All Questions
1,203 questions
10
votes
0
answers
371
views
How large must the characteristic of $k$ be, for the cohomology of the Lie algebra $\mathfrak{sl}_n(k)$ to be exterior as in characteristic zero?
$\DeclareMathOperator\SU{SU}$In this question, all Lie algebra cohomology is of the form $H^*(\mathfrak{g}; k)$, with $k$ the trivial one-dimensional representation of $\mathfrak{g}$. All Lie algebra ...
- 1,335
4
votes
0
answers
108
views
Shafarevich conjecture for Abelian varieties over global function fields
Let $S$ be a finite set of places of a global function field $K$. Are there finitely many Abelian varieties over $K$ with good reduction outside $S$? What if we exclude isotrivial families?
- 679
4
votes
0
answers
135
views
Nilpotent orbits in characteristic $0$ vs. positive characteristics
Let $G_\mathbb{C}$ be a connected reductive group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}_{\mathbb{C}}$. For any algebraically closed field $k$, let $G_k$ denote the connected reductive group ...
- 2,751
11
votes
1
answer
381
views
Chromatic representation theory of the symmetric groups?
We know that in characteristic 0, the group ring of the symmetric group $\Sigma_n$ splits via one idempotent for each partition of $n$.
In characteristic $p$, I believe the analogous statement is that ...
- 64k
4
votes
0
answers
183
views
Characters of finite fields - Reference request
Let $\mathbf{F}_q$ ($q=p^f$) be a finite field. We are interested in the characters $\chi: \mathbf{F}_q\rightarrow \mathbf{K}$ ($\chi(0)=0$) where the $ \mathbf{K}$ is an alg.closed field of ...
- 41
2
votes
1
answer
292
views
One unexpected observation related to algebraic curves and their Jacobians
Let $C$ be a projective irreducible algebraic curve over an algebraically closed field of characteristic $0$ (for simplicity). Assume that there is also a cover $\varphi\!: C \to E$ to an elliptic ...
- 1,833
2
votes
1
answer
220
views
Looking for an example of a point $P$ on an abelian variety $X$ such that no curve on $X$ contains all multiples of $P$
Is there an example of an abelian variety $X$ defined over a number field $K$, with $\dim X > 1$, and a $K$-rational point $P$ on $X$, such that no curve $C$ on $X$ (say defined over a number field)...
- 658
2
votes
1
answer
141
views
Endomorphism ring of the Jacobian of a generic smooth plane quartic
Let $C$ be an arbitrary smooth plane quartic defined over a number field $K$. Assume $C$ is not hyperelliptic, and denote by $J$ the Jacobian of $C$. How does $\text{End}(J)$ look like for a generic ...
- 2,907
2
votes
0
answers
193
views
Real structure(s) of a Shimura curve ("complex conjugation" of abelian surfaces)
For a complete lattice $L \subseteq \mathbb{C}^2$ let $A_L$ denote the complex abelian algebraic surface that is isomorphic (as a complex manifold) to the complex torus ${\mathbb{C}^2}/{L}$ (this ...
- 103
2
votes
0
answers
179
views
Is the Weil restriction of an elliptic curve self-dual?
$\DeclareMathOperator\res{res}$Let $K=\mathbb{Q}(\sqrt{-3})$, and let $$p\equiv 1\pmod 3$$
be a prime split in $K$. Assume that
$$p=\omega*\overline\omega,\quad\text{where}\quad\omega\equiv 1\pmod 3.$$...
- 390
1
vote
0
answers
166
views
Reference request showing that a very general Abelian variety $ A $ of genus $ g>1 $ has cyclic class group with ample generator
In Example of a $ \mathbb{Q} $-factorial, CM normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not CM I asked for an example of a Cohen Macaulay, normal, ...
- 912
1
vote
1
answer
184
views
Is multiplication by $d$ on the Jacobian of a nodal curve étale?
Let $k$ be an alg.closed of char$k=0$ and let $A$ be an abelian variety over $k$. This
Lemma on stacks project states that $[d]\colon A\to A$ is étale. In particular, when $A$ is the Jacobian of a ...
