All Questions
1,203 questions
37
votes
3
answers
5k
views
Is there a nice proof of the fact that there are (p-1)/24 supersingular elliptic curves in characteristic p?
If $k$ is a characteristic $p$ field containing a subfield with $p^2$ elements (e.g., an algebraic closure of $\mathbb{F}_p$), then the number of isomorphism classes of supersingular elliptic curves ...
- 45.7k
27
votes
4
answers
3k
views
Have people successfully worked with the full ring of differential operators in characteristic p?
This question is inspired by an earlier one about the possibility of using the full ring of differential operators on a flag variety to develop a theory of localization in characteristic $p$. (Here ...
- 57.6k
4
votes
1
answer
746
views
Vanishing of $\text{Ext}^2$ sheaf from abelian variety to multiplicative group
Does anyone know a proof or reference for the following statement? Or if it's false (which seems unlikely to me), a counterexample?
Let $k$ be a field (maybe we need it to be perfect) and $A$ an ...
- 3,605
15
votes
7
answers
2k
views
Examples of rational families of abelian varieties.
I'd like to know examples of non-trivial families of abelian varieties over rational bases (e.g. open subschemes of the projective line P^1).
One can generate many examples as Jacobians of rational ...
- 10.5k
1
vote
0
answers
213
views
Isogeny and canonical isomorphism of global highest differential forms
Let $A$ and $B$ be Abelian Varieties of dimension $d$ over a local field $K$. Let $\phi : A \rightarrow B$ be an isogeny and $\phi^{\prime}$ its dual.
Recall that one has a canonical isomorphism $...
13
votes
0
answers
749
views
Rings whose Frobenius is flat
Let $R$ be a ring of characteristic $p>0$. The (absolute) Frobenius is the map of rings $F_R:R\rightarrow R$ defined by $x\mapsto x^p$.
I am interested in rings for which $F_R$ is flat (hence ...
anon1432
1
vote
0
answers
242
views
Harmonic forms on a complex torus
Let $T=\mathbb{C}^3/\Lambda$ be a complex torus of our interest and $L$ be a holomorphic line bundle on $T$, I am interested in $H^{0,2}_{\bar\partial_L}(T,L)$, i.e., the $(0,2)$ harmonic forms taking ...
- 954
1
vote
0
answers
164
views
From a factor of automorphy on an abelian variety to a divisor
Given a complex abelian variety $A = V/\Gamma$ (for $\Gamma$ being a lattice in the complex vector space $V$), one knows how to describe a holomorphic line bundle in terms of factors of automorphy: By ...
- 12.1k
2
votes
0
answers
540
views
Polarizations in algebraic and symplectic geometry
In context of Abelian varieties there are a couple of equivalent ways to
introduce the polarization of a algebraic variety. One way is to
choose a line bundle $\mathcal{L}$ which satisfies certain ...
- 6,006
8
votes
1
answer
2k
views
Why is, for a group scheme of finite type, "smooth" (resp. irreducible) equivalent to "geometrically reduced" (resp. geometrically irreducible)?
I have some questions about two statements from Bosch's "Algebraic Geometry and Commutative Algebra" about algebraic varieties (page 479). Since I still don't have the permission to add images I quote ...
- 6,006
19
votes
1
answer
977
views
Lang's Jacobian identity: slicker, elementary proof?
In Jeffrey Lang, A Jacobian identity in positive characteristic, J. Commut. Algebra, Volume 7, Number 3 (2015), pp. 393--409, the following result is proven:
Theorem 1. Let $p$ be a prime. Let $\...
- 33.8k
3
votes
0
answers
87
views
Two pairings on the group $K(L)$ associated with a non-degenerate line bundle $L$ on an abelian variety
Let $A$ be an abelian variety over a field and let $L$ be a non-degenerate line bundle on $A$.
Then $L$ gives rise to a morphism $\lambda:A\to A^*$ from $A$ to its dual.
As usual, let $K(L):=\ker(\...
- 3,076
4
votes
2
answers
460
views
Are Chow groups invariant under universal homeomorphisms?
