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Results tagged with abelian-varieties
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user 82179
Abelian varieties are projective algebraic varieties endowed with an Abelian group structure. Over the complex numbers, they can be described as quotients of a vector space by a lattice of full rank. They are analogs in higher dimensions of elliptic curves, and play an important role in algebraic geometry and number theory.
3
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Commutation of endomorphisms of abelian varieties
Remark. The condition that $\bigcup_{r \geq 0} \ker(\phi^r)(k)$ is Zariski dense is not needed. Indeed, if it is not satisfied, replace $\phi$ by $[\ell] \circ \phi$ for $\ell$ invertible in $k$. Sinc …
- 28.8k
16
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Points of abelian varieties over purely transcendental extensions
This follows from the following well-known lemma.
Lemma. Let $A$ be an abelian variety over $k$. Then any map $f \colon \mathbb P^1 \to A$ is constant.
Proof 1. The map $f$ induces a map on the Alba …
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Stabilizers in abelian varieties are also abelian? reference request
This is not true: it does not need to be connected even if $X$ is smooth connected over an algebraically closed field $K$ of characteristic zero. Indeed, if there is an isogeny $\pi \colon A \to B$ an …
- 28.8k
1
vote
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Reduction step to $k=\bar{k}$ in the proof of rigidity lemma
One way to argue is as follows: given a morphism $f \colon X \times Y \to Z$, consider its graph $\Gamma_f \subseteq X \times Y \times Z$. Let $W$ be the scheme-theoretic image of $\Gamma_f$ under the …
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5
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Is multiplication by $d$ on the Jacobian of a nodal curve étale?
Lemma. Let $G$ be a commutative group scheme of finite type over a field $k$, and let $d$ be a positive integer invertible in $k$. Then the multiplication by $d$ map $[d] \colon G \to G$ is finite éta …
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1
vote
$p$-divisibility of Picard groups
Inspired by the $\mathbf G_m$ case (Kummer theory), here is a positive result if you assume that $E(k)$ contains full $\ell$-torsion:
Lemma. Let $k$ be a field and $\ell$ a prime invertible in $k$, su …
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4
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Is the set of points on an abelian surface which project to rational points on the Kummer su...
Varieties over a field $k$ can be understood by their $\bar k$-points with the Zariski topology, together with the $\operatorname{Gal}(\bar k/k)$-action. Any morphism $f \colon X \to Y$ of $k$-varieti …
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11
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Do all simple factors of jacobians of curves come from correspondences?
This argument is mostly contained in t3suji's comments, but with some of the proofs somewhat expanded.
As a convention (consistent with that of the theory of Chow motives), all actions of corresponde …
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4
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Zeta function of Abelian variety over finite field
This is not an answer.
Here is a possible strategy, that was too long for a comment. I briefly thought it gave a full answer, but there is a lot of stuff missing.
If $\alpha_1,\ldots,\alpha_{2g}$ ar …
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4
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On families of supersingular abelian surfaces over the projective line
I will answer the question for the specific family of abelian surfaces as constructed by Moret-Bailly [MB]. There might be other types of examples for which the answer is different (?). We will recall …
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3
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Néron models vs integral models
Note that sheaf pushforward and preheaf pushforward agree, so this a question about categories, not sites.
Lemma. If $X$ is a finite type $k$-scheme with $\dim X > 0$, then $j_*h_X$ is not re …
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2
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Reference for torsion-freeness of the group of correspondences on a smooth projective variety
It's not just torsion-free, we can actually compute it in terms of the Picard and Albanese variety of $T$, by the following classical result:
Lemma. If $X$ and $Y$ are smooth projective varieties over …
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1
vote
Accepted
Cup products and correspondences
It is not true that $a^*(\alpha \smile \beta) = a^*\alpha \smile a^*\beta$:
Example. Let $X = \mathbf P^2$, and let $a$ be the Künneth projector onto $H^2(\mathbf P^2) \subseteq H^*(\mathbf P^2)$. Exp …
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A constructive proof of the theorem of the cube
This is not really an answer, but a rephrasing together with some comments on why this is difficult. I end with one example where you can actually compute something (purely algebraically) on $E \times …
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7
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Étale group schemes and specialization
If $k \to \ell$ is any regular field extension (i.e. $k$ is algebraically closed in $\ell$), then $X(k) \to X(\ell)$ is a bijection when $X \to \operatorname{Spec} k$ is étale. Indeed, it suffices to …
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