All Questions
1,203 questions
2
votes
0
answers
162
views
Faltings' height theorem for isogenies over finite fields
For an Abelian scheme over a ring of integers in a number field, Faltings has a theorem that describes how the Faltings' height changes through an isogeny. There are multiple references for this ...
- 7,746
28
votes
3
answers
2k
views
Intuitive pictures in characteristic p
This is a tough one, but does anyone know of any images that recall characteristic p geometry (over algebraically closed fields) in some sense? It is not enough if it is some picture that can be also ...
- 2,215
5
votes
2
answers
281
views
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 ...
3
votes
1
answer
426
views
Purely inseparable isogeny
How to prove purely inseparable isogeny between two abelian varieties is radical (universally injective)? Purely inseparable morphism means the extension between the two function fields is purely ...
- 91
1
vote
0
answers
325
views
Compatiblity of completion and fibre products. (Formal completion and formal groups)
Let $S$ be a scheme (not necessarily locally noetherian), $X$ a smooth separated group scheme over $S$, and $\hat{X}$ be the formal completion along with the identity section.
Then does the group ...
- 1,364
2
votes
1
answer
184
views
Is there a way to find any $\mathbb{F}_2(t)$-point on the elliptic curve $\mathcal{E}$?
Consider the ordinary elliptic curves
$$
E\!:y_1^2 + x_1y_1 = x_1^3 + 1,\qquad E^\prime\!: y_2^2 + x_2y_2 = x_2^3 + x_2^2 + 1
$$
over the field $\mathbb{F}_2$. They are quadratic twists to each other....
- 1,833
1
vote
1
answer
179
views
Is the Jacobian isogenous over $\mathbb{F}_p$ to the direct product of the elliptic curves?
Let $\mathbb{F}_p$ be a finite field such that $p \equiv 1 \ (\mathrm{mod} \ 3)$ and $p \equiv 3 \ (\mathrm{mod} \ 4)$. Consider the Jacobian of the hyperelliptic curve $C\!: y^2 = (x^3 + b)(x^3-b)$, ...
- 1,833
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
3
votes
0
answers
232
views
Lifting a Frobenius endomorphism under an étale morphism
Let $X$ be a smooth affine scheme over $\mathbb{Z}/{p^2}$ that is a complete intersection, say $X$ is the spectrum of $\mathbb{Z}/{p^2}[x_1,...x_n]/(f_1, ... f_r)$, where $n-r$ is the dimension of $X$....
- 111
9
votes
1
answer
415
views
Analogue of the original Birch–Swinnerton-Dyer conjecture for abelian varieties
$\newcommand{\Q}{\Bbb Q}
\newcommand{\N}{\Bbb N}
\newcommand{\R}{\Bbb R}
\newcommand{\Z}{\Bbb Z}
\newcommand{\C}{\Bbb C}
\newcommand{\F}{\Bbb F}
\newcommand{\p}{\mathfrak{p}}
$
Let $A$ be an abelian ...
- 1,742
4
votes
3
answers
610
views
General curves of genus 3 as plane sections of Kummer surfaces
Is it true that a general curve of genus 3 is a plane section of an appropriate Kummer surface in $\mathbb P^3$? By Kummer surface I mean image of a principally polarized Abelian surface w.r.t. the ...
- 1,939
1
vote
0
answers
256
views
Deformation of $p$-divisible group
To try to understand the deformation of $p$-divisible group more explicit, I am thinking given a connected $p$-divisible group $G_0$ on $\overline{\mathbb{F}_q}$, Choose a deformation $G$ of $G_0$ ...
- 397
3
votes
1
answer
252
views
Is this unipotent group, over characteristic 2, connected?
Let $E_{ij}(x)\in \mathrm{Mat}_{7\times7}(\overline{\mathbb{F}}_2)$ be the matrix with zeros everywhere, except for the value $x$ at $ij$. Set $$a(x)=1+E_{12}(x)+E_{34}(x)+E_{56}(x),\quad b(y)=1+E_{23}...
- 4,593
18
votes
1
answer
1k
views
Do all simple factors of jacobians of curves come from correspondences?
For this question I will let the overly ambiguous word curve mean: smooth projective and connected curve over $\mathbb C$ (or equivalently a smooth compact Riemann-Surface).
Let $C$ be a curve over ...
- 2,224
0
votes
0
answers
165
views
Elliptic curves and archimedean place
here is my question :
Let $K$ be any field, $ E \to spec(K)$. Let $ v \in M_K $ an archimedean place.
We know that $ \overline{K_v} \simeq \mathbb{C}$ and there exists $\tau_v \in \mathbb{H}$ such ...
7
votes
1
answer
812
views
Galois action on $p$-adic Tate module of Abelian variety over finite field semisimple?
Let $A,B$ be positive dimensional Abelian varieties over a finite field and $p$ be an arbritrary prime. By Zarhin, Homomorphisms of abelian varieties over finite fields http://www.math.nyu.edu/~...
user19475
9
votes
1
answer
1k
views
Coderivations of $S(V)$ correspond to linear maps $S(V) \to V.$ Only over characteristic $0$?