Let $f:X\to Y$ be a universal homeomorphism of schemes, $R$ a coefficient ring. Which assumptions on $f$ and $R$ are suffient to ensure that the pullback map $f^*$ of $R$-linear Chow groups is ...
- 16.9k
6
votes
1
answer
348
views
Extensions of (semi-)abelian schemes
Let $S$ be a regular Noetherian scheme, and let $U\subset S$ be the complement of a divisor. If $A\to B$ is an isogeny of abelian schemes over $U$, and $A$ extends to a semi-abelian scheme over $S$, ...
- 63
2
votes
0
answers
274
views
Generic rank of proper pushforward of the trivial line bundle
Given a proper surjective morphism $f:X\rightarrow Y$ where $X$ and $Y$ are smooth projective varieties. The proper pushforward $f_!$ is the homomorphism that sends the class of a coherent sheaf $M$ ...
- 5,901
9
votes
1
answer
456
views
Isomorphic Jacobian Varieties Just Like Abelian Varieties — Torelli's Theorem
Torelli's theorem states:
Let $R$, $R'$ be compact Riemann surfaces of genus $g$, $J(R)$, $J(R')$ their Jacobian varieties, $\Theta$, $\Theta'$ their respective theta divisors. The Riemann surfaces ...
- 560
12
votes
2
answers
883
views
Residues of $\frac{1}{\prod_{i=1}^n (x-P_i)^{e_i}}$
This is a problem occurring in my research about deformations of $\mathbb{Z}/p^n$-covers over a ring of power series. Given an algebraically closed field $k$ of characteristic $p>0$, suppose $1< ...
- 245
6
votes
1
answer
373
views
For an abelian scheme, $R^pf_* \Omega^q$ is locally free and its formation is compatible with any base change
Let $k$ be a field, $\bar{R} \to R$ a local homomorphism of artinian local rings with the residue fields $k$, $I$ its kernel, $A/R$ an abelian scheme, and $\mathscr{T}$ its tangent sheaf.
Let $A_0 = A ...
- 1,364
15
votes
2
answers
596
views
When is the etale cohomology of $\mathrm{Sym}^n(X)$ isomorphic to the $\Sigma_n$-invariants in the étale cohomology of $X^n$?
Suppose $X$ is a smooth projective variety defined over an arbitrary algebraically closed field $k$, and consider the action of $\Sigma_n$ on the $n$-fold product $X^n$. Is it true that $H_{\acute{e}t}...
- 233
1
vote
0
answers
97
views
$p$-power torsion points of abelian varieties along $p$-adic Lie extensions
Let $p$ be a prime and $K$ be a number field. Let $K_\infty$ be a uniform $p$-adic Lie extension of dimension $l$ over $K$ with unique intermediate fields $K_n$ of degree $p^{nl}$ over $K$. We ...
- 81
3
votes
0
answers
145
views
Richelot isogenies in characteristic $2$
I am interested in Richelot isogenies between ordinary abelian surfaces in characteristic $2$. If I am not mistaken, the corresponding theory is developed in Article "J.-B. Bost, J.-F. Mestre, ...
- 1,833
2
votes
0
answers
175
views
Covering abelian varieties over finite fields with the product of curves
Question. Given an $n$-dimensional abelian variety $X$ over a finite field, is it possible to find smooth projective curves
$C_1,\ldots, C_n$ such that there exists a finite regular map
$C_1\times \...
- 5,901
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
11
votes
1
answer
340
views
Lifting a splitting of an Abelian variety to characteristic 0
Let $R$ be the ring of integers in a (complete) algebraic closure of $\mathbb Q_p$ with maximal ideal $\mathfrak p$. Suppose I have an Abelian surface $\mathcal A/R$ such that over every $R/\mathfrak ...
- 7,746
4
votes
1
answer
269
views
Fourier transform on finite groups in characteristic $p>0$
Is there a Fourier theory for finite groups in characteristic $p>0$? Assume that $p$ divides the order $|G|$ of finite groups (or just work with $p$-groups), i.e., in a modular representation-...