- 123
1
vote
1
answer
212
views
The ideal in $\mathbb{Z}[x]$ of all vanishing polynomials of a curve automorphism
Let $C$ be a projective irreducible algebraic curve of genus $g$ over an algebraically closed field of characteristic $0$ (for simplicity). Given an automorphism $\alpha \in \mathrm{Aut}(C)$ of order $...
- 1,833
1
vote
0
answers
52
views
Abelian surface with CM by a prescribed quartic field
Given a quartic field $K$, is it possible to exhibit an explicit abelian surface $A$ defined over a number field with CM by the field $K$?
For example, let's take the non-Galois CM-field $K=\mathbb Q(...
- 23
4
votes
2
answers
448
views
$p$-divisibility of Picard groups
Let $p$ be a prime number and let $k$ be a field with $char(k)\neq p$ such that all finite extensions have degree coprime to $p$. (For example, we can take $k=\mathbb{R}$ and $p\neq 2$ or let $k$ the ...
- 337
5
votes
1
answer
160
views
Derived subalgebra of a restricted Lie algebra
Let $L$ be a restricted Lie algebra over a field of characteristic $p>0$. It is well known that the commutator subalgebra $[L,L]$ is not necessarily restricted (that is, closed under the $p$-map).
...
- 630
3
votes
0
answers
296
views
Explicit family of polynomials describing embedded torus in complex projective space
This question is cross-posted (with modifications) from MSE. The original question is probably unfit for MathOverflow (although a professor I asked said that this is very nontrivial), but I'm hoping ...
- 1,763
11
votes
0
answers
374
views
Example of abelian variety over finite field which doesn't lift
What is an example of an abelian variety over a finite field $\mathbb{F}_p$ which doesn't lift to $\mathbb{Z}_p$? This question seems to hint that they should exist, but no example is given.
Note that ...
- 21.4k
7
votes
1
answer
483
views
A "comprehensive" family of abelian varieties
I'm looking for a family of abelian varieties $A\rightarrow S$ over a base that is finite type over $\mathbb{Q}$ (or $\mathbb{Z}$) that is "comprehensive" in the following sense: for every ...
- 333
2
votes
0
answers
127
views
Classification of restricted Lie algebras of reductive groups
$\DeclareMathOperator\Lie{Lie}$Let $G/K$ be a reductive group over a field $K$. In characteristic $0$ the Lie algebra is invariant under base change of fields, so to understand $\Lie(G)$ it is enough ...
- 461
2
votes
0
answers
178
views
Product subvariety of a simple abelian variety
Terminology: A subvariety $V$ of an abelian variety $A/\mathbb{C}$ is called a product if there are integral closed subvarieties $U,W\subset A$ such that $\dim U,\dim W>0$ and the sum morphism $U\...
- 615
7
votes
0
answers
174
views
Failure of injectiveness of maps between cotangent spaces of abelian varieties
Let $p$ be a prime and $K$ a finite extension of $\mathbb Q_p$ with ramification index $e$. Let $\mathcal O_K$ be the ring of integers of $K$ and $k$ its residue field and the unique maximal ideal. ...
- 2,224
3
votes
0
answers
171
views
Grothendieck-Messing in characteristic 0?
Let $A$ is an abelian scheme over a base scheme $S$. Let $S \rightarrow S'$ be a thickening defined by an ideal of square zero (for example).
If $p$ is locally nilpotent on $S$, then Serre-Tate and ...
- 261
2
votes
0
answers
189
views
Tangent space to the moduli space of abelian varieties
Letting $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties. I saw without reference that the tangent space of $\mathcal{A}_g$ at a point $t$ could be canonically identified ...
- 91
4
votes
1
answer
435
views
Étale group schemes and specialization
If $A$ is an abelian variety over a finite field $\mathbf{F}_q$, then $A(\mathbf{F}_q)$ (resp. $A(\overline{\mathbf{F}}_q)$) is a finite (resp. infinite torsion) group, but $A(\mathbf{F}_q(t))$ is a ...
user501361
4
votes
1
answer
275
views
If $A$ is an abelian variety over $k$ and $S$ is a $k$-scheme, what's $A'(S)$ geometrically?