Definition. Let $k$ be a commutative ring. Let $V$ be a $k$-module. We turn the symmetric algebra $\mathrm{S}\left(V\right)$ of $V$ into a graded Hopf algebra by defining the comultiplication
\begin{...
- 33.8k
11
votes
2
answers
653
views
Abelian variety with prescribed endomorphism ring
Consider the cyclotomic field $L={{\mathbb{Q}}}(\zeta_8)={{\mathbb{Q}}}(\sqrt{2},i)$, where $\zeta_8$ is a primitive 8-th root of unity. Let $\Lambda={{\mathbb{Z}}}[\zeta_8]$ denote the ring of ...
- 14.2k
13
votes
1
answer
675
views
Is an abelian variety with a Galois invariant, rank one submodule of its Tate module, CM?
Let $A$ be an absolutely simple abelian variety over a number field $K$. Assume that, for some prime $p$, the Tate module $T_p A$ has a submodule of rank one, invariant under the absolute Galois group ...
- 30.5k
9
votes
1
answer
1k
views
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....
- 528
24
votes
3
answers
3k
views
Products of primitive roots of the unity
Let $m>2$ be an integer and $k=\varphi(m)$ be the number of $m$-th primitive roots of the unity. Let $\Phi = \{ \xi_1, \ldots, \xi_{k/2}\} $ be a set of $k/2$ pairwise distinct primitive $m$-th ...
- 8,838
1
vote
0
answers
194
views
Nef divisors on abelian varieties are pullbacks of ample ones
It is well known that for any polarization ( that is, ample line bundle) $L$ on an abelian variety $A$, there is an isogeny $\phi\colon A \to B$ to another abelian variety with a principal ...
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 ...
- 3,076
6
votes
1
answer
535
views
Quaternion algebra actions on $\ell$-adic cohomology
Let $E$ be a supersingular elliptic curve over $\mathbf{F}_p$, and $H$ its endomorphism algebra $\text{End}(E)\otimes_{\mathbf{Z}}\mathbf{Q}$, a quaternion algebra (non split at $p$ and $\infty$).
...
user132229
4
votes
0
answers
169
views
Extensions of fraction field and residue field
Let $A \subset B$ be integrally closed local domains, $K(A), K(B)$ be fields of fractions, and $k(A),k(B)$ be residue fields. How to prove $[K(B):K(A)] \ge [k(B):k(A)]$?
This question should be easy ...
- 393
1
vote
1
answer
230
views
On a refinement of Mordell's conjecture for curves
Let $C$ be an algebraic curve of genus $g \geq 2$, defined over $\overline{\mathbb{Q}} \subset \mathbb{C}$. It is then defined over a finite extension $K$ of $\mathbb{Q}$. We assume that $C(K) \ne \...
- 27k
-2
votes
1
answer
282
views
Existence of divisor in the Jacobian of smooth curve of genus two whose intersection with theta divisor is 1
Let $C$ be a smooth projective curve of genus $2$ and $J$ denotes the Jacobian of $C$. Let $\theta$ be the image of $C$ under the abel Jacobi map.
Is there exist a divisor $D$ in $J$ such that $D.\...
- 137
2
votes
1
answer
172
views
What is the quotient $E \!\times\! E^\prime / G$?
Consider a finite field $\mathbb{F}_p$ such that $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$, $\mathbb{F}_{p^2}$-isomorphic elliptic curves (of $j$-invariant $0$)
$$
E\!:y_1^2 ...
- 1,833
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 ...
- 663
2
votes
0
answers
167
views
Chow variety of 1-cycles on abelian surface
It is an easy exercise to show that on a K3 surface, a smooth genus $g$ curve moves in a $g$-dimensional linear system. Nearly the same exercise shows that on an abelian surface, the corresponding ...
- 1,701
6
votes
0
answers
130
views
Sections of infinite order of elliptic surfaces
Let $X\to \mathbb{P}^1$ be a non-isotrivial elliptic surface over $\mathbb{C}$ with a section and with $X$ a smooth projective connected surface over $\mathbb{C}$. Let $\sigma:\mathbb{P}^1\to X$ be a ...
- 513
9
votes
1
answer
381
views
Lifting of families of curves to characteristic 0
Let $k$ be a finite field, $X_0$ be a smooth affine variety over $k$ and $C\rightarrow X$ a smooth projective family of curves of genus $\geq 2$.
By a result of Elkik we can always lift $X_0$ to a ...
- 728
5
votes
0
answers
400
views
Is $h^1(X,O_X)$ always equal to the dimension of the Albanese?
Let $X$ be a projective integral scheme over $\mathbb{C}$.
If $X$ is smooth, then $\mathrm{h}^1(X,\mathcal{O}_X)$ is the dimension of the Albanese variety of $X$. Probably, even if $X$ is normal, ...
- 353
15
votes
2
answers
845
views
Elements of arbitrary large order in the first Galois cohomology of an elliptic curve
Let $E$ be an elliptic curve over $k=\mathbb{Q}$. Consider $H^1(k,E)$.