- 41
0
votes
1
answer
480
views
Why does MAGMA claim that the automorphism group of a curve is trivial?
I have been trying to compute the Automorphism group of a curve using MAGMA with no success. This is what I have tried: I have tried to compute the Automorphism group of the curve $y^3=x^4-x$ and no ...
- 269
0
votes
0
answers
172
views
Pullback of algebraic $K$-theory along the surjection of abelian varieties
Given a surjective homomorphism of abelian varieties $f:A\rightarrow B$ where $\text{dim}(A)>\text{dim}(B)$, does $f^*$ induce a rational injection of algebraic $K$-theory? According to the ...
- 5,901
4
votes
0
answers
213
views
Computing homology class of curve in product of elliptic curves
I have a smooth, projective curve $X/\mathbb{C}$ of genus $g$, embedded in a product of elliptic curves $A = \prod_{i=1}^g E_i$. Since $H_*(A; \mathbb{Z})$ with the Pontryagin product is isomorphic to ...
- 1,856
3
votes
0
answers
197
views
How to write down an explicit equation of given degree yielding a smooth hypersurface in a projective space?
Let F be a field of positive characteristic $p$ and let $d,n$ be two positive integers.
Can we explicitly write down an equation defining a smooth hypersurface $X_d⊂\mathbb P^n_F$ of degree d ?
This ...
- 417
3
votes
1
answer
634
views
Given a semi-abelian scheme, is the set of points such that the fibres are abelian varities open?
Let $\pi:\mathcal{A}\rightarrow C$ be a semi-abelian scheme, i.e. $\mathcal{A}$ is a smooth separated commutative group scheme over $C$ via $\pi$ with geometrically connected fibres, such that each ...
- 452
5
votes
2
answers
478
views
Isogeny components of Jacobians of étale covers
Let us work over $\mathbb{C}$. Fix $X$ a smooth projective curve of genus at least $2$. For every simple abelian variety $A$, it is easy to come up with a ramified covering $Y\to X$ with a non-...
- 1,000
10
votes
0
answers
294
views
Relation between Faltings height and height on moduli space
Let $E$ be an elliptic curve over a number field $K$. The difference between the semistable Faltings height $h_F(E)$ of $E$ and the height $h(j_E)$ of the $j$-invariant of $E$ can be bounded in terms ...
- 101
3
votes
1
answer
282
views
Frobenius splitting for an excellent, non $F$-finite, $F$-pure hypersurface
Let $p$ be an odd prime. Let $k$ be a field of characteristic $p$ such that $[k:k^p]=\infty$ (i.e. $k$ is not $F$-finite ) .
Also assume that $-1$ is not a square in $k$ . Consider the homogeneous ...
- 642
9
votes
0
answers
439
views
Uncountably many non-isomorphic Tate modules
Do there exist uncountably many abelian surfaces with good reduction over $\mathbb{Q}_p$ with pairwise non-isomorphic rational $p$-adic Tate modules?
If we took $l$-adic Tate modules there would be ...
user158636
48
votes
5
answers
15k
views
Algebraically closed fields of positive characteristic
I'm taking introductory algebraic geometry this term, so a lot of the theorems we see in class start with "Let k be an algebraically closed field." One of the things that's annoyed me is that as far ...
- 12.6k
1
vote
1
answer
441
views
About the type of a polarization of an abelian variety
The following is a question I posted about a week ago on Maths stackexchange there, but it didn't bring any discussion nor comment. For this reason I am posting it here also.
Let $X$ be an abelian ...
- 769
10
votes
1
answer
563
views
Are there infinitely many real multiplication fields of abelian surfaces over $\mathbb Q$?
Do there exist infinitely many real quadratic fields $F$ such that there is an abelian surface $A$ over $\mathbb Q$ whose ring of endomorphisms, tensored with $\mathbb Q$, is $F$?
Do there exist ...
- 149k
26
votes
4
answers
6k
views
Why do people think that abelian varieties are the hardest case for the Hodge conjecture?
