All Questions
1,203 questions
19
votes
1
answer
2k
views
The Infinitesimal topos in positive characteristic
This question was inspired by and is somewhat related to this question.
In his article "Crystals and the de Rham cohomology of schemes" in the collection "Dix exposes sur la cohomologie ...
- 19.4k
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 ...
- 63.9k
2
votes
1
answer
190
views
Induced action on Prym variety
Let $C$ be a smooth projective curve of genus $g$ with an involution $\iota: C \to C$. We have the quotient map $\pi: C \to C/\iota$, with $C/\iota$ a smooth curve of genus $h$.
The pullback map $\pi^...
- 149k
2
votes
0
answers
117
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 \...
- 443
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.
...
- 19.4k
5
votes
2
answers
391
views
Abelian variety with CM defined over real numbers
Is there an abelian variety $A/\mathbb R$ of dimension $n$ such that $End_{\mathbb R}(A)\otimes \mathbb Q$ contains a field $K$ of degree $[K:\mathbb Q]=2n$? ($End_{\mathbb R}(A)$ is the ring of $\...
- 1,566
4
votes
1
answer
840
views
Nakai-Moishezon theorem for abelian varieties
In Birkenhake and Lange's book, they prove a version of the Nakai-Moishezon theorem for complex abelian varieties that says that if $L_0$ is an ample line bundle on a complex abelian variety $X$ of ...
- 781
8
votes
0
answers
201
views
Monodromy groups that are profinitely dense in Sp(2g,Z)
$\DeclareMathOperator\Sp{Sp}$Assume $g\geq 2$. It is known that there exist finitely generated subgroups of $\Sp(2g,\mathbb{Z})$ of infinite index that surject onto all finite quotients of $\Sp(2g,\...
- 63.9k
8
votes
0
answers
688
views
An Azumaya algebra from a vector bundle, and a construction of Belov-Kanel and Kontsevich
Let $S/k$ be a scheme over a perfect field $k$ of characteristic $p>0$.
In Automorphisms of the Weyl Algebra, Belov-Kanel and Kontsevich write down the map
$$\alpha: H^0(\Omega^1_{S/k}/d\mathcal O) ...
- 9,650
5
votes
1
answer
534
views
Ordinary abelian varieties and Frobenius eigenvalues
Say $A_0$ is an ordinary abelian variety over ${\mathbf{F}}_q$. Call $\mathcal{A}$ the canonical lift of $A_0$ over $R := W({\mathbf{F}}_q)$. It carries a lift of the $q$-th power map on $A_0$. We ...
- 149k
4
votes
0
answers
189
views
If the Frobenius endomorphism of a characteristic $p$ ring is epimorphic, is it surjective?
MO question 19282 is about integral epimorphisms of commutative rings, and a counterexample is given to surjectivity. What about the case of the Frobenius endomorphism of a commutative, characteristic ...
- 4,492
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
3
votes
1
answer
245
views
Can non-geometrically reduced reduced subschemes happen for reductive groups?
The title is meant to be punchy, but also a tongue-in-cheek acknowledgement of the prevalence of ‘reduce’-derived words in this area. (Unfortunately, I overlooked the fact that the question in the ...
- 12.9k
6
votes
2
answers
417
views
Distribution of dimensions of factors of the Jacobian of X_0(p)
Let X_0(p) be the modular curve of level p where p is prime. The Jacobian variety J_0(p) has a natural family of quotients defined over Q with dimensions summing to dim(J_0(p)), each quotient ...
- 2,224
4
votes
0
answers
144
views
How often is the rank of J_0(p)^- zero
As mentioned in this answer there is a conjecture by
Kimball Martin that, formulated slightly informally, has the following special case.
Conjecture:
On average $J_0(p)$ has 2 simple components when ...
- 2,224
4
votes
0
answers
262
views
de Rham Bloch-Ogus theory in positive characteristic
In their famous paper Gersten's conjecture and the homology of schemes, one of the results that Bloch and Ogus prove is that the second page of the coniveau spectral sequence for $X$ smooth over a ...