Let $A$ be an abelian variety over a field $k$ with group operation $m\colon A\times A\to A$, and let $A'$ be the dual abelian variety. I know that $A'(k)$ is isomorphic to the subgroup $\operatorname{...
- 773
4
votes
1
answer
244
views
Néron model, torsion and ramification
Let $B$ a discrete valuation ring, say for simplicity with residue field of characteristic $0$, and $K$ its quotient field. Assume that I have an abelian variety $A$ over $K$ and let $A'$ be its Néron ...
- 3,031
20
votes
0
answers
408
views
Ado's theorem and the reduction to positive characteristic
The synopsis: proofs of Ado theorem in positive characteristic are simple, and in characteristic $0$ are difficult. Can one infer the characteristic $0$ case from the positive characteristic case?
The ...
- 2,934
10
votes
1
answer
410
views
Torsion points of abelian variety as zeros of a section of a vector bundle?
Let $A$ be an abelian variety over $\mathbb{C}$ and let $X_m$ the subset of nontrivial $m$-torsion points on $A$. Can we realize $X_m$ as the zero locus of a global section of a suitable vector bundle ...
- 3,031
79
votes
12
answers
13k
views
Is there a high-concept explanation for why characteristic 2 is special?
The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, ...
- 118k
1
vote
0
answers
125
views
Abelian varieties with endomorphism structure
Let me stick to principally polarised abelian varieties $X$ over $\mathbb C$.
I have seen several definitions of what it means for $X$ to have real multiplication by a totally real field $F$:
There ...
2
votes
1
answer
170
views
Automorphism of positive characteristic field
Suppose $K$ is a field with $\text{char}(K) \geq 0$. Let $L$ be a cyclic extension of $K$ with degree $2n$. We consider the generator $\sigma$ of the Galois group $\text{Gal}(L/K)$.
I am interested in ...
- 923
1
vote
1
answer
484
views
Why an elliptic curve can be defined as an abelian variety of dimension 1?
Now we define an elliptic curve as "a smooth projective curve of genus one with a specified base point". (A little question by the way: Is the requirement "with a specified base point&...
- 65
2
votes
0
answers
113
views
From rational to integral generators of Néron–Severi group
Suppose I've found rational generators for the Néron–Severi group $\mathrm{NS}(A)$ for an abelian
variety over $\mathbb{C}$. How would I check if they are integral generators for $\mathrm{NS}(A)$. Are ...
- 91
3
votes
0
answers
227
views
Tate isogeny theorem over varieties?
Let $X$ be a nice scheme, $\pi:E\to X$ an elliptic curve, and $\ell$ a prime invertible on $K$. Then we can consider the "Tate module" $(R^1\pi_*\mathbb{Z}_{\ell})^\vee=\hbox{''}\varprojlim\...
- 371
4
votes
0
answers
143
views
Road map for learning about the computational/general theory of modular curves/isogenies of abelian varieties for cryptography
I am a graduate math/crypto student. So I've had some free time last year and I heard about elliptic curves in cryptography and how a resilient cryptosystem got demolished by a spectacular attack ...
- 41
1
vote
0
answers
152
views
Vanishing of pushforwards under Fourier-Mukai
In the book Fourier-Mukai and Nahm transforms in geometry and mathematical physics page 85 corollary 3.5 there is the following claim:
First some notation.
Let $X$ be an abelian variety of dimension $...
- 303
7
votes
0
answers
242
views
Mumford's definition of an abelian variety's $Pic^0$
I'm not sure whether this is a research-level question, but upon skimming through Mumford book of Abelian Varieties I noticed he gives this definition
$$
\begin{equation}
\label{eq}
\text{Pic}^0(A)=\{\...
- 71
6
votes
1
answer
308
views
Is there a separable isogeny between any two isogenous abelian varieties?
