All Questions
1,203 questions
5
votes
2
answers
635
views
Uniruled degenerations of abelian varieties
Suppose I have a smooth projective variety $X$ over $\mathbb{C}$ with $K_X$ semiample, and consider the fiber space $f:X\to Y$ given by $|\ell K_X|$, for some $\ell>0$ large, where $Y$ is a normal ...
CommunityBot
- 1
3
votes
1
answer
270
views
Restriction of separable map
If $f: X\to Y$ is a separable map between varieties that is a bijection on closed points, is it true that $f$ remains separable when restricted to an integral subscheme $Z\subset X$?
If we drop the ...
- 4,111
5
votes
1
answer
598
views
A surjective morphism of abelian varieties induces an epimorphism on the torsion subgroups
Let $f:A\to B$ be a surjective morphism of abelian varieties (over an algebraically closed field).
Why is it true that $f$ induces an epimorphism on the points of finite order $A_{\mathrm{tors}}\to ...
- 66.3k
6
votes
1
answer
551
views
Fields generated by torsion points of CM elliptic curves
I'm using the same setup as Corollary 1.7 on p. 44 of de Shalit manuscript (Iwasawa theory of elliptic curves with complex multiplication).
I think there is a mistake in his Corollary 1.7 and I'm ...
- 7,609
3
votes
1
answer
351
views
$\mathbb Q_p$ étale local sytem in characteristic $p>0$
Let $k$ a field with $char(k)=p>0$, separable closure $k^{sep}$ and $f:X\rightarrow Y$ a smooth projective morphism of smooth variety over $k$.
1)Is it true that there exists a (EDIT) dense open ...
- 728
2
votes
1
answer
298
views
supersingular Abelian scheme
By a supersingular Abelian scheme, I mean an Abelian scheme which is fibrewise a supersingular Abelian variety, i.e. isogenous to a product of supersingular elliptic curves (F. Oort, Subvarieties of ...
CommunityBot
- 1
2
votes
1
answer
526
views
extending homomorphisms of Abelian schemes
Let $S$ be an integral scheme with function field $K = K(S)$. Let $\mathscr{A}, \mathscr{B}$ be Abelian schemes over $S$. Let $L/K$ be a separable field extension. Given $f_L \in \mathrm{Hom}(\mathscr{...
CommunityBot
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7
votes
1
answer
239
views
Commutation of endomorphisms of abelian varieties
Let $A$ be an abelian variety over an algebraically closed field $k$.
Let $\phi:A\to A$ be an étale isogeny (over $k$). Suppose that the set $\cup_{r\geq 0}({\rm ker}\,\phi^{\circ r})(k)$ is
...
- 28.9k
3
votes
1
answer
283
views
Polarization on Prym varieties
Let $C\rightarrow C'$ be a double cover of curves, the restriction of the polarisation of $J_C$ to the Prym varieties $P$ attached to this double cover, gives a polarization on $P$,
Does this ...
CommunityBot
- 1
8
votes
0
answers
636
views
Weil pairing and Tate module for $p$-torsion in characteristic $p$
Let $A/\mathbf{F}_{p^n}$ be an Abelian variety with $\bar{A} = A \times_{\mathbf{F}_{p^n}} \bar{\mathbf{F}}_{p^n}$.
If $\ell \neq p$ is prime, there is a natural isomorphism $$\mathrm{H}^2_{\mathrm{...
CommunityBot
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3
votes
1
answer
343
views
Two spaces attached to mod 2 level 9 modular forms--a conjectural Hecke isomorphism
MOTIVATION
Nicolas and Serre have analyzed the structure of the space of mod $2$ modular forms of level $1$, viewed as a "Hecke-module". They show that for each $p>2$, the operator $T_p$ acting on ...
- 5,422
1
vote
0
answers
67
views
Basis of homomorpshims of abelian varieties with minimal degree
Let $A, B$ be simple abelian varieties of dimension $g$ defined over a finite field $k$. We know that $Hom_k(A, B)$ is a free $\mathbb{Z}$-module of dimension $2g$.
