All Questions
1,203 questions
5
votes
2
answers
281
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Singular abelian surfaces that can be defined over $\mathbb Q$
An abelian surface $A$ is called singular if it has maximal Picard number $\rho(A) = 4$.
By work of Shioda-Mitani, any singular abelian surface $A$ is the product $A = E_1 \times E_2$ of two ...
- 20.4k
5
votes
1
answer
208
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A generalization of Witt's theorem for quaternion algebra isomorphism
Let $Q$ be a quaternion $k$-algebra (namely, a dimension 4 $k$-central simple algebra).
Then it is possible to (canonically) attach a smooth projective conic $C_Q\subseteq \mathbf{P}_k^2$ to $Q$: if ...
- 1,602
8
votes
1
answer
1k
views
Monodromy groups of families of abelian varieties: a reference request
In Serre's letter to Vigneras of 2 Oct 1986, he summarizes a course he's giving in Paris, explaining how to control the image of the mod-l Galois representations attached to abelian varieties. In ...
CommunityBot
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12
votes
2
answers
1k
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Modularity of higher dimensional abelian varieties
In another question I asked about strategies for giving an effective version of the Shafarevich conjecture for abelian varieties over $\mathbb{Q}$.
For elliptic curves, one can give a proof using ...
7
votes
0
answers
262
views
Factors of the Jacobian of modular curves
Let $J_1(p)$ be the Jacobian of the modular curve $X_1(p)$ for p an odd prime. We know that $J_1(p)$ is isogenous to a direct sum of abelian varieties $\oplus_{f}A_f$ where the sum runs over Hecke/...
4
votes
0
answers
468
views
Quaternion algebras in characteristic 2
Let $k$ be a field and let $Q$ be a quaternion algebra over $k$.
It is well known that, if $\mathrm{char}\,k\neq 2$, one can define $Q$ as the $k$-algebra of dimension $4$ generated by elements $x,y$ ...
- 2,493
3
votes
1
answer
531
views
Can an abelian variety dominate a variety of general type?
Let $X$ be a projective (not necessarily smooth) normal variety of general type over $\mathbb{C}$. Let $A$ be an abelian variety and let $A\to X$ be a surjective morphism.
Is $X$ zero-...
- 2,427
2
votes
0
answers
94
views
The quotient of a superspecial abelian surface by the involution
Let $E_i\!: y_i^2 = f(x_i)$ be two copies of a supersingular elliptic curve over a field of odd characteristics. Consider the involution
$$
i\!: E_1\times E_2 \to E_1\times E_2,\qquad (x_1, y_1, x_2, ...
- 1,833
3
votes
0
answers
409
views
Non algebraizable formal abelian schemes
I'd like to collect a good number of examples of formal schemes over $\text{Spf}(\mathbf{Z}_p)$, whose special fiber is a projective variety over $\mathbf{F}_p$, but that are not algebraizable.
If ...
CommunityBot
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3
votes
0
answers
78
views
Under what conditions are superspecial abelian surfaces isomorphic over a finite field?
Let $E_1$, $E_2$, $E_3$, $E_4$ be supersingular elliptic curves over a finite field $\mathbb{F}_{p^2}$, where $p$ is an odd prime. There is a well known theorem stating that over the algebraic closure ...
- 1,833
5
votes
2
answers
254
views
How to prove that a certain curve does not lie in a proper abelian subvariety of abelian variety
Let $E$ be an elliptic curve defined over $\mathbb{Q}$ by the equation
$$E: y^2=x^3-Ax+B=:f(x).$$
Consider the abelian variety $E^3:=E \times E \times E \subset \mathbb{P}^2 \times \mathbb{P}^2 \...
- 519
2
votes
1
answer
207
views
Subschemes in group action
Let $k$ be a field, of any characteristic. Let $G$ be a smooth group scheme over $k$ and let $X$ be a smooth scheme of finite type over $k$. Let $Y\subseteq X$ be a smooth subscheme of $X$, and let $...
- 23.4k
3
votes
2
answers
412
views
Generic Mumford Tate group and algebraic points
I will stick with a concrete example for this question, but it should probably be cast in a more general framework.