Question: Let $k$ be an algebraically closed field, and $A,B$ abelian varieties over $k$. Suppose there exists an isogeny $A\to B$. Does this imply existence of a separable isogeny $A\to B$?
Known ...
- 1,403
0
votes
0
answers
281
views
Néron–Severi group of Abelian surfaces
Suppose that an Abelian surface $A$ is isogenous to the product of two elliptic curves $E \times E'$. When can we say that the Néron–Severi group is generated by the classes of these two elliptic ...
- 91
7
votes
1
answer
288
views
Automorphic classification of different types of abelian surfaces
For elliptic curves over $\mathbb{Q}$ the Mumford-Tate group is either $\mathrm{GL}_2$ or $\mathrm{Res}_\mathbb{Q}^F (\mathbb{G}_m)$ if it has CM with the imaginary quadratic field $F$. In this case ...
- 190
2
votes
1
answer
160
views
Finding the mistake in an argument concerning $F$-finite $F$-split local Cohen--Macaulay ring of dimension $1$
Let $R$ be a commutative Noetherian ring, and $\phi: R \to R$ be a ring homomorphism. For an $R$-module $M$, let $^{\phi}M$ be the $R$-module defined via restriction of scalars via $\phi$, i.e., as ...
- 412
6
votes
1
answer
391
views
Tate–Shafarevich group of Jacobian of Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$
$C/ \Bbb{Q}: 3X^3 + 4Y^3 + 5Z^3 = 0$ is known to be a nontrivial element of the Tate–Shafarevich group of the elliptic curve $E/\Bbb{Q}:X^3 + Y^3 + 60Z^3 = 0$. It is also an example of an abelian ...
- 1,541
2
votes
0
answers
82
views
Is there any work on the intersection loci of the universal theta divisor with torsion sections?
Let $Y$ be a Siegel modular variety of some non-stacky level and genus $g$, carrying over it a universal principally polarized family of dimension-$g$ abelian varieties $A\to Y$. Inside $A$, with fine ...
- 2,054
6
votes
1
answer
771
views
A regular, geometrically reduced but non-smooth curve
Can anyone give an example of a projective, regular, geometrically reduced but non-smooth curve ?
Of course, the base field should be imperfect.
In Exercise 4.3.22 of Qing Liu's book Algebraic ...
- 620
1
vote
0
answers
184
views
Calculation of de Rham cohomology of abelian varieties/ jacobian varieties
It's known that for a elliptic curve like $E:y^2=x(x-1)(x-t)$ we have a basis $\frac{dx}{y}, \frac{xdx}{y}$ for $H_{dR}^1(E)$. But find such a basis is not an easy thing. I wonder for a general ...
- 785
5
votes
1
answer
312
views
Conductor at 2 of abelian surfaces with real multiplication
Let $A/\mathbb{Q}$ be an abelian surface such that $\text{End}_{\mathbb{Q}}(A)\otimes \mathbb{Q}$ is a real quadratic field $E$. I am interested in bounding the conductor of $A$ at $2$.
Let $\mathfrak{...
- 984
1
vote
0
answers
70
views
Simplicity of abelian varieties and localization
Let $A$ be an abelian variety defined over a number field $K$. Let $v$ be a place of $K$ and denote by $K_v$ the $v$-adic completion of $K$ with respect to $||\cdot||_v$.
Assume $A$ is simple, is it ...
- 2,907
3
votes
1
answer
216
views
How to determine the type of a divisor on a product of elliptic curves?
I already asked this on Math.SE, but didn't receive an answer yet.
Say $E_1, \dotsc, E_n$ are elliptic curves (everything over $\mathbb C$), and $D \subset E_1 \times \dotsc \times E_n$ is an ...
- 1,286
2
votes
0
answers
288
views
A gap in a proof of Orlov’s result on the group of autoequivalences of the derived category of an abelian variety
Let $X$ be an abelian variety over an algebraically closed field of charactristic $0$. In this paper, Orlov showed that there is a short exact sequence $$0\to \mathbb Z \oplus X \times \hat X \to\...
- 256