Is it always possible to have a $\...
- 389
4
votes
0
answers
240
views
On the class of Shimura data of Hodge type that cover a given Shimura datum of abelian type
A Shimura datum $(G,X)$ is of Hodge type if there exists an injective morphism of Shimura data $(G,X) \hookrightarrow (\mathrm{GSp}_{2g}, \mathfrak{H}^{\pm})$, for some integer $g$.
Let $(G',X')$ be ...
- 133
2
votes
2
answers
563
views
morphism of abelian variety
Let $f: A \rightarrow B$ be a morphism of abelian varieties defined over a finite field $k$. Let $G$ be a finite group of $A$ and $\pi:A\rightarrow A/G$ the quotient morphism.
Looking at just the ...
- 196
4
votes
0
answers
267
views
Dimension of the moduli space of abelian varieties with a prescribed endomorphism algebra
Let $D$ be a division algebra over a number field $K$, and consider abelian varieties $A$ over the complex numbers, of dimension $g$ with an action of (an order of) $D$. Is it known when this set is ...
- 66.3k
2
votes
0
answers
286
views
Does the sheaf of locally exact differential forms splitting in positive characteristic
Let k be an algebraically closed field of characteristic $p>2$, $X$ a smooth projective curve of genus $g>1$ over $k$, and $F_X:X\rightarrow X$ be the absolute Frobenius morphism. Let $B^1_X$ be ...
- 85
1
vote
1
answer
238
views
Endomorphisms of abelian varieties defined over finite fields
Let $A$ be an abelian variety defined over a finite field $k$, and $End(A)$ be it ring of endomorphisms defined over an algebraic closure $\overline{k}$ of $k$. Suppose that for an integer $M$ ...
- 2,665
2
votes
1
answer
143
views
Fiber of the Prym map in dim 2
This must be very classical, but I can't find a reference.
Is there an explicit description of the (generic?) fibers of the Prym map $\mathcal{R}_3 \to \mathcal{A}_2$?
By this I mean the map ...
- 66.3k
6
votes
0
answers
343
views
Are all stabilizer groups of the co-adjoint action smooth?
Let $k$ be a (non-archimedean) local field of positive characteristic $p$ and $\mathfrak{n}$ be any finite-dimensional nilpotent Lie algebra over $k$ with nilpotence length $l<p$. It is well-known ...
CommunityBot
- 1
6
votes
0
answers
467
views
Torsionfree crystalline cohomology implies torsionfree etale cohomology?
Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$. Let $W=W(k)$ be the ring of Witt vectors of $k$.
Assume that the crystalline cohomology $H^2_{...
- 91
15
votes
0
answers
579
views
Geometry underlying a comparison of Dieudonné theories
Maybe these hypotheses aren't necessary, but for me $\mathbb G$ will be a smooth formal group of dimension 1 and finite height over a perfect field $k$.
There are several presentations of the ...
- 6,308
1
vote
1
answer
241
views
locally closed orbits in metric Hausdorff topology
I learned the following fact from Bruhat and Tits's paper "Homomorphismes “abstraits” de groupes algebriques simples" Section 3.18 that
Let $k$ be a local field. Suppose that a $k$-group $H$ acts $k$...
- 1,702
1
vote
0
answers
187
views
Unitary dual of the Heisenberg group over non-archimedean local fields of characteristic two
What is the unitary dual of the Heisenberg group over non-archimedean local fields k of characteristic two? This is well-known for the real Heisenberg group and in fact, when local fields have ...
- 1,702
10
votes
2
answers
761
views
Applications of alterations
In 1995, de Jong proved the existence of regular alterations in arbitrary characteristic. I would like to have a little survey of important applications of this theorem, e.g. things you could do if ...
