All Questions
1,203 questions
3
votes
1
answer
320
views
Explicit period lattices for abelian surfaces
Given an explicit description (as an intersection) of an abelian surface $A$ is there an algorithm for computing the period lattice of the surface? For the specific examples that I am interested in, ...
- 715
11
votes
1
answer
730
views
What is the Brauer group of the moduli space of (p.p.) abelian varieties?
What is the Brauer group of the moduli space of principally polarized abelian varieties of a given dimension? I am primarily interested in the "open" moduli space, i.e. not a compactification. The ...
- 30.5k
10
votes
0
answers
323
views
The mod 3 reduction of some powers of delta
Let f in Z/3[[x]] be the mod 3 reduction of the Fourier expansion of the normalized weight 12 cusp form delta for the full modular group. The exponents appearing in f are all 1 mod 3. Fix
k>0 and ...
- 5,422
31
votes
1
answer
5k
views
Modern proof of Serre's open image theorem?
Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. Serre's open image theorem (which appears in his book 'Abelian $l$-Adic Representations and Elliptic Curves'...
- 1,905
2
votes
1
answer
250
views
Can we control the size of the intersection of two abelian subfactors of an abelian variety ?
Let $A$ be an abelian variety over an algebraically closed field $K$, and $B$ an abelian subvariety. We know (for instance, from Milne course on AV) that there is an abelian "cofactor" $B'\subset A$ ...
- 277
14
votes
1
answer
1k
views
Frobenius splitting of Fano varieties
Dear MO,
Question 1. Do you know of an example of a Fano variety which is not Frobenius split?
Background
(1) A variety $X$ in characteristic $p$ is called Frobenius split if there is a "$p$-th ...
- 16.2k
15
votes
6
answers
3k
views
Characteristic zero and characteristic $p$ in algebraic geometry
Are there non-trivial (i.e. excluding concepts that can be defined only for $p>0$) statements in algebraic geometry that hold for all fields of characteristic $p$ for all prime $p$ but are known ...
user16974
1
vote
1
answer
851
views
Serre-Tate canonical lifts for finite fields
A result of Serre-Tate states that we can canonically lift an ordinary abelian variety over a perfect field $k$ of positive characteristic to an abelian scheme over the ring of Witt vectors of $k$ and ...
- 13
4
votes
0
answers
282
views
Does semi-stable reduction behave well with Weil restriction of scalars
Let $A$ be an abelian variety over a number field $K$ with semi-stable reduction over $O_K$.
Does the Weil restriction $\textrm{Res}_{K/\mathbf{Q}}A$ of $A$ to $\mathbf{Q}$ have semi-stable reduction ...
- 1,213
11
votes
1
answer
761
views
Mordell-Weil group of the universal abelian scheme
Let $n>2$ and let $k$ be either $\bf Q$ or a finite field whose characteristic is prime to $n$. Let $A_{g,n}$ be the moduli scheme, which represents the functor, which with every $k$-scheme $S$
...
- 3,076
1
vote
0
answers
162
views
Construction of RM abelian variety from eigenform
Let $f$ be a normalized eigenform of weight $2$ level $N$. If the Fourier coefficients of $f$ generate a totally real field $F$, then we associate to $f$ a system of $\ell$-adic Galois representations ...
- 15.4k
0
votes
2
answers
336
views
Does the self-product of a $g$-dimensional abelian variety contain an abelian variety of dimension smaller than $g$ at some point
Let me be more precise than the title. (This will be my last attempt to do something with abelian varieties. Sorry for all the basic questions. The answers have been great!)
Let $A$ be a simple ...
- 1,213
0
votes
1
answer
298
views
Is any simple abelian variety covered by a non-simple abelian variety
Let $A/k$ be a simple abelian variety.
Does there exist a non-simple abelian variety $B/k$ and a finite homomorphism $f:B\to A$ over $k$?
I don't need $f:B\to A$ to be etale.
- 1,213
3
votes
1
answer
492
views
Are abelian varieties degree two covers of some projective space
Let $A$ be an abelian variety over a field $k$ of dimension $g\geq 2$.
There exists a finite morphism $A\to \mathbf{P}^g_k$. Here's the question.
Does there exist a finite morphism $A\to \mathbf{P}^...
- 1,213
0
votes
0
answers
263
views
Frobenius eigenvalues of abelian variety
Let $A$ be an abelian variety over a finite field $\mathbb{F}_q$ and $x_i$ the Frobenius eigenvalues on $H^1$. Does $x_i \mapsto q/x_i$ permute the $x_i$, and why? It should follow from Poincare ...
- 417
6
votes
2
answers
709
views
Ample vector bundles on complex tori
Let $X$ be a $n$-dimensional complex torus and $\omega$ a Kähler form on $X$. Then, it is well known that a real $(1,1)$-class $[\alpha]\in H^{1,1}(X,\mathbb R)$ is a Kähler class if and only if for ...
