Questions tagged [abelian-varieties]

In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions.

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What is the dimension of the coproduct of two abelian varieties?

Let's say I have two abelian varieties, $A$ and $B$, and I take their coproduct. My intuition says that $\dim(A+B) = \dim(A) + \dim(B)$. Is this accurate?
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If $E$ has additive reduction at $v$ then $H^0(I_v,E[p^\infty])$ is finite

Let $E$ is an elliptic curve defined over a number field $F$, $p$ a prime of $\mathbb{Z}$ and $v$ a valuation of $F$ that does not lie over $p$. Call $F_v$ the completion of $F$ with respect to $v$ ...
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About the type of the polarization of an abelian variety

Let $X$ be an abelian variety of dimension $g$ over an algebraically closed field $k$ and consider $\lambda:X\rightarrow \hat X$ a polarization of $X$ of degree $d$. Assume that $d$ is prime to the ...
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Relations between good ordinary reduction for $E/\mathbb{Q}$ and for $E/K$ with $K$ number field [duplicate]

Reading some articles and Silvermann's "The arithmetic of elliptic curves" I found these two different definitions for good ordinary reduction. Silvermann, with Theorem V.3.1, says that an elliptic ...
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A theorem of Lutz about the structure of the points of an elliptic curve over a finite extension of $\mathbb{Q}_p$

Reading the article of Greenberg "Iwasawa Theory for Elliptic Curves", he cites (p.13) a theorem of Lutz that says: Theorem: Let $E/K$ be an elliptic curve defined over a finite extension $K$ of $\...
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What is the $k^{al}$-points of a variety over $k$?

Let $X=\mathrm{Spec}(\mathbb{Q}[x,y]/(y^2-x^3-x-1))$. So $X$ is an elliptic curve over $\mathbb{Q}$ given by function $y^2=x^3+x+1$. Now I wonder what is meaning of a $\mathbb{C}$ points of $X$? I ...
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Frobenius endomorphism of Abelian varieties

Let $A$ be an abelian variety over $\mathbb{F}_q$ with $q=p^n$, such that all of it's endomorphisms are defined over $\mathbb{F}_q$. Then $End(A)\otimes \mathbb{Q}$ is of Albert type (IV) with center $...
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relation between Riemann theta function and Jacobi theta function

So we know Jacobi 3rd theta function can be defined using different summations such as: \begin{equation} \theta_{3}(a,b)=1+2\sum_{m=1}^{\infty}b^{m^2}\cos(2ma) \end{equation} and I also know that ...
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Existence of non-degenerate line bundles on abelian schemes

Let $X$ be an abelian scheme over some base scheme $S$. Even without projectivity hypothesis, we may talk of the dual abelian scheme $\hat X$ of $X$. Given a line bundle $\mathcal L$ over $X$, the ...
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How do Fourier-Mukai equivalences not contradict reconstruction theorems?

Apparently there is a thing called Fourier-Mukai equivalence in which the derived categories of coherent sheaves of two distinct schemes (e.g. an abelian variety and its dual) can be equivalent. On ...
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Polarization of abelian variety made by the sum of two divisors

Let $X$ be an abelian variety of dimension $n$, and let $L$ be a polarization, that is, an ample line bundle on $X$, with $\chi(L)=3$. In my specific case, I have that $L=\mathcal{O}_X(\Theta + D)$, ...
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How do you identify the tangent space of the Jacobian variety with differential forms of the first kind?

I know that the Jacobian variety of a Riemann surface $X$ can be defined as $$Jac(X) = (\Omega_{hol}^{1}(X))^{*}/\text{H}_1(X, \mathbb{Z}).$$ My question is how do you canonically identify its tangent ...
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Isomorphic elliptic curves (abelian varieties) over $\mathbb{C}$ and $\overline{\mathbb{Q}}$

For two elliptic curves $E_1, E_2$ defined over $\mathbb{Q}$, we assume $E_1$ and $E_2$ are isomorphic over $\mathbb{C}$, then how to prove they are isomorphic over $\overline{\mathbb{Q}}$? Also, can ...
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Proving the immersion part of an Embedding

Trying to see the proof of embedding the Jacobian of a Compact Riemann Surface $X$ using Theta functions. So, using the Theta divisor we have the corresponding line bundle say $L$, we want to prove ...
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What the exact definition of coherent $G-$ sheaf?