Let $Sym_g(\mathbf{C})$ be the space of symmetric matrices of order $g$ with ...
- 35.2k
6
votes
0
answers
284
views
Is there a finite number of supersingular genus 2 curves?
Consider an algebraically closed field $k$ of prime characteristic. It is widely known that up to $k$-isomorphism there is a finite number of supersingular elliptic curves over $k$. Let $C$ be a ...
- 1,833
8
votes
1
answer
750
views
flat/crystalline cohomology of abelian variety
Let $A/k$ be an abelian variety over an algebraically closed field and $\ell \neq \mathrm{char}\,k$.
In http://jmilne.org/math/articles/1986b.pdf, Theorem 15.1(b) it is proved that $$H^r_{et}(A, R) = \...
CommunityBot
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7
votes
2
answers
468
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Extending the Abel-Jacobi map over the DM-compactification $\overline{\mathcal{M}}_2$?
Let $\mathcal{M}_2$ be the moduli space of genus two curves and $\mathcal{A}_2$ the moduli space of principally polarized abelian surfaces. Then the Abel-Jacobi map gives an open embedding $\mathcal{M}...
- 13.6k
1
vote
0
answers
150
views
Translates of a line bundle on a complex $n$-torus
Suppose $\mathbb T:=V/\Gamma$ is a complex $n$-torus (i.e., $V$ is an $n$-dimensional $\mathbb C$-vector space and $\Gamma$ is a rank $2n$ lattice in $V$). Fix a holomorphic line bundle $L\in\text{Pic}...
- 1,761
9
votes
1
answer
833
views
Endomorphism ring of simple ordinary abelian variety
Is there an example of an ordinary and simple abelian variety $A$ over an algebraically closed field $K$ (of characteristic $p>0$) such that ${\rm End}(A)$ is not commutative? Note that the answer ...
- 790
2
votes
0
answers
198
views
schemes vs varieties in abelian varieties and maximal subscheme where line bundle is trivial
This is a very elementary question as I am just learning about abelian varieties by reading Mumford's book,
Let $X$ be a complete variety (irreducible and reducible over an algebraically closed field)...
- 785
19
votes
3
answers
2k
views
Bhargava's work on the BSD conjecture
How much would Bhargava's results on BSD improve if finiteness of the Tate-Shafarevich group, or at least its $\ell$-primary torsion for every $\ell$, was known? Would they improve to the point of ...
CommunityBot
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2
votes
1
answer
295
views
Chow groups modulo homological equivalence for abelian varieties
Let $X$ be an abelian variety over a field $k$.
Let $A^p_{\rm hom}(X)$ be the $p$-th Chow group of cycles modulo homological equivalence ($\ell$-adic, if $k$ is of char $p$).
Do we have $$A^p_{\rm ...
- 149k
3
votes
0
answers
58
views
Is the isogeny class 1109.a of abelian surfaces in the LMFDB complete?
The LMFDB lists the Jacobian of the genus-2 curve 1109.a.1109.1 (http://www.lmfdb.org/Genus2Curve/Q/1109/a/1109/1) as being isolated in its rational-isogeny class. However, the LMFDB does not purport ...
- 61
9
votes
1
answer
1k
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deformation theory in positive characteristic
The idea "Formal deformation theory in characteristic zero is controlled by a differential graded Lie algebra (dgla)" goes back to Goldman-Millson, Deligne, Drinfeld among others; see Lurie's ICM talk....
- 13.6k
37
votes
4
answers
12k
views
Finite extension of fields with no primitive element
What is an example of a finite field extension which is not generated by a single element?
Background: A finite field extension E of F is generated by a primitive element if and only if there are a ...
- 366
6
votes
0
answers
154
views
Descent via an explicit isogeny (genus 2)
This question is related to a previous question posted by me here answered by Prof. M. Stoll. 5-Descent or ($\sqrt{5}$-Descent?) on certain genus 2 Jacobians.
Here I ask some technicalities of a ...