- 7,902
1
vote
0
answers
522
views
Component group of Neron model of a parametrized abelian variety
Let $A$ be an abelian variety of dimension $2$ over a $p$-adic field $K$ with (additive) valuation $v$. Assuming $A$ has multiplicative reduction, the theory of $p$-adic theta functions gives us an ...
- 15.4k
7
votes
0
answers
207
views
Unicritical rational functions on curves in characteristic $p$
Let $k$ be an algebraically closed field of positive characteristic $p$, and let $X_{/k}$ be a smooth projective connected curve. Let $x_0$ be a point of $X(k)$.
How precisely can one describe the ...
- 1,199
0
votes
0
answers
359
views
Chern class of line bundle inducing a principal polarisation
What can one say about the Chern class $c_1(\mathcal{L})$ of a line bundle $\mathcal{L}$ on an Abelian variety $A$ inducing a (principal) polarisation $A \to A^\vee$?
Why am I asking this? My ...
user19475
3
votes
2
answers
472
views
Induced maps of an automorphism of a curve on the tangent ot its jacobian and on its differential forms
Let $C$ be a smooth projective curve of genus $g$ over a field $k$ and $J$ be its jacobian (defined over $k$). Let $\sigma: C \rightarrow C$ be a $k$-automorphsm of $C$. This automorphism $\sigma$ ...
- 1,109
10
votes
0
answers
492
views
Obstructions to deforming an abelian variety
Are abelian varieties unobstructed? That is, given an abelian scheme $X \to \mathrm{Spec}R $ for $R$ a local artinian ring and $\mathrm{Spec} R \to \mathrm{Spec} S $ a nilpotent thickening, can we ...
- 25.6k
4
votes
1
answer
918
views
Heisenberg group in characteristic two
I have seen two constructions called the Heisenberg group. If $k$ is a field of characteristic not equal to $2$ and $V$ is a $2n$-dimensional vector space over $k$ with symplectic form $\omega : V \...
- 3,605
4
votes
2
answers
1k
views
Is the absolute Galois group of the field of Laurent series in positive characteristic finitely generated?
If $K$ is an algebraically closed field of characteristic $p>0$, then $K((t))$, the field of Laurent series with coefficients in $K$, has infinitely many Galois extensions of degree $p$. Indeed, ...
- 3,362
3
votes
1
answer
1k
views
books (or notes) on complex multiplication
This would be a vague question, but I still want to ask here. Do you have any recommended book on complex multiplicaton. I know only 2 books: Shimura's book ...
- 1,109
2
votes
1
answer
434
views
Weil reciprocity on abelian varieties and biextensions?
I was once told, by someone who would likely be right about such things, that the version of Weil reciprocity for abelian varieties (as in Lang, Abelian Varieties) should come out of consideration of ...
- 12.6k
2
votes
1
answer
359
views
kernel of an isogeny and coker of its induced map on the Tate module
In a proof in Milne's note "Abelian Variety" (top on p.52), I saw an equality:
$ \mathrm{Ker}(\beta)(l) = \mathrm{Coker}(T_{l}(\beta))$, here $\beta$ is an (separable) isogeny of an abelian variety $A/...
- 1,109
1
vote
0
answers
238
views
Classification of fibres in pencils of curves of genus two
For the case of characteristic positive, there exist some clasification of families of curves of genus two over a elliptic curve?.
- 11
1
vote
0
answers
231
views
Where can I find a copy of Serre's Cours au college de France 1985-1986?
Hi,
I was wondering: where might I be able to find a copy of this work online?
And are there any other resources for the proof of the open image theorem for abelian varieties with endomorphism ring ...
- 15.4k
3
votes
1
answer
231
views
upper bounds for the ranks of the minus parts of modular jacobians
Let $p$ be a prime, and $J^-(p)$ be the maximal quotient of the Jacobian of the modular curve $X_0(p)$ on which the involution acts by $-1$.
Is anything known or conjectured about upper bounds for ...
- 1,832
4
votes
3
answers
795
views
Quotients of Tate modules
Let $p$ be a prime number, let $K$ denote a finite extension $\mathbb{Q}_{p}$ and let
$\overline{K}$ be an algebraic closure of $K$. Let $A$ be an ellitpic curve over
$K$ and denote by $T_{p}A$ its ...
- 41
0
votes
1
answer
162
views
Terminology about Abelian varieties over finite fields
Is there a standard meaning for ordinary and supersingular Abelian varieties over finite fields? If so, where can I find it (together with basic properties about them)?
- 1,271
1
vote
1
answer
373
views
etale covers of line bundles on an abelian variety
subj: etale covers of line bundles on an abelian variety
Is there an explicit decryption of finite
etale covers of a line bundle $L$ on an abelian variety and its associated C*-bundles
$L^o = L \...
- 468
6
votes
1
answer
1k
views
Albert classification of rational endomorphism rings of simple Abelian varieties over finite fields
Recall the Albert classification of rational endomorphism rings with involution of simple Abelian varieties over arbitrary fields:
Type I: totally real, trivial involution
Type II and III: quaternion ...
user19475
2
votes
0
answers
279
views
deRham cohomoloy of CM liftings of Jacobians
Let $k$ be field of characteristic $p>0$ and $W=W(k)$ the ring of Witt vectors of $k$. We call a smooth curve over $k$, ordinary, when the Jacobian of $J(X)$ of $X$ is an ordinary abelian variety. ...
