All Questions
117 questions
3
votes
1
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671
views
Coherent cohomology of an abelian scheme and base change
Let $f\colon A \rightarrow S$ be an abelian scheme of dimension $d$. I would like a reference or an argument for the fact that $R^1f_* \mathcal{O}_A$ is locally free of dimension $d$ and that its ...
- 2,663
3
votes
0
answers
312
views
field of definition of isogenies of abelian varieties
Let $A$ be an abelian variety over a field $k$, and let $N$ be a finite subgroup of $A$. Suppose that $N$ is also defined over $k$, or at least that all Galois automorphisms fixing $k$ leave $N$ ...
- 1,298
3
votes
0
answers
305
views
Abelian varieties by Moonen and van der Geer: proof of rigidity lemma
I try to understand a reduction step in the proof of rigidity lemma as proved in Moonen's and van der Geer's Abelian varieties (Lemma 1.11 on page 12- if the link not work the draft version is online ...
- 6,006
2
votes
0
answers
139
views
Subgroups-ideals correspondence for abelian varieties over $\mathbf{F}_p$
The question I have arose while reading Waterhouse's Thesis (Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 1969 521–560.), and motivates another question I recently asked.
...
- 2,406
2
votes
0
answers
2k
views
What's the best reference for Abelian varieties?
I am curious about learning about Abelian varieties, specifically how they are in some ways generalizations of elliptic curves.
I know of the two sources: https://www.jmilne.org/math/CourseNotes/AV....
- 101
2
votes
1
answer
801
views
Canonical lifts from $\mathbb F_q$ and CM-theory
One knows that (ordinary) Jacobians of hyperelliptic curves over a finite field $\mathbb F_q$ (mostly of genus 1 (elliptic curves) and 2) are extensively studied by cryptographers, as a platform for ...
- 647
2
votes
1
answer
184
views
Is there a way to find any $\mathbb{F}_2(t)$-point on the elliptic curve $\mathcal{E}$?
Consider the ordinary elliptic curves
$$
E\!:y_1^2 + x_1y_1 = x_1^3 + 1,\qquad E^\prime\!: y_2^2 + x_2y_2 = x_2^3 + x_2^2 + 1
$$
over the field $\mathbb{F}_2$. They are quadratic twists to each other....
- 1,833
2
votes
1
answer
446
views
A curve in an abelian surface and its image in the Kummer surface
This is related to a question that I already asked Curve through the 16 singular points of a Kummer surface. I am new to Abelian varieties and I am trying to understand more about them.
Let $X=J(C)$ ...
- 643
2
votes
1
answer
771
views
Why is the Tate local duality pairing compatible with the Cartier duality pairing?
This question is a follow up to Why is the norm map dual to restriction under Tate local duality?
Let $A$ and $B$ be dual abelian schemes over a base scheme $S$. For an integer $n \ge 1$, consider ...
- 1,103
2
votes
1
answer
583
views
Nef divisors on abelian varieties
The following question stems from a question I already asked on MO:
Nakai-Moishezon theorem for abelian varieties
I would like to prove that if $L_0$ is an ample line bundle on an abelian variety $A$...
- 663
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{...
user19475
2
votes
2
answers
834
views
Shimura datum of family of fake elliptic curves
Suppose we have a PEL type $(H,\phi ,*;T,O,V)$ where H is a rational nonsplit quaternion algebra, $\phi$is an embedding of Q-algebra $\phi : H-->M(2,R)$, and * is a positive anti involution of H; ...
- 709
1
vote
0
answers
299
views
Application of Galois descent
I not understand an assumption done at the beginning of the proof of Rigidity lemma in Moonens and van der Geers book about Abelian variaties (page 12). Here is it:
Question: Why the assumption $k= \...
- 6,006
1
vote
0
answers
539
views
Ext of Tate-modules of abelian varieties
Let $K$ be a local field (in fact, finite extension of $\mathbb{Q}_p$) and let $A$ and $B$ be abelian varieties over $K$.
Associated to $A$ and $B$ are the Tate-modules $T_p(A)$ and $T_p(B)$.
Both ...
- 1,841
1
vote
1
answer
285
views
a problem about ideals of polynomial rings
Let $\{f_n\}_{n=1}^\infty\in \mathbb{C}[x,y]$ be a sequence of polynomials given by the following expressions
$$
f_n(x,y)=\sum_{i=0}^{[\dfrac{n}{2}]}(-1)^{n-i}{{n-i}\choose i}x^{n-2i}y^i.
$$
Let $(...
- 1,990
0
votes
2
answers
2k
views
non discrete valuation ring [closed]
Hi,
I am looking for examples of non-discrete valuation rings. Could you help me?
Thanks
- 141
0
votes
1
answer
271
views
Proof of rigidity lemma
I have problems to understand a proof in this paper by Pierrick Dartois on Abelian varieties:
Theorem 1.13 (rigidity lemma). Let $ \varphi: X \times_k Y \to Z$ be a morphism of $k$-schemes. Assume ...
- 6,006