11
votes
1
answer
508
views
Points of abelian varieties over purely transcendental extensions
I heard about the result in the theory of abelian varieties which says the following: given an abelian variety $X$ defined over a field $k$ and a purely transcendental extension $k\subset L\subset L'$ ...
- 28.8k
5
votes
1
answer
448
views
Tate-Shafarevich group over number fields
Let $A$ be an abelian variety over a number field $K$, $\text{Sha}(A/K)$ its Tate-Shafarevich group, $\ell$ a prime.
Is it known that the $\ell$-primary torsion subgroup $\text{Sha}(A/K)\{\ell\}$ is ...
- 26.1k
2
votes
0
answers
98
views
Why do "large" opens of abelian surfaces have "small" canonical bundle?
Let $A$ be an abelian surface, and let $S$ be a finite set of points. Let $U=A\setminus S$. Note that $U$ is a "large" open of $A$.
Let $B\to A$ be a proper birational surjective morphism with $B$ ...
- 57
3
votes
0
answers
112
views
Indecomposablity in purely inseparable extensions
Let $k$ be a field of characteristic $p$ (e.g. the separable closure of $\mathbb{F}_p(t)$) and consider the extension $k(x)/k$ where $x^p\in k$ but $x$ does not. Consider a (finitely generated) ...
- 63.9k
2
votes
0
answers
176
views
Trivial Tate modules
Let $A$ be an abelian group, and $p$ a prime.
I'll call $$T_p(A) := \text{Hom}_{\mathbf{Z}}(\mathbf{Q}_{p}/\mathbf{Z}_{p}, A).$$
If $A$ is finite, then $T_p(A)$ is trivial, but the converse is not ...
CommunityBot
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4
votes
0
answers
115
views
Relations between an projective variety and galois cohomology
Let $f_1, \cdots, f_k$ be homogeneous polynomials over $\mathbb{Q}[x_0, \cdots, x_n]$. They define an projective variety $X$ over $\mathbb{P}^n(\mathbb{C})$, namely their set of zeros $$X = Z(f_1, \...
- 423
4
votes
1
answer
388
views
subvarieties of abelian varieties over number fields
How "special" are closed subvarieties of abelian varieties over number fields? (Dimension 1 is easy.)
For example: Are there interesting families of varieties of general type which are not closed ...
- 9,497
0
votes
1
answer
151
views
Behaviour of (principal) polarizations of (singular) surfaces under birational maps
Assume we have two p.p. simple abelian surfaces $(A_i,D_i)$, i=1,2, over $\mathbb{C}$ with the following commutative diagram:
$\require{AMScd}
\begin{CD}
A_1 @>{birational}>> A_2\\
@V{2:1}VV ...
CommunityBot
- 1
7
votes
1
answer
540
views
Mod 2 modular forms in levels 5 and 25--how to account for this Hecke isomorphism?
The space $P1$ of my earlier question 203755 "Two spaces attached to mod 2 level 9 modular forms...", is essentially the space of mod 2 level 3 modular forms. That such a space should appear inside ...
- 5,422
8
votes
1
answer
414
views
Sha finiteness vs $\ell$-primary torsion
Where do I find a proof of the fact that over global function fields of characteristic $p>0$, finiteness of the Tate-Shafarevich group of an abelian variety is equivalent to finiteness of its $\ell$...
CommunityBot
- 1
2
votes
0
answers
254
views
Global sections of higher direct images
If $f : X\to V$ is a smooth proper map of smooth schemes, what are the global sections of
$R^if_{fppf, *}\mu_p$
$R^if_{fppf, *}\mathbb{G}_{\rm m}$
I was reading Milne's book "Arithmetic duality", ...
CommunityBot
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2
votes
1
answer
202
views
Identification of cohomology sheaf in the definition of the Kodaira-Spencer morphism for abelian schemes
Let $p:A \to S$ be a projective abelian scheme, where $S$ is some smooth scheme over a base field $k$. Then we have the Kodaira-Spencer morphism
$$
\kappa : T_{S/k} \to R^1p_*T_{A/S}
$$
where $T_{S/k}$...