- 637
17
votes
4
answers
2k
views
What are supersingular varieties?
For varieties over a field of characteristic $p$, I saw people talking about supersingular varieties.
I wanted to ask "why are supersingular varieties interesting". However, as I don't want to ask an ...
- 2,040
2
votes
0
answers
179
views
Notation for a canonical quotient of an abelian variety in positive characteristic
This is a light question about notation, but I received no answer in Stackexchange.
Let $k$ be an algebraically closed field of characteristic $p>0$ and let $A=A_{/k}$ be an ordinary abelian ...
- 797
3
votes
0
answers
742
views
Explicit description of O^{cris}_n in Fontaine/Messing
Let $k$ be a perfect field of characteristic $p$, $W(k)$ the Witt ring and $K$ its quotient field. In their article "$p$-adic periods and $p$-adic etale cohomology" Fontaine and Messing give in II.1.4 ...
1
vote
0
answers
316
views
Weil pairing as an algebraic cycle?
Is there an algebraic cycle corresponding to the Weil pairing on an abelian variety (of dim>1)? Ideally I'd like to see an example as explicit as possible, e.g.
an explicitly given variety of dim>1 ...
- 1,905
1
vote
1
answer
557
views
complex deformations of abelian varieties
Let $A$ be an abelian variety defined over $\mathbf{C}$ (of dimension $>1$) and let $\Theta_A$ be the holomorphic tangent sheaf of $A$.
Question. How does one compute $H^1(A,\Theta_A)$ ?
If $A$ ...
- 7,609
5
votes
0
answers
397
views
false elliptic curves and principal polarizations
Hi,
Let $\Delta$ be a quaternion algebra over $\mathbf Q$ and let $\mathcal O_\Delta$ be a maximal order in $\Delta$.
Recall that a false elliptic curve over a field $K$ is a pair $(A/K,i)$ ...
- 2,842
2
votes
0
answers
172
views
Dual Honda systems
Hello,
There is an equivalence of categories between p-divisible groups over the ring of Witt vectors $W(k)$ and the category of "Honda systems", that is couples $(M,L)$ formed by a Dieudonné module $...
- 51
3
votes
0
answers
405
views
lifting abelian varieties
Hello,
Let A be an abelian variety over $k=\overline{\mathbb{F}_p}$. This abelian variety is endowed with a principal polarization and an action by $\mathcal{O}_E$, the ring of integers of an ...
- 51
3
votes
1
answer
980
views
Construction of Kummer map for abelian variety
Let $A$ be an abelian variety over the rational numbers $\mathbf{Q}$. Let $V=T_p A \otimes \mathbf{Q}_p$ be the $\mathbf{Q}_p$-Tate module of $A$. Let $G$ be the absolute Galois group of $\mathbf{Q}$. ...
- 1,213
4
votes
1
answer
900
views
Algebraic relationships between elliptic functions
If $f$ and $g$ are two elliptic functions with the same periods, then there exists an algebraic relationship of the form $P(f,g)=0$, where $P$ a polynomial of two variables with constant coefficients.
...
2
votes
0
answers
321
views
CM abelian variety from an algebraic Hecke character?
Hi,
Given an algebraic Hecke character $\chi$ of a number field $k$ there should be a
"rank 1 CM-motive" $M$ with
$\overline Q$-coefficients such that $L(s,M) = L(s,\chi)$. This follows from the ...
- 2,842
8
votes
1
answer
982
views
Is there a really big ring of differential operators in characteristic p?
$k$ is a field of characteristic $p$.
$k[t]$ has canonical first-order differential operator $\partial$
As an endomorphism of $k[t]$, $\partial^p=0$.
First way to fix it:
Use the divided power ...
4
votes
1
answer
796
views
Deformation space of non-ordinary abelian varieties
It is a well known result of Serre and Tate that if $A$ is an ordinary abelian variety over a field $k$ of characteristic $p>0$, then the deformation space $\mathcal{M}$ of $A$ to an abelian ...
- 395
15
votes
0
answers
403
views
Does every Abelian variety have a finite resolution by Jacobians?
One knows that every Abelian variety is a quotient of a Jacobian. Does every Abelian variety have a finite resolution by Jacobians?
user19475
10
votes
6
answers
2k
views
Proofs in the same vein as Ax-Grothendieck
I would like to see other examples of (ideas of) proofs and results in the same vein as the proof of the Ax-Grothendieck theorem. To explain what I mean by "in the same vein", I will quote from the ...
- 23.4k
2
votes
1
answer
570
views
Generalization of singular moduli
$j$-invariants of CM curves $E$ over (say) the complex numbers are known as singular moduli. As the theory of complex multiplication explains, singular moduli are algebraic integers of great ...
- 2,406