All Questions
1,203 questions
8
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8k
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What does "supersingular" mean?
Are supersingular primes and supersingular elliptic curves related?
(this was essentially a subquestion in my earlier question, but still looks sufficiently different to me to deserve a separate post)...
- 15.1k
8
votes
2
answers
657
views
Adjoining torsion points from abelian varieties
Let $L/\mathbb{Q}$ be the field generated over $\mathbb{Q}$ by all of the (projective) coordinates of all of the torsion points of all abelian varieties defined over $\mathbb{Q}$. Is $L$ algebraically ...
- 11.3k
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 $A_1, A_2$ ...
- 1,742
8
votes
2
answers
1k
views
Lefschetz on étale fundamental group for quasi-projective varieties
If $X$ is a smooth projective variety of dimension at least $3$ over $\mathbb{C}$, Lefschetz's Hyperplane theorem says that for every hyperplane section $H$
$$\pi^1(H)\to\pi^1(X)$$
is an isomorphism, ...
- 483
8
votes
2
answers
2k
views
etale cohomology of an abelian variety and its dual
Let $A$ an abelian variety over a field $k$ and $A^{*}$ the dual abelian variety.
How can we relate the étale cohomology of $A$ with etale cohomology of $A^{*}$?
- 3,472
8
votes
1
answer
774
views
Abelian varieties with good reduction everywhere over function fields
There is a famous theorem due to J.-M. Fontaine,
Il n'y a pas de variété abélienne sur Z
(and independently by V.A. Abrashkin) that there are no abelian varieties over Z. I was wondering whether ...
- 83
8
votes
1
answer
1k
views
obstruction to smooth lifting of smooth schemes
According to general theory, for a square zero thickening defined by an ideal I: SpecA -> SpecA', there is an obstruction of lifting a smooth scheme X over A to a smooth scheme over A' living in H^2(X,...
- 5,052
8
votes
3
answers
762
views
possible CM-types of abelian varieties
Fix a CM-field $K$ of degree $2g$. Let $A$ be a polarized abelian variety of dimension $n$ over $\mathbb{C}$, with an isomorphism $\theta : K \to End_{\mathbb{C}}(A) \otimes_{\mathbb{Z}} \mathbb{Q}$. (...
- 1,832
8
votes
1
answer
2k
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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
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
8
votes
1
answer
943
views
Automorphisms over finite field that do not lift to an automorphism in characteristic zero
My main question is the following: is there an automorphism of the affine space $\mathbb{A}^n$ (automorphism of an algebraic variety) defined over a finite field which does not lift to an automorphism ...
- 7,722
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 $\ell$...
user113452
8
votes
1
answer
774
views
A variant on characteristic $p$ de Rham cohomology
I was thinking about de Rham cohomology in characteristic $p$, and in particular the recent question about Poincare residues, and I came up with the following construction.
Let $k$ be a perfect field ...
- 156k
8
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1
answer
818
views
To what extent does Poincare duality hold on moduli stacks?
Poincare duality gives us, for a smooth orientable $n$-manifold, an isomorphism $H^k(M) \to H_{n-k}(M)$ given by $\gamma \mapsto \gamma \frown [M]$ where $[M]$ is the fundamental class of the manifold,...
- 6,290
8
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1
answer
331
views
If C is a fusion category over a field of nonzero characteristic and dim C = 0, is Z(C) ever fusion?
If $C$ is a fusion category and $\dim(C) \neq 0$ (the latter is automatic in characteristic zero, but not in nonzero characteristic), then the Drinfel'd center $Z(C)$ is fusion. More generally, if $C$...
- 28.1k
8
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1
answer
2k
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About the Serre-Tate theorem
It is somehow a general principle that the (infinitesimal) local behavior of a representable moduli functor $X$ at some point $x$ is closely related to the deformation problem of the structure ...
- 1,305
8
votes
1
answer
808
views
Automorphisms of curves in positive characteristic
It is well known that over an algebraically closed field of characteristic zero a general curve (for an open subset of $M_g$) of genus $g\geq 3$ is automorphism-free.
Is this result still true over ...
user68440
8
votes
3
answers
570
views
Variations on a theme of O'Bryant, Cooper and Eichhorn concerning power series over $\mathbb Z/2\mathbb Z$
Define 2 power series over the field $\mathbb Z/2\mathbb Z$ by $f=1+x+x^3+x^6+\dots$, the exponents being the triangular numbers, and $g=1+x+x^4+x^9+\dots$, the exponents being the squares. Write $f/g$...