In this answer Daniel Loughran writes: "I'm pretty sure that this cohomology group has elements of arbitrarily large order". I ...
- 14.2k
8
votes
1
answer
603
views
absolute Galois group of the function field of a curve over $\mathbb{F}_p^{alg}$
In their 2008 paper "Torelli theorem for curves over finite fields" Bogomolov, Korotiaev and Tschinkel mention in the beginning of Section 9 that absolute Galois groups of curves over $\mathbb{F}_p^{...
- 4,070
5
votes
2
answers
505
views
n-th root of unity in n-th division field of abelian variety?
Let $K$ be a number field and $A/K$ an abelian variety over it.
Can it be that $K(A[n])$ does not contain a primitive $n$-th rooth of unity?
If the answer is yes is it always possible to ...
- 53
2
votes
0
answers
100
views
Composing equal characteristic and mixed characteristic deformations
Suppose $k$ is an algebraically closed field of characteristic $p>0$ and $A$ is a $k$-point on a moduli space of certain objects over $k$. Suppose, moreover, that $A$ deforms to a $k[[t]]$-point $B$...
- 245
7
votes
3
answers
1k
views
Families of abelian varieties on the line (or more generally simply connected varieties)
I'm curious whether the following is true:
Question 1: Let $V/\mathbb{C}$ be a smooth connected variety such that $V^\text{an}$ is simply connected. Then, is every abelian scheme $f:\mathscr{A}\to ...
- 787
1
vote
0
answers
51
views
Do we have $\cap_{P\in S}\left(\mathcal{O}_P+(1-\tau)(K)\right)=\left(\cap_{P\in S}\mathcal{O}_P\right)+(1-\tau)(K)$?
Let $\mathbb{F}$ be a finite field and let $q$ be its number of elements. Let $(C,\mathcal{O}_C)$ be a geometrically irreducible smooth projective curve over $\mathbb{F}$ and let $K=\mathbb{F}(C)$ be ...
- 1,479
6
votes
0
answers
126
views
For which (g,q) does there exist a supersingular curve?
We say a curve over a finite field $\mathbb F_q$ is supersingular if it's Frobenius eigenvalues (on $H^1(X,\mathbb Z_\ell)$) are of the form $q^{1/2}\alpha$ for $\alpha$ a root of unity.
As far as I ...
- 7,746
2
votes
0
answers
122
views
A Lefschetz style formula for the $\ell^\infty$ torsion of an Abelian variety over a finite field
Let $A/\mathbb F_q$ be an abelian variety over a finite field. Define $A_\ell = A[\ell^\infty](\mathbb F_q)$, the $\ell^n$ ($n\geq 0)$ torsion points defined over the base field. I can assume $\ell \...
- 7,746
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
3
votes
0
answers
305
views
Abelian varieties by Moonen and van der Geer: proof of rigidity lemma
I try to understand a reduction step in the proof of rigidity lemma as proved in Moonen's and van der Geer's Abelian varieties (Lemma 1.11 on page 12- if the link not work the draft version is online ...
- 6,006
5
votes
1
answer
219
views
Largest ranks achieved by abelian varieties of fixed dimension
This is a follow-up to this earlier question on elliptic curves: Largest rank assumed by infinitely many elliptic curves
Let $g \geq 1$ be an integer. For each $g$, what is known about the largest ...
- 27k
4
votes
1
answer
836
views
Which hypersurfaces in $\mathbb{P}^n$ are abelian varieties?
Over an algebraically closed field $k$, which smooth hypersurfaces $X \subset \mathbb{P}^n$ are abelian varieties?
If $n=2$, then the smooth hypersurfaces of degree 3 (i.e. elliptic curves) are ...
- 53
18
votes
2
answers
938
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?
- 790
3
votes
1
answer
399
views
How to produce low genus curves on abelian surfaces?
I would like to find "simple" complex algebraic curves (i.e. low genus) on a complex abelian surface $A$ (which are not just abelian subvarieties or translates of them). For example, a genus 2 curve, ...
- 4,164
2
votes
0
answers
195
views
Structure of non-big divisors in an abelian variety
Let $A$ be an abelian variety over $\mathbb{C}$. If $A$ has an effective non-big divisor, then $A$ is not simple. (In a simple abelian variety, every non-zero effective divisor is ample.)
What can ...
- 513
24
votes
2
answers
2k
views
Have we ever proved any non-solvable case of reciprocity without the Langlands program ?
The reciprocity of the title is the following not completely well-posed problem:
Fix $P(X)$ a monic irreducible polynomial of degree $n$, with coefficients in $\mathbb Z$. "Describe"
(in some sense) ...
- 26.1k
4
votes
0
answers
222
views
Mordell-Weil group of an abelian variety on the perfect closure of a finitely generated field
This question is closely related the question Over which fields does the Mordell-Weil theorem hold?
I consider the following question:
(1) Let $K$ be a finitely generated field extension of $\...
- 26.1k