All Questions
1,203 questions
3
votes
0
answers
40
views
p-torsion in the Tate-Shafarevich group of supersingular elliptic curves
Let $E$ be a supersingular elliptic curve over $\mathbb{F}_p(t)$. Is something known on the $p$-torsion of the Tate–Shafarevich group in this case? In particular, I would like to know if (or if known ...
- 103
1
vote
0
answers
89
views
Generic reducedness of geometric generic fibre
Let $f:X\to Y$ be a surjective morphism between two projective schemes over a field of characteristic $p>0$. Also assume that $X$ is smooth,$Y$ smooth & irreducible and $f_*\mathcal{O}_X=\...
- 5,986
3
votes
1
answer
323
views
Which abelian varieties over a local field can be globalized?
As the title says, if $\mathcal{A}$ is an abelian variety over $\mathbb{Q}_p$, is there a criterion as to if I should expect there to exist $A$ over $\mathbb{Q}$ such that
$$\mathcal{A}\cong A\times_{\...
- 4,428
2
votes
1
answer
125
views
Questions about elliptic curves with level-$n$ structure
Let $n$ be a positive integer, which is $4$ or a prime number $l$.
Let $E$ be an elliptic curve defined over a number field $K$.
Assume that all the $n$-torsion points of $E$ are defined over $K$, i.e....
- 77
0
votes
0
answers
82
views
How can complex abelian varieties degenerate to tropical abelian varieties
There is a similar interesting question here
which has not been answered. I therefore ask this question in the hope to get an answer. I wonder how a family of complex abelian varieties can exactly ...
- 85
0
votes
0
answers
190
views
About Chern classes via Atiyah class
I am trying to understand a construction of the Chern classes of a vector bundle $\mathcal{E}$ via the Atiyah class, like is done in this text and here in section 1.4. I am interested in the case ...
- 348
7
votes
1
answer
193
views
Constructing non-split $\mathbf{G}_m$-extensions of elliptic curves
I would like to find examples of abelian surfaces $A$ over a DVR $R$ with residue field $k$, so that the special fibre of $A$ is a non-split $\mathbf{G}_{m/k}$-extension of an elliptic curve over $k$.
...
- 265
5
votes
1
answer
290
views
Why does a line bundle on an abelian variety give a group extension only if it is algebraically trivial?
If $X$ is an abelian variety and $L$ is a line bundle, deleting the zero-section one obtains a diagram
$$
0 \to \mathbb G_m \to Y \stackrel{\phi}{\to} X \to 0
$$
where $\mathbb G_m = \phi^{-1}(0)$,
...
0
votes
0
answers
125
views
Néron-Tate height on abelian varieties and PDEs
Let $A$ be an abelian variety over some number field $K$. We know the $\mathbb{C}$-points of $A$ form a complex analytic manifold, so $A(\mathbb{C})$ is a smooth manifold in fact. It then makes sense ...
3
votes
0
answers
111
views
Weil pairing for an abelian variety uniformized as torus modulo lattice over a non-archimedean field
Suppose I have an abelian variety $A$ with split toric reduction over a non-archimedean field $K$, so that $A$ can be realized as $T / \Lambda$ where $\Lambda$ is a lattice in the torus $T$. Letting $...
- 1,298
1
vote
1
answer
279
views
Moduli space of complex and anti-complex tori?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SL{SL}$By Will Sawin's answer to Moduli Spaces of Higher Dimensional Complex Tori the moduli space of complex $d$-tori is $X ...
- 341
3
votes
1
answer
114
views
The cyclic twist of elliptic curve is a principally polarized abelian variety
Let $L/K$ is a cyclic extension of degree $p$, and let $E/K$ be an elliptic curve.
Let $E^L$ be the kernel of the map $Res^L_{K}(E) \rightarrow E$, where $Res^L_{K}(E)$ is the Weil-restriction.
Is the ...
- 85
1
vote
0
answers
82
views
Behavior of translation functors in characteristic $p$
Let $G$ be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic $p>0$, and let $\mathfrak g$ be the Lie algebra of $G$. Let $U_\chi(\mathfrak g)$ ...
- 2,974
5
votes
1
answer
150
views
Does an abelian variety $A$ have a model over a finite field if its $p$-divisible group $A[p^{\infty}]$ does?
Let $A$ be an abelian variety over an algebraically closed field $k$ of characteristic $p>0$. Let $X := A[p^{\infty}]$ be the associated $p$-divisible group. Assume that $X$ admits a model over a ...
- 769
3
votes
1
answer
159
views
Reference request: generalized Jacobian variety for higher dimensional variety
Let $X\subset \mathbf{P}^n$ be a hypersurface such that the singular locus of $X$ consists of a single ordinary double point. I'm trying to find a reference to the "generalized" intermediate ...
- 389
2
votes
1
answer
159
views
Complexification of Néron models of Abelian varieties
Let $A$ be an abelian variety of dimension $g$ over the quotient field $K$ of a DVR $R$ which is a subfield of the complex field $\mathbb{C}$. Then by a result of Grothendieck, we know that there is a ...