- 5,422
8
votes
1
answer
1k
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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 ...
- 23k
8
votes
1
answer
982
views
Is there a really big ring of differential operators in characteristic p?
$k$ is a field of characteristic $p$.
$k[t]$ has canonical first-order differential operator $\partial$
As an endomorphism of $k[t]$, $\partial^p=0$.
First way to fix it:
Use the divided power ...
8
votes
1
answer
952
views
Rank 2 vector bundle on a product of elliptic curves
Let $E$, $F$ be two complex elliptic curves, and $A=E \times F$. Let us denote by
$\pi_E \colon A \to E, \quad \pi_F \colon A \to F$
the natural projections. For all $p \in F$ let us write $E_p$ ...
- 66.3k
8
votes
1
answer
536
views
Endomorphism algebras of abelian surfaces with real multiplication
Given an abelian variety $A$ over a field $F$, one may consider the ring of endomorphisms $End(A)$, the ring of $F$-rational maps $A \to A$ respecting the group structure on $A$. We may also consider ...
- 3,625
8
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1
answer
758
views
Do all 0-dimensional Shimura Varieties show up (as CM points) in $\mathcal{A}_g$?
Question: Let $S$ be a 0-dimensional Shimura variety. Does $S$ necessarily admit a morphism (in the category of Shimura varieties) to $\mathcal{A}_g$ for some $g\geq 1$? Here $\mathcal{A}_g$ is the ...
- 2,834
8
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1
answer
747
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Deligne's exterior power
In "Catégories Tannakiennes", Deligne defines the $n$th exterior power of an object $A$ of an abelian tensor category $\mathcal{C}$ as the image of the morphism
$$p : A^{\otimes n} \to A^{\otimes n}, ...
- 63.1k
8
votes
1
answer
172
views
Non-degenerate points on a Jacobian surface
Let $C$ be a (hyperelliptic) genus $2$ curve over a number field $K$ with a $K$-rational Weierstrass point $\infty$. We embed $C$ in its Jacobian $J$ via $\infty$.
Question: Is there a quadratic ...
- 93
8
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1
answer
168
views
Index of the endomorphism ring of an abelian surface
For an abelian surface $A/\mathbb{Q}$ such that $R:=\mathrm{End}_{\mathbb{Q}}(A)$ is an order in a real quadratic field $K$ (so a $\mathrm{GL}_2$-type surface), is there a bound on the index $[O_K : R]...
- 173
8
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1
answer
750
views
flat/crystalline cohomology of abelian variety
Let $A/k$ be an abelian variety over an algebraically closed field and $\ell \neq \mathrm{char}\,k$.
In http://jmilne.org/math/articles/1986b.pdf, Theorem 15.1(b) it is proved that $$H^r_{et}(A, R) = \...
user19475
8
votes
1
answer
823
views
Does complex multiplication for higher dimensional abelian varieties give some generalization of class field theory?
I am currently learning some aspects of the theory of complex multiplication for elliptic curves, and the relationship with class field theory.
As I understand it, there is a very special class of ...
- 3,058
8
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1
answer
2k
views
Poincaré line bundle
I am being stuck by the proof of the existence of Poincaré line bundle of complex torus in Griffiths-Harris. Here is the question:
Let $M$ be a complex torus and $M'$ be the complex torus dual to $M$...
- 750
8
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1
answer
1k
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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 ...
- 19.2k
8
votes
0
answers
261
views
Simultaneous rank jumping of elliptic curves over number fields
Here's something fun that I've been wondering about for a while out of curiosity. It has to do with the rank $\mathrm{rk}(E(K))$ of an elliptic curve $E$ over a number field $K$, and especially how it ...
- 9,497
8
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0
answers
201
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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,\...
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0
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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
8
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0
answers
688
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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
$$\alpha: H^0(\Omega^1_{S/k}/d\mathcal O) ...
- 2,368
8
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0
answers
196
views
Simple abelian varieties of $\mathrm{GL}_2$ type with positive rank and large dimension
$\DeclareMathOperator\GL{GL}$I would like know if there are known constructions of simple abelian varieties of $\GL_2$ type of arbitrarily large dimension and positive Mordell-Weil rank, whose rank is ...