- 2,054
1
vote
0
answers
78
views
Elliptic fibrations on some Kummer surface in characteristic $2$
In the question I ask about one elliptic fibration on the surface
$$
K\!: y^2 + x_1x_2y = (x_1x_2)^2(x_1 + x_2 + 1) + (x_1 + x_2)^2.
$$
over a finite field $\mathbb{F}_q$ of characteristic $2$ such ...
- 1,833
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
3
votes
1
answer
188
views
Maximal closed subscheme stable under the action of a finite connected group scheme
Let $k$ be a field of characteristic $p>0$, $X$ a smooth projective $k$-variety and $Y\subseteq X$ a closed irreducible subvariety. Let $G$ be a connected finite $k$-group scheme acting on $X$.
...
- 7,377
3
votes
1
answer
233
views
Is the Ueno fibration smooth?
Let $A$ be an abelian variety over $\mathbb{C}$ and let $X\subset A$ be a closed subvariety. Let $X\to Y$ be the Ueno fibration. (That is, $Y$ is of general type and a closed subvariety of $A/B$ where ...
- 313
5
votes
1
answer
419
views
Lifting $\mathfrak{sl}_2$-triples
Let
$k$ be an algebraically closed field,
$G$ a (smooth, connected) reductive algebraic group over $k$,
$H$ a (smooth, connected) reductive group of semisimple rank 1, and
$T$ a maximal torus in $H$.
...
- 631
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 ...
- 12.9k
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 ...
- 4,492
4
votes
1
answer
237
views
Mirror partners of some Calabi-Yau threefolds
I don't have experience in mirror symmetry, hence I am not sure that my question is of research level. Sorry in advance.
Let $k$ be an algebraically closed field of characteristic $\neq 2, 3$. ...
- 426
2
votes
1
answer
608
views
Do we have Hodge symmetry for char $p$?
Let $X$ be a smooth projective variety over a field $k$. Let $h^{p,q}=dim_k H^q(X,\Omega_{X/k}^p)$ be the Hodge numbers.
If $k$ is of char $0$, by Lefschetz principle, we always have Hodge symmetry, i....
- 28.8k
4
votes
1
answer
336
views
What is the involution on the moduli space of genus 3 curves induced by the Torelli map
Let $M_g$ be the moduli space of genus $g$ curves, $A_g$ be the moduli space of principally polarized dimension $g$ abelian varieties. They have dimensions $3g-3,g(g+1)/2$ respectively. The Torelli ...
- 40.3k
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
8
votes
2
answers
704
views
Isomorphic endomorphism algebras implies isogenous (for abelian varieties over finite fields)?
$\newcommand{\F}{\mathbb{F}}
\newcommand{\End}{\mathrm{End}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}$
I would like to know if the following is true:
Proposition A : Let $A_1, A_2$ ...
- 5,050
14
votes
1
answer
570
views
Weil pairing on abelian varieties and etale Chern classes
Given a line bundle $L$ on an abelian variety $A/k$, there is an associated Weil pairing $e_L\colon\bigwedge^2V_pA\to\mathbb Q_p(1)$, where $p$ is a prime different from the residue characteristic of ...
- 5,050
7
votes
1
answer
400
views
Does perfect fraction field imply perfect residue field?
Let $A$ be a local integral domain of characteristic $p$. Let $K$ be the fraction field and let $k$ be the residue field of $A$. If $K$ is perfect, is $k$ necessarily perfect?
Thoughts:
If $A$ is ...
- 28.2k
9
votes
2
answers
910
views
Example of a smooth projective family of varieties in characteristic $p$ where the Hodge numbers jump
Let $\mathbb F$ be an algebraic closure of the field of order $p$. Let $S=\textrm{Spec}(\mathbb F[[z]])$ with special point $s$ and generic point $\eta$. I'm looking for an example of a smooth ...
- 2,607
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
4
answers
3k
views
"Albanese" schemes: When does an "initial abelian scheme" exist under a given scheme?
For a variety V, its Albenese variety Alb(V) is a variety with a map V → Alb(V) which factors uniquely into any map from V to an abelian variety. Can we say something similar for an arbitrary ...