- 85
2
votes
0
answers
43
views
The vector space dimension of Selmer group of abelian variety
Let $A/K$ be an abelian variety with $\mathbb{Z}[\mu_p] \subset End_K(A).$
Let $\pi$ be the prime of $p$, i.e. $(p)=(\pi^{p-1})$.
I want to obtain the relation of Selmer groups $Sel_\pi(A/K)$ and $...
- 85
1
vote
0
answers
104
views
Reference about the semiabelian variety associated to a stable curve
If a stable curve $C$ of genus $g$ is such that its dual graph has Betti number $1$, then the Jacobian of $C$ is a semiabelian variety of torus rank $1$. It is described by Mumford that the data of a ...
1
vote
0
answers
87
views
Birational geometry of special divisor varieties and double covers of curves [closed]
Let $\pi: \tilde{C} \to C$ be an étale double cover of a smooth non-hyperelliptic curve $C$. Associated to this cover is a principally polarized abelian variety $(P, \Xi)$, called the Prym variety, ...
user538396
2
votes
0
answers
95
views
p-adic uniformization pairing
I am studying the paper of R. Greenberg and G. Stevens, $p$-adic $L$-functions and $p$-adic periods of modular forms.
I am currently trying to understand the definition of $\mathcal{L}$-invariant for ...
- 21
2
votes
1
answer
132
views
What is the polarization type of the push-forward of the Poincaré-bundle to the Jacobian of a curve?
$\DeclareMathOperator{\Jac}{Jac}\DeclareMathOperator{\Pic}{Pic}$Let $C$ be a smooth curve of genus $g > 0$, and consider the Picard torus $\Pic^d(C)$ of line bundles of degree $d$. Let $\mathcal P$ ...
- 1,286
2
votes
0
answers
96
views
Confusion about the Lefschetz standard conjectures for abelian varieties in the integral setting
Let $(A,\theta)$ be a principally polarized abelian variety of dimension $d$ over a number field $k$. By the hard Lefschetz theorem
$$H^2(A,\mathbb{Q}_{\ell}) \xrightarrow{\theta^{d-2}} H^{2d-2}(A,\...
- 679
1
vote
0
answers
82
views
Bounding the Bloch-Kato Selmer group of a twisted symmetric power of a Tate module
Let $E$ be an elliptic curve over $\mathbb{Q}$, with good reduction at $p$, and let $V = H^1_{et}(\overline{E}, \mathbb{Q}_p)$ be (the dual of) its (rationalized) Tate module.
Let $S^nV$ denote its $n$...
- 2,907
0
votes
0
answers
88
views
Geometry of prym locus
The celebrated solution to the Schottky problem provides a beautiful geometric characterization of Jacobians among all principally polarized abelian varieties (ppavs). One might hope for a similarly ...
1
vote
0
answers
219
views
What is the relation between two abelian varieties that have the same formal group?
Consider two abelian varieties $A$ and $B$ over the $p$-adic number field $\mathbb{Q}_p$. Let $\hat{A}$ and $\hat{B}$ be the associated formal groups of $A$ and $B$, respectively. Assume that $\hat{A}=...
- 195
2
votes
0
answers
127
views
Function field of abelian varieties
Notation I consider smooth projective varieties over $\mathbb{C}$. $A$ is an abelian variety and $\hat{A}$ is its dual. $k(X)$ is the field of functions of a variety $X$.
Context I try to understand ...
0
votes
0
answers
111
views
Albanese map and curve
Let $S$ be a complex projective integral separated smooth surface (as a scheme). I consider the albanese map $\alpha : S \mapsto A$. I suppose $\alpha(S)$ is a smooth curve of genus $h^{1}(\mathcal{O}...
- 73
3
votes
0
answers
192
views
How can I prove this stronger version of Fedder's Criterion?
I was reading Fedder's original paper which proved what is now known as "Fedder's criterion". I noticed that the abstract stated something which is a priori stronger than what is proved in ...
- 317
1
vote
0
answers
195
views
When does the formal group of an abelian variety possess integral coefficients?
I am looking for a sufficient condition that ensures the formal group associated with an abelian variety has integral coefficients.
In precise, let $A$ be an abelian variety over a number field $K$ ...
- 195
5
votes
1
answer
344
views
Surjection onto endomorphisms of multiplicative group of a field
Let $k$ be an algebraically closed field of characteristic $p > 0$. Denote by $k^\times$ the multiplicative group of $k$. There is a ring homomorphism given by restriction to $k^\times$
$$
\mathbb{...
- 53
1
vote
0
answers
127
views
Does the torsion points of abelian varieties transfer to their formal group laws (upon suitable choice of coordinates)?
Let $A$ and $B$ be two abelian varieties over any algebraically closed field. Let $A[p^n]$ and $B[p^n]$ denotes the set of $p$-power torsion points of $A$ and $B$. Assume that $A[p^n]$ and $B[p^n]$ ...
