All Questions
1,203 questions
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
2
votes
1
answer
106
views
Does a polynomial $P(X,Y)$ that specializes to a polynomial $P(x_0,Y)$ with distinct roots in $\overline{k}$ have distincts roots in $\overline{k(X)}$
Let $k$ be a field, $P\in k[X][Y]$ be a monic polynomial of degree $n$ in $Y$.
I am looking for a simple proof of the following fact.
"If there exists $x_0\in k$ such that $P(x_0,Y) \in k[Y]$ has ...
- 12.9k
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
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
2
votes
1
answer
373
views
Can an abelian surface be bielliptic
Is an abelian surface containing an elliptic curve a bielliptic surface?
Suppose I have an abelian surface $A$ over the complex numbers that contains an elliptic curve $E$. Then
$A \to A/E$ is an ...
- 39.3k
4
votes
1
answer
746
views
Vanishing of $\text{Ext}^2$ sheaf from abelian variety to multiplicative group
Does anyone know a proof or reference for the following statement? Or if it's false (which seems unlikely to me), a counterexample?
Let $k$ be a field (maybe we need it to be perfect) and $A$ an ...
- 773
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 ...
- 63.9k
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
7
votes
1
answer
303
views
Explicit equations for the universal vector extension of an elliptic curve
The universal vector extension $E$ of an abelian variety $A$ is an algebraic group, an extension of $A$ by a vector group $0 \to V \to E \to A \to 0$, such that any other extension of $A$ by a vector ...
- 30.5k
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 ...
- 5,050
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
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
$$p=\omega*\overline\omega,\quad\text{where}\quad\omega\equiv 1\pmod 3.$$...
- 47.4k
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
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
13
votes
3
answers
1k
views
Faltings height in short exact sequences
Let $K$ be a number field and $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ a short exact sequence of abelian varieties over $K$. Let $h(A)$ denote the logarithmic Faltings height (...
- 1,116
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
6
votes
1
answer
268
views
Hochschild cohomology of an Azumaya algebra
Let $k$ be a field. Given a commutative $k$-algebra $Z$ and an associative algebra $A$ that is Azumaya over $Z$, do we have an isomorphism of Hochschild cohomologies: $HH^*(A) \cong HH^*(Z)$?
This is ...
- 63.9k
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
4
votes
1
answer
238
views
Homogeneous polynomials cutting out complex abelian varieties
This is an update to a previous question of mine. The more clarified questions, results and definitions make me feel like this warrants a separate post instead of a large edit of the original one.
...
- 3,272
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 ...
- 63.9k
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
4
votes
2
answers
319
views
Is the set of points on an abelian surface which project to rational points on the Kummer surface a subgroup?
Let $C$ be a hyperelliptic curve of genus 2 defined over $\mathbb{Q}$, let $J$ be its Jacobian, and let $X$ be the Kummer surface associated to $J$ (i. e. $X$ is the singular Kummer surface which ...
- 4,745
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)...
- 149k
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 $...
- 149k
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 ...
- 149k
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\...
- 9,497
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 ...
- 28.8k
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
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
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 ...
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
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 ...
- 6,377
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
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 ...
- 52.7k
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
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
2
answers
1k
views
Hochschild Kostant Rosenberg theorem for varieties in positive characteristic?
Is there is a known version of the HKR theorem as proved in say Swan's paper "Hochschild Cohomology of Quasiprojective Varieties" in positive characteristic? I assume something is known about this as ...
- 2,368
3
votes
0
answers
132
views
How does the number of connected components of the Néron model change in a family of abelian varieties?
Given an elliptic curve $E/\mathbb{Q}_p$, it is known that the component group of the Néron model of $E$ is cyclic of order $-v(j(E))$ when $E$ has split multiplicative reduction, and in all other ...
- 333
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
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 ...
- 20.5k
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 ...
- 10.5k
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).
...
- 12.9k
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
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
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
3
votes
0
answers
91
views
Can you find a Darboux basis for any skew integral form on a full-rank lattice in $ℂ^n$ so that the first $n$ vectors are $ℂ$-linearly independent?
Any skew bilinear form $\omega$ on $\mathbb{Z}^{2n}$ can be brought into the form
\begin{equation}
\begin{pmatrix}
0 & \Delta \\
-\Delta & 0
\end{pmatrix},
\quad
...
- 1,446
3
votes
0
answers
256
views
What is a twisted D-module?
Let $X/\mathbb{C}$ be an abelian variety, $Y$ be the dual abelian variety, and $P$ be the Poincaré bundle on $X\times Y$. On p.207, Correction to “Sheaves with connection on abelian varieties” (by M. ...
- 615
1
vote
0
answers
88
views
Bad primes of twists of modular curves $X_E^{-1}(p)$
I am interested in a good reference for reading about primes of bad reduction for the modular curve $X_E^{-}(N)$, which is a twist of $X(N)$ parametrizing all elliptic curves $E'$ whose $N$-torsion ...
- 637
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 ...
- 50.7k