- 16.2k
3
votes
1
answer
519
views
Néron models vs integral models
Let $X$ be a smooth projective $k$-scheme, $k$ being a number field. Let $\mathcal{O}_k$ be the ring of integers of $k$.
Fix a large enough category of schemes $\text{Sch}/k$ containing $X$, and ...
- 28.8k
2
votes
0
answers
69
views
A question about abelian varieties
For an abelian variety $A$ over a global field $K$, $\mathcal{A}$ its Néron model on $C$, either a smooth projective geometrically irreducible curve over a finite field, or the spectrum of the ring of ...
CommunityBot
- 1
3
votes
0
answers
176
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Component groups of commutative group schemes
I'm interested in the following question.
Suppose $P$ is a smooth commutative group scheme over a global field $k$, such that $P$ is separated and locally of finite type.
Suppose, in addition, $P^0$ ...
CommunityBot
- 1
7
votes
3
answers
927
views
Lefschetz fixed-point theorem for the Frobenius map
Where can one find a proof of Lefschetz fixed-point theorem for the Frobenius map on elliptic curves over algebraic closures of $F_{p}$ ?
This could immediately follow if their coholomogies (for the ...
- 156k
11
votes
2
answers
1k
views
Cotangent complex of perfect algebra over a perfect field
Let $A$ be a perfect $\kappa$-algebra over a perfect field $\kappa$ of positive characteristic $p$. Then the algebraic (= classical) cotangent complex $L_{A/\kappa}^{\operatorname{alg}}$ is known to ...
- 17.9k
4
votes
2
answers
340
views
On families of supersingular abelian surfaces over the projective line
Let $k=\mathbb{F}_q$. I recently learned that there are non-isotrivial families $f:X\to \mathbb{P}^1_k$ of supersingular abelian surfaces. In particular, the Kodaira-Spencer map of this family is non-...
- 9,497
3
votes
1
answer
127
views
Triviality of torsors after a field extension of bounded degree
Let $G$ be an abelian variety defined over a ring $R$. Is there a natural number $n$ such that, for any field $k$ over $R$ and any $G_k$-torsor $T$, there exists an extension $L/k$ of degree $n$ for ...
- 47.4k
4
votes
0
answers
184
views
Weil Pairing and Galois descent
One way the Weil pairing for an Abelian Variety $A/k$ is phrased is the following (for simplicity, let me only deal with the multiplication by $m$ map ($[m]: A\to A$) instead of arbitrary isogenies):
...
- 7,746
4
votes
1
answer
509
views
Intersection number of divisors on abelian surfaces and its invariance under translation by 2-Torsion points
I am reading Fulton's but I cannot find a useful result for my problem that seems to be something well known.
Let $\mathcal{J}$ be the Jacobian of a hyperelliptic curve of genus $2$. Consider $D_1,...
- 66.3k
3
votes
1
answer
404
views
Mumford-Tate groups of abelian surfaces
For elliptic curves, one may easily compute Mumford-Tate groups; there are just two cases:
1) $E$ has no complex multiplication, and the Mumford-Tate group of $E$ is $GL_2$
2) $E$ has complex ...
- 5,050
11
votes
4
answers
3k
views
Torsion of an abelian variety under reduction.
Let $p$ be a prime. Suppose you have an Abelian scheme $A$ over $Spec\ \mathbb{Z}_p$. How do you prove that if $q$ is another prime, then the $q$-torsion of $A$ injects into the torsion of $A_p$, ...
- 65.4k
4
votes
0
answers
210
views
Universal vectorial bi-extension as a scheme
In 'The universal vectorial Bi-extension and p-adic heights' Coleman works with the pullback of the Poincaré biextension of an abelian variety A to its universal vectorial extension and claims this is ...
- 63.9k
9
votes
1
answer
383
views
What is the essential image of $AbVar$ in $p-div$?
Given an abelian variety $A$ over a base scheme $\text{Spec } \mathcal{O}_{K_p}$, we define the functor $P$ as taking $A \mapsto \text{colim}_n A[p^n]$, its associated $p$-divisible group. What is the ...
- 3,539