I am learning Milne's Abelian Varieties. It gives the definition of coherent $G$-sheaf as following: Let $V$ be a variety over $k$, $k$ is a field. Let $G$ be a finite group acting on $V$ (on the ...
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$p$th roots of unity in a characteristic $p$ field

Let $\mu_n$ denote the group scheme of $n$-th roots of unity over a field $k$. Let $p$ be the characteristic of $k$. I've read that if $(n,p) = 1$, $\mu_n$ is the discrete group isomorphic to the $n$...
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“tensor products of elliptic curves”

Disclaimer: I know next to nothing about elliptic curves (EC) and abelian varieties. The question is motivated by EC cryptography. Question. Suppose I have two elliptic curves $E/k$ and $E'/k$. Is ...
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Étale fundamental group acts by similitude on Tate modules of abelian variety

Note: the below question doesn't make sense. I have mistakenly mixed two different $\pi_1$ and came up with a false claim. See my last comment. Let $X$ be an abelian variety over an algebraically ...
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Generalisation of the Rank of an Abelian Variety

If $A$ is an abelian variety over a number field $K$, then the set of $K$-rational points $A(K)$ is a finitely generated group by the Mordell-Weil Theorem. By the classification of finitely generated ...
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(In)dependence of the conductor of a Galois representation and the choice of l

In the work of A. P. Ogg, Elliptic Curves and Wild Ramification, he proves that the conductor of an elliptic curve is independent of the choice of $\ell$. That is, for example, if $E$ is an elliptic ...
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Abelian covers of smooth algebraic varieties

For each smooth algebraic variety $X$ over some field $F$, does there exist an abelian variety $Y$ defined over $F$ (or some extension of $F$) with an unramified (or étale) morphism $Y \rightarrow X$? ...
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Kernel of multiplication-by-n map between abelian varieties

My question is about Remark 7.3 from Milne's note on abelian varieties: https://www.jmilne.org/math/CourseNotes/av.html (ver. 2.0). Let $A$ be an abelian variety over a field $k$; Let $A_n(k):=\ker([...
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When $\phi_{\mathcal L}=0$ for $\mathcal L$ a line bundle over an abelian scheme $X/S$

Let $X\rightarrow S$ be a projective abelian scheme. To a line bundle $\mathcal L$ on $X$, we associate its Mumford line bundle $\Lambda(\mathcal L):= \mu^{\star}\mathcal L\otimes p_1^{\star}\mathcal ...
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Is the Poincaré sheaf symmetric?

The following discussion is based on the content of FGA explained about the Picard scheme. This is mostly formal: I am trying to find a good way to think about the Poincaré sheaf. Let us consider $\...
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Lie algebra of a commutative group scheme is abelian

I'm reading the proof of the fact that if $G$ is a commutative group scheme, its Lie algebra is abelian in Mumford's book abelian varieties chapter 11. It seems to me that an easier proof should work: ...
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Lower Bound of $x(P)$ in terms of coefficient $a, b$ of an Elliptic Curve

According to Hall–Lang conjecture there are absolute constants $C$ and $κ$ such that for every elliptic curve $E/Q$ given by a Weierstrass equation $E:=y^2=x^3+ax^2+b$ with $a, b \in \mathbb{Z}$ and ...
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Is multiplication by $n$ always an isogeny on an abelian scheme?

Given $n$ a non-zero integer, it is known that multiplication by $n$ on an abelian variety (defined over any field $k$) is an isogeny. The proof of this fact uses the existence of an ample symmetric ...
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Degree of an isogeny of abelian varieties related to the rank of its kernel

Let $\alpha:X\rightarrow Y$ be an isogeny between two abelian varieties over a field $k$, of degree $d$ and separable degree $d_s$. We may assume $k$ to be algebraically closed if necessary. In ...
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47 views

Tangent vector of a scheme

I am reading Mumford's book abelian varieties. On page 97, he defined a tangent vector at $x \in X$ (where $X$ is a scheme of finite type over an algebraically closed field $k$ and $x$ is a point of $...
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Bound on Coefficients of Elliptic Curve

Problem: For a given point $Q:=(x_j,y_j)$ on 2-D, is there always an elliptic curve $E:=y^2=x^3+ax^2+b$ with a point $P:=(x_i,y_i)$ on it, such that $nP=Q$ and $(|x_i|)^k >ab$, where $k\...
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Is scalar multiple of points of finite order less than 17 always?