CommunityBot
- 1
2
votes
1
answer
259
views
Isogeny from kernel in higher dimensional abelian varieties
Is there any kind of generalization of Vélu formulae for Jacobians?
The question technically is:
Given a hyperelliptic curve $H/\overline{\mathbb{F}}_q$ and its jacobian $J_H$ of dimension $2$, if $...
5
votes
0
answers
354
views
Line bundles of characteristic $0$ on abelian varieties
Maybe what I'm asking is well-known to the experts, however I was not able to find a suitable reference. Any pointer to the literature will be appreciated.
For the notation and teminology, I refer ...
- 66.3k
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$ ...
- 66.3k
12
votes
1
answer
750
views
Vanishing theorems in positive characteristic
In the paper
Deligne, Pierre; Illusie, Luc (1987), "Relèvements modulo $p^{2}$ et décomposition du complexe de De Rham", Inventiones Mathematicae 89 (2): 247–270, doi:10.1007/BF01389078
I found the ...
- 66.3k
7
votes
0
answers
374
views
Arbitrarily non-degenerate Hodge to de Rham spectral sequence
It is true that for any $n$ there exists a compact complex manifold which Frolicher spectral sequence does not degenerate at the $n$-th page(https://arxiv.org/pdf/0709.0481.pdf).
Does the analogous ...
- 7,377
4
votes
1
answer
622
views
finitness of syntomic/fppf cohomology with coefficients in a finite flat group scheme
Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathscr{A}/X$ be an Abelian scheme.
Is then $H^1_\mathrm{SYN}(X,\mathscr{A}[p]) = H^1_\mathrm{fppf}(X,\mathscr{...
CommunityBot
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5
votes
2
answers
1k
views
Weil Conjectures for Grassmannians
To establish the Weil conjectures for $n$-dimensional projective space over a finite field is elementary. Does there exist a simple direct proof of the conjectures for finite field Grassmannians?
- 2,218
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 ...
CommunityBot
- 1
3
votes
1
answer
265
views
Geometric (or at least non-cohomological) proof of Lefschetz trace formula for curves
There is an isomorphism between (rational) correspondences on a curve $C/\mathbb{F}_p$ orthogonal to the "valence zero" ones (i.e. orthogonal under intersection pairing to $\{*\}\times C$ and $C\times ...
- 66
15
votes
0
answers
517
views
Zariski vs etale torsors over abelian varieties
Question. Let $A$ be an abelian variety (say, over the complex numbers), $G$ an algebraic group, $c$ a class in $H^1_{\rm et}(A, G)$. Denote the multiplication by $N$ map on A by $m_N:A\to A$. Does ...
- 16.2k
0
votes
1
answer
204
views
Prime divisors on the Jacobian of a genus 2 curve over $\mathbb{F}_q$ under the $n$ map
Let $H$ be a hyperelliptic curve geometrically irreducible of genus 2 over $\mathbb{F}_q$ with a rational point $\infty$ given by the model $y^2=f(x)$, where $f$ is monic of degree 5.
Consider the ...
CommunityBot
- 1
12
votes
9
answers
6k
views
Proofs of Mordell-Weil theorem
I would like to ask if there exist pedagogical expositions of the Mordell-Weil theorem (wikipedia). What parts of number theory (algebraic geometry) one should better learn first before starting to ...
- 4,008
2
votes
2
answers
321
views
What are sufficient and necessary conditions to be a generalized Zariski surface over a finite field?
Let $X$ be an absolutely irreducible reduced surface over a finite field $k$ of characteristic $p$. What are sufficient and necessary conditions for $X$ to be a generalized Zariski surface over $k$ (...
CommunityBot
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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 ...
- 9,497
1
vote
0
answers
149
views
Smoothability of stable curves in mixed characteristic
Let $R$ be a complete DVR with residue field $k$ algebraically closed of characteristic $p$ and fraction field $K$ of characteristic zero. Let $C$ be a stable curve (in the sense of Mumford-Deligne) ...