- 195
17
votes
2
answers
2k
views
How to think of algebraic geometry in characteristic p?
How does a working mathematician usually think about algebraic geometry in characteristic $p$? For the sake of concreteness, and to make things more "geometric" (whatever that means), let's ...
2
votes
0
answers
212
views
Correspondences and Albanese
$\DeclareMathOperator\Alb{Alb}\DeclareMathOperator\CH{CH}$I'm wondering how the Albanese functor interacts with correspondences. Specifically if $X$, $Y$ are say smooth projective schemes over an ...
- 4,428
3
votes
0
answers
135
views
Abel's theorem for cubic threefold
The classical Abel's theorem for curves states that the fiber of Abel-Jacobi map
$$Sym^kC\to J(C),\ (x_1,\ldots,x_k)\mapsto \sum_{i=1}^k\int_{x_0}^{x_i}$$
as a linear functional on $H^0(C,\Omega_C)$ ...
- 1,803
2
votes
0
answers
103
views
Existence of supersingular abelian variety over $\mathbb F_p$ with $\mathcal O_E$-action where $E/\mathbb Q$ is quadratic imaginary ramified at $p$
Let $E/\mathbb Q$ be a quadratic totally imaginary number field with ring of integers $\mathcal O_E$. Let $p > 2$ be an odd prime number which ramifies in $\mathcal O_E$. Let $\mathcal P$ be the ...
- 769
4
votes
1
answer
215
views
Atkin-Lehner involution on the modular abelian varieties
Let $f$ be a CM modular forms with coefficients in the imaginary qudaratic field $K=Q(\sqrt{-3}))$ which correspond to an elliptic curve $E$ defined over $K$. Then Shimura constructed an abelian ...
- 390
1
vote
2
answers
197
views
What are the finite étale coverings of a quasi-hyperelliptic surface?
Let $X$ be a quasi-hyperelliptic surface in characteristic 3 where the canonical bundle $K_X$ is trivial.
Question: Is there a finite étale covering $Y \rightarrow X$ such that
$Y$ is an abelian ...
- 9,501
4
votes
1
answer
288
views
Characterizing principal polarizations of abelian surfaces
Suppose $X$ is a complex abelian variety of dimension 2. Then I believe the ring of endomorphisms $\mathrm{End}(X)$, tensored with $\mathbb{C}$, is isomorphic to a subalgebra $M_2(\mathbb{C})$ of $2 \...
- 22.3k
0
votes
0
answers
80
views
Elliptic curve over global function field: poles of $j$-function & ramification of torsion fields [duplicate]
Let $E/ \Bbb C(t)$ be an elliptic curve over $ \Bbb C(t)$ with nonconstant $j$-invariant $j_E \in \Bbb C(t)-\Bbb C$ and $p>2$ some prime such that it is bigger than an order of a pole $v$ of $j_E$. ...
- 5,986
2
votes
1
answer
200
views
Books and lecture notes about Moduli spaces of Abelian varieties
Following this question, I would like to ask about books and lecture notes for Moduli spaces of Abelian varieties. I suppose that Mumfords book "Geometric Invariant theory" treats it but it ...
- 224
2
votes
1
answer
176
views
Isomorphisms $S^d(S^m(V)^*) \cong \Lambda^d(S^{m+d-1}(V)^*)$
$\DeclareMathOperator\SL{SL}$Let $K$ be an algebraically closed field. Let $V$ be a 2-dimensional vector space over $K$. The group $G=\SL_2(K)$ acts naturally on $V$ by left-multiplication on column ...
- 185
1
vote
1
answer
154
views
Divisors on product abelian fourfolds
Given a principally polarized abelian surface $A$ with CM of signature $(1,1)$ by an imaginary quadratic number field $K$, I am interested in studying the Néron-Severi group $\text{NS}(A\times A)$. ...
- 91
1
vote
1
answer
142
views
Grössencharakter or Galois representation associated to a CM elliptic curve in characteristic $p$
When $E$ is an elliptic curve over a number field $L$, with complex multiplication by a maximal order in an imaginary quadratic field $K$, Silverman’s Advanced Topics in the Arithmetic of Elliptic ...
- 531
4
votes
1
answer
291
views
Effective version for Silverman’s specialization theorem
In his paper, Silverman proves Theorem C (page 208, the indexes of the pages don't match the file), which says that the set $$\{t\in C^0(\bar{K})\mid \sigma_t \text{ is not injective}\}$$ is a set of ...
- 463
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 ...
- 91
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 ...
- 658
3
votes
0
answers
91
views
Mattuck's Theorem for abelian varieties for a non-locally compact field
Let $A$ be an abelian variety of dimension $d$ defined over a complete ultrametric field $K$ of dimension $0$. Let us put on $A(K)$ the topology induced by the one of $K$ (for example, following ...
- 95
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 ...
- 2,521
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 ...
- 658
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.
...
- 1,763