- 2,224
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0
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471
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Sheaf whose singular support is not Lagrangian
For constructible sheaves $\mathcal F$ on real analytic manifolds $X$, there is a notion of the singular support $SS(\mathcal F)$ which is a radially invariant singular Lagrangian subset of the ...
- 18.7k
8
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0
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636
views
Weil pairing and Tate module for $p$-torsion in characteristic $p$
Let $A/\mathbf{F}_{p^n}$ be an Abelian variety with $\bar{A} = A \times_{\mathbf{F}_{p^n}} \bar{\mathbf{F}}_{p^n}$.
If $\ell \neq p$ is prime, there is a natural isomorphism $$\mathrm{H}^2_{\mathrm{...
user19475
8
votes
0
answers
518
views
$p$-adic uniformisation of abelian varieties
In the paper $p$-adic L-functions and $p$-adic periods of modular forms of Greenberg and Stevens $\S3$ page $420$ they make the following statement:
Let $A$ over $\mathbf{Q}_p$ be an abelian variety ...
- 121
8
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0
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381
views
Degeneration of wildly ramified local monodromy representations - near or far from Deligne?
Suppose we have a surface $S$ mapping to a curve $C$ and a finite cover $Y/S$ that is ramified at a divisor $D$. For each point $x \in C$ we get a ramified cover of the curve $S_x$, and we can study ...
- 149k
8
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0
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405
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Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s
It is well known (cf. Dolgachev) that there is a beautiful notion of mirror symmetry for lattice-polarized K3 surfaces. That is, if we are given a rank $r$ lattice $M$ of signature $(1, r - 1)$ and a ...
- 6,290
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0
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503
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Points of minimum Arakelov height and harmonic arithmetical varieties
Added. (28/2) To put it less pompously (and more vaguely, less concretely), I wanted to relate the impression that it is the general rule that an Arakelov (i.e., geometric) height on an arithmetical ...
8
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0
answers
244
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Corresponding notion of unramified for motives (or de Rham cohomology)
The etale cohomolgoy of a variety $X$ over a number field $K$ is a Galois representation of $\mathrm{Gal}(\overline K/K)$ with some properties coming from $X$, e.g., it is unramified outside $S$ if $X$...
- 381
8
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0
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873
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Resolution of singularities in positive characteristic
I am currently trying to make some small parts of the minimal model program work for some very explicit varities in positive characteristic. I have such a variety $X_1$ and I know that there is a ...
- 783
8
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0
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649
views
Automorphic representations attached to abelian varieties
Let $A$ be an abelian variety defined over $\mathbb{Q}$, of dimension $d$. It is widely expected that there is an automorphic representation $\pi_A$ of $GL(2d)/\mathbb{Q}$ whose L-function agrees ...
- 13.1k
7
votes
3
answers
752
views
Does every linear group admit a subgroup of dimension 1?
Suppose that $G$ is a linear group of positive dimension, defined over some field $k$. Is that true, that $G$ admits a (closed) one-dimensional subgroup?
I'm pretty much sure this is true in ...
- 267
7
votes
3
answers
908
views
Canonical liftings of endomorphisms of ordinary abelian varieties
I am looking for a reference to the following ``well known" fact.
Let $k$ be a perfect field of prime characteristic $p$ and $W(k)$ its ring of Witt vectors. Let $A_0$ be an ordinary abelian variety ...
- 5,050
7
votes
2
answers
541
views
Is the symmetric product of an abelian variety a CY variety?
Let $n>1$ be a positive integer and let $A$ be an abelian variety over $\mathbb{C}$. Then the symmetric product $S^n(A)$ is a normal projective variety over $\mathbb{C}$ with Kodaira dimension zero ...
- 353
7
votes
2
answers
1k
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Explicit way to construct simple complex tori/abelian varieties of dimension at least 2
The following question was motivated by one of the earliest exercises of Complex Abelian Variaties by Birkenhake and Lange during my presentation last year.
It can be shown that any complex torus $X$...
- 1,671
7
votes
4
answers
736
views
Simply connected quasi-projective varieties in positive characteristic
I am looking for examples of non-projective (quasi-projective) varieties $X$ defined over a field of positive characteristic, which have trivial étale fundamental group.
It is well known that the ...
- 4,450
7
votes
3
answers
1k
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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