- 4,492
18
votes
2
answers
1k
views
Albanese variety over non-perfect fields
It is a result of Serre (Morphismes universels et varietes d'albanese) that the Albanese (abelian) variety, i.e. an initial object for morphisms to (torsors over) abelian varieties, exists for any ...
- 4,492
2
votes
1
answer
282
views
Finite quotients of the absolute Galois group of $\mathbb{Q}$ via torsion of elliptic curves
Let $E/\mathbb{Q}$ be an elliptic curve and let $p$ be a prime. Then there is an action of the absolute Galois group of $\mathbb{Q}$ on $E[p]$ that factors through a finite quotient.
Does any finite ...
- 149k
19
votes
1
answer
2k
views
Perfectoid approach to resolution of singularities in char $p$
Since perfectoid techniques have built a bridge between char $0$ and char $p$ worlds, it is conceivable that they can be applied to resolution of singularities in char $p$ using their successful ...
- 21.3k
4
votes
1
answer
199
views
Jacobians $\mathbb{F}_q$-isogenous to the direct square of an ordinary elliptic $\mathbb{F}_q$-curve of $j$-invariant $0$
Consider an ordinary elliptic curve $E_b\!: y^2 = x^3 + b$, of $j$-invariant $0$ over a finite field $\mathbb{F}_q$, such that $\sqrt{b} \not\in \mathbb{F}_q$.
Question. What are some examples of ...
- 66.3k
4
votes
1
answer
638
views
perfect fields in positive characteristic
Let $k$ be an infinite perfect field in positive characteristic $p$, i.e. every element of $k$ is a $p$th power. I am interested in properties of finite fields that can be extended to $k$. For example:...
- 168
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
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
5
votes
0
answers
328
views
Ramification behavior of field given by adjoining $p$-torsion point on formal group of abelian variety
Setup. Let $p > 2$ be a prime, let $K$ be the completion of the maximal unramified extension of $\mathbb{Q}_p$, and fix an algebraic closure $\overline{K}$ of $K$. Let $A/K$ be an abelian variety ...
- 998
8
votes
0
answers
300
views
Which algebra is $\mathbb{F}_p \otimes_{HH^{\cdot} (\mathbb{F}_p)} \mathbb{F}_p$?
Let $HH^{\cdot}(\mathbb{F}_p)$ be the Hochschild cohomology of $\mathbb{F}_p$ over $\mathbb{Z}$, which as a $E_1$ ring is simply $\mathbb{F}_p[x]$ with $x$ in cohomological degree $2$. Then it's clear ...
- 2,356
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
4
votes
1
answer
280
views
Semisimplicity of the étale cohomology mod $p$
Let $X$ be a smooth projective variety over a field $k$. Then if $\ell\neq \text{char} k$, $k$ is finite, and $X$ is an abelian variety it was shown by Weil that the $\ell$-adic cohomology of $X_{k^{...
- 86
3
votes
3
answers
423
views
Are there characteristic-dependent Betti numbers in characteristic not equal to two?
Let $\Delta$ be a finite simplicial complex on $n$ vertices. Let $S = \mathbb{k}[\mathbf{x}]$ be a polynomial ring over a field $\mathbb{k}$ in $n$ variables and $I$ be the Stanley-Reisner ideal of $\...
- 150
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
11
votes
1
answer
618
views
Canonical lift of the deformation of an ordinary abelian variety
If $A/k$ is a principally polarised ordinary abelian variety ($k$ a perfect field of characteristic $p$, we may assume it is finite for simplicity), we have a canonical lift $\hat{A}/W(k)$.
Now if I ...
- 16.2k
2
votes
0
answers
61
views
The intersection form on a Jacobian
$\DeclareMathOperator{\End}{End}$
Let $J=\mathop{Jac}(C)$ be a Jacobian, $\Theta$ the theta divisor. Since it is principal, it induces a bijection between the Néron-Severri group $NS(J)$ and the ...
- 645
8
votes
1
answer
172
views
Non-degenerate points on a Jacobian surface
Let $C$ be a (hyperelliptic) genus $2$ curve over a number field $K$ with a $K$-rational Weierstrass point $\infty$. We embed $C$ in its Jacobian $J$ via $\infty$.
Question: Is there a quadratic ...
- 149k
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