- 2,323
1
vote
0
answers
118
views
Poincare complete reducibility with an endomorphism ring
Let $A$ be an abelian variety over a field of characteristic $0$ and let $R \subset \mathrm{End}(A)$ be a (commutative, if that matters) subring. Suppose that $B \subset A$ is an $R$-stable abelian ...
- 2,663
10
votes
1
answer
845
views
How to compute the formal group law of a Shimura variety (using its invariant differentials)?
I have a 3 dimensional abelian variety whose formal group law breaks into a formal summand where one of the pieces is one-dimensional.
I am desperately wondering how to compute the $p$-series of ...
- 52.7k
2
votes
0
answers
345
views
Examples of semi-stable models of curves
Let $R$ be a discrete valuation ring with fraction field $K$ of characteristic zero and residue field $k$ of characteristic $p>0$. Assume $k$ is algebraically closed. I want to produce examples of ...
- 2,323
2
votes
0
answers
248
views
What are the point-like objects in $D^b(X)$ when $X$ is an abelian variety?
Let $X$ be a projective variety over an algebraically closed field $k$ and $D^b(X)$ be the derived category of bounded complexes of coherent sheaves on $X$. Let $S$ be the Serre functor on $D^b(X)$. ...
- 9,019
9
votes
1
answer
546
views
Morphisms for good reduction are maps respecting filtration
Please see edits below!
So, let $A,A'/K$ be abelian varieties where $K$ is a $p$-adic local field with residue field $k$. Suppose further that they have good reduction with models $\mathscr{A},\...
CommunityBot
- 1
4
votes
1
answer
272
views
Finiteness of cohomology with finite coefficients
Let $G$ be a finite abelian group and let $S$ be a variety over $\mathbb{F}_p$. It is natural (I think) to expect that the cohomology group $H^i(S,G)$ is finite. But with respect to which cohomology?
...
- 4,111
2
votes
0
answers
187
views
How can I find a basis of $H^0(Y, -K_Y)$ for the del Pezzo surface $Y$ of degree 5?
Consider the surface $X \subset \mathbb{P}^2_{x, y, z}\times\mathbb{P}^1_{t, s}$:
$$
s^3y^2 + t^3yz = (t+s)x^2 + tsxz + t(t^2 +s^2)z^2
$$
over a finite field $k = \mathbb{F}_{2^d}$, $\mathrm{gcd}(d, 6)...
- 1,833
1
vote
0
answers
102
views
Base locus of the Eigen spaces of global sections of totally symmetric line bundle
Let $X$ be an abelian surface over complex numbers. Let $L$ be a totally symmetric line bundle of type $(r,r)$ for $r\geq 2$.
The involution $i$ on $X$ gives an action on $H=H^0(X,L)$ thus giving a ...
- 179
8
votes
1
answer
1k
views
Semisimplicity of Frobenius on *integral* Tate module
Let $K$ be a number field and $A/K$ an Abelian variety; let $l$ be a (rational) prime. Do there exist infinitely many primes $\mathfrak{p}$ of $K$ such that the Frobenius at $\mathfrak{p}$ acts ...
- 149k
11
votes
3
answers
552
views
When $k = \mathbb{F}_q$ finite field, $X$ always has $k$-rational point, and so $A \simeq X$?
Let $k$ be an arbitrary field. Let $(A, e)$ be an abelian variety over $k$, and let $X$ be a torsor for $A$, i.e. $X$ is a proper smooth $k$-variety, and there is an $A$-action acting $:A \times X \to ...
- 47.4k
4
votes
0
answers
173
views
Are there methods to compute the induced action of Frobenius map on the Neron-Severi group of a supersingular abelian surface over a finite field?
Let $A$ be a supersingular abelian surface over a finite field $\mathbb{F}_q$. In that case the Neron-Severi group $NS(A\otimes\overline{\mathbb{F}_q})$ is the lattice of rank $6$. Are there methods ...
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