Today, I heard that people think that if you can prove the Hodge conjecture for abelian varieties, then it should be true in general. Apparently this case is important enough (and hard enough) that ...
- 16k
8
votes
0
answers
196
views
Simple abelian varieties of $\mathrm{GL}_2$ type with positive rank and large dimension
$\DeclareMathOperator\GL{GL}$I would like know if there are known constructions of simple abelian varieties of $\GL_2$ type of arbitrarily large dimension and positive Mordell-Weil rank, whose rank is ...
- 2,224
4
votes
0
answers
130
views
Castelnuovo–Mumford regularity and wedge powers in positive characterisitc
A vector bundle on $\mathbb{P}^n$ is said to be $r$-regular if
$$H^i(\mathbb{P}^n,F(r-i))=0$$ for all $i>0$.
It is always true that if $F$ is $r$-regular and $G$ is $s$-regular (both vector bundles)...
- 2,356
18
votes
1
answer
1k
views
On Tate's "Endomorphisms of Abelian Varieties over Finite Fields", sketch of proof of main result?
Let $k$ be a field, $\overline{k}$ its algebraic closure, and $A$ an abelian variety defined over $k$, of dimension $g$. For each integer $m \ge 1$, let $A_m$ denote the group of elements $a \in A(\...
user97565
2
votes
0
answers
96
views
Linear forms and the second Voronoi decomposition
This is not my area of expertise, so forgive me if the question is a bit naive. Given a collection of vectors $v_1,\ldots,v_d$ in $\mathbb{R}^n$ (with $d\geq n$), there is a corresponding set of ...
- 147
1
vote
0
answers
125
views
Different ways to construct the isogeny category of abelian varieties
Let $k$ be a field and let $\mathbf{AV}_{/k}$ be the category of abelian varieties over $k$. I'm interested in different definitions of the isogeny category of $\mathbf{AV}_{/k}$.
Of course, the ...
- 804
5
votes
0
answers
160
views
The image of a curve under the multiplication endomorphism of its Jacobian
Let $X$ be a complex smooth projective curve of genus $g\geq 2$. Embed $X$ in its Jacobian
${\rm{J}}(X)\cong{\rm{Div_0}}(X)/{\rm{Div_p}}(X)$ where ${\rm{Div_0}}(X)$ is the group of degree zero ...
- 3,599
9
votes
2
answers
868
views
Nakano vanishing in positive characteristic
Let $X$ be a smooth projective variety defined over a field $k$.
In characteristic zero, the following is a special case of the (Kodaira-Akizuki-)Nakano vanishing theorem:
$(\ast) \quad$ $\mathrm H^...
- 1,072
3
votes
1
answer
673
views
Endomorphisms of abelian varieties with real multiplication
Let us work over $\mathbb{C}$ to make life easier.
I've came across to the following definition. Let $F$ be a totally real number field of degree $g$, with ring of integers $\mathcal{O}_F$. An ...
user120787
3
votes
1
answer
317
views
What is known about lower etale cohomology of unirational varieties?
Which information is currently known about $H^1_{et}(X,\mathbb{Z}_l)$ and $H^2_{et}(X,\mathbb{Z}_l)$, where $X$ is a smooth unirational variety over an algebraically closed field of finite ...
- 16.9k
11
votes
2
answers
1k
views
Representations of $\mathrm{SL}(2)$ in characteristic 2
$\DeclareMathOperator\SL{SL}$In characteristic zero one can use the Clebsch-Gordan rule to decompose tensor products of $\SL(2)$-modules. In characteristic $p$, things are more complicated.
I am ...
- 605
3
votes
1
answer
251
views
Do Neron-Severi groups of smooth projective unirational varieties contain $\ell$-torsion?
Let $X$ be a smooth projective unirational variety over an algebraically closed field of characteristic $p>0$, and $\ell\neq p$ a prime. My question: can the Neron-Severi group of $X$ contain (non-...
- 16.9k
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 ...
- 2,011