil restriction of abelian schemes along finite étale (resp. finite locally free) morphisms
Q: Is there a simple proof of the fact that the Weil restriction of an abelian scheme along a finite étale morphism is an abelian scheme ?
Details: Let $S$ be a scheme and $f:S'\r …
1
vote
2answers
162 views
Moduli Spaces of Higher Dimensional Complex Tori
I know that the space of all complex 1-tori (elliptic curves) is modeled by $SL(2, \mathbb{R})$ acting on the upper half plane. There are many explicit formulas for this action.
S …
1
vote
0answers
106 views
About Alexeev and Nakamura’s paper “on Mumford’s construction of degenerating abelian varieites”
Is there anyone familiar with this paper? It seems to me it contains "some" typos and even some small elementary mistakes, which makes my reading very slow. Of course the key reaso …
2
votes
1answer
292 views
Tate conjecture for abelian varieties over a finitely generated extension of an algebraically closed field
Let $K$ be a finitely generated extension of an algebraically closed field of characteristic zero, and $A,B$ abelian varieties over $K$.
Then is $Hom_K(A,B)\otimes \mathbb{Z_l} \ …
12
votes
1answer
592 views
Status of Grothendieck’s conjecture on homomorphisms of abelian schemes
In [1] Grothendieck posits the following:
Conjecture. Let $S$ be a reduced connected scheme, locally of finite type over Spec($\mathbf{Z}$) or a field $k$, $A$ and $B$ two abelian …
1
vote
0answers
219 views
Decomposition theorem for principally polarized abelian varieties in positive characteristic.
In characteristic zero we have the following decomposition theorem for polarized abelian varieties: it gives an isomorphism between a PPAV and a product of PPAV's of lower dimensio …
2
votes
1answer
155 views
Is the canonical height of a totally p-adic point on an abelian variety bounded away from zero?
Inspired by the result of Schinzel and Smyth that a totally real number other than $0$ and $\pm 1$ has height at least $\frac{1}{2}\log \Big( \frac{1+\sqrt{5}}{2} \Big) = 0.240659\ …
11
votes
1answer
304 views
The torsion point count in higher dimension
It is an easy consequence of the Serre open image theorem that for the torsion point count on elliptic curves, the following possibilities arise.
If $E/\bar{\mathbb{Q}}$ is an el …