According to the theorem of Mazur (1977), The torsion subgroup of the group of rational points $E(Q)$ on an elliptic curve must be one of the following 15 groups: $C_N$ with $1 ≤ N ≤ 10$ or $N = 12,$...
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Restriction of $\mathcal O_X(D)$ to a prime divisor occuring in $D$

Let $X$ be a regular variety over a field $k$, and consider $D=\sum_{i=1}^nn_iD_i$ a divisor on $X$, with $n_i\in \mathbb Z\backslash\{0\}$ and $D_i$ a prime divisor. Consider the invertible sheaf $\...
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Maximal closed subscheme over which a line bundle is trivial

The following statement comes from the book "Abelian varieties" by Mumford, at the very beginning of chapter 10. All varieties/schemes are defined over a field $k$. Let $X$ be a complete variety, $...
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What do we mean when we say two elliptic curves are $\mathbb{F}_q$ isomorph?

One text i'm reading has the following definition So i guess $I(t)$ is the set of elliptic curves with $\#E(\mathbb{F}_q) = q + 1-t$, and $N(t)$ is the number of non-equivalent curves. But how is a $\...
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Mordell's Theorem and $nP=Q$ for given $P,Q$

Let $E$ be an elliptic curve given by an equation $E : y^2 = x^3 + Ax + B$ with $A, B \in Q$. Then the group of rational points E(Q) is a finitely generated abelian group. In other words, there is a ...
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Why the discriminant of an elliptic curve being this value has this implication?

The text "Nonsingular Plane Cubic Curves over Finite Fields" by Schoof has the following proposition: The full proof is here: I can't understand this step of the proof: I know why the part in blue ...
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Smallest Integer Solution of an Elliptic Curve

Description: An elliptic curve is defined as $y^2=x^3+ax^2+b$, here, $a, b$ are integers and $(x',y')$ is the point on curve with smallest possible integers coordinate of the elliptic curve. Question:...
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Generic element of linear system over complex torus is reduced

Let $X$ be a complex torus with a fixed positive definite line bundle $L$. I'd like to show that the generic element of $|L|$ is reduced. I tried to use the fact that given $x_1 , \dots x_n$ points ...
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Is the polarization defined by the divisor is defined up to translation?

Let $A$ be an abelian variety, $D \subset A$ an effective divisor inducing the principal polarization on $A$. Question: Does divider $D + p= \{q+p \mid q \in D \}$, for $p \in A$ induce the same ...
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Self product $E \times E$ of an elliptic curve $E$.

Suppose that the self product $E \times E$ of an elliptic curve $E$ contains a compact Riemann surface $C$ of genus $2$. For points $a, b \in E$, we have to $D = a \times E+E \times b$ is ample ...
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Application of Stein factorisation on rigidity lemma

Let $X,Y$ Noetherian schemes and $f:X \to Y$ proper map. The Stein factorisation factorizes $f$ as $X \xrightarrow{g} Spec \text{ } f_* \mathcal{O}_X \xrightarrow{h} Y$ with $h$ finite and $g$ has ...
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Reference request: Supersingular locus of moduli space of abelian varieties

Let $A_{g}\otimes \mathbb{F}_{p}$ be the moduli space of principally polarised abelian varietes of dimension $g$ and $V_{0}$ the subset corresponding to abelian varieties whose $p$-rank is $0$. I ...
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Extend rational map $f: V \dashrightarrow X$ to Abelian variety

I try to understand the proof of an extension theorem proved in Moonen's and van der Geer's draft Abelian varieties(in online accessible notes Theorem 1.18 on page 14): (1.18) Theorem. Let $X$ be an ...
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103 views

Reduction to $k=\bar{k}$ in rigidity lemma

I try to understand a reduction step the proof of rigidity lemma as proved in Moonen's and van der Geer's Abelian varieties (Lemma 1.11 on page 12): Lemma. Let $X$, $Y$ and $Z$ be algebraic varieties ...
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Are abelian schemes flat?

Let $\pi: \mathcal A \to S$ be an abelian scheme, i.e. a proper, smooth group scheme over a scheme $S$ whose geometric fibers are connected and of dimension $g$. Is $\pi$ necessarily flat? What is ...
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Quotient of abelian variety by finite group scheme

Let $X$ be an abelian variety over a field $k$. Let $G$ be a finite group scheme over $k$ acting on $X$ and denote $X/G$ to be the geometric quotient of $X$ by $G$, which always exists in this case. ...
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Weil pairing exact sequence split for an elliptic curve

Let $E$ be an elliptic curve over $\mathbb{Q}$ with a p-torsion point $P$ lying in $E(\mathbb{Q})$, where $p$ is a prime number. Let $e(\ ,\ )$ denote the Weil pairing. Choose a point $Q$ such that $...
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1answer
54 views

$\text{End}_{\mathbb{Q}}(E)=\mathbb{Z}$

Let $E$ be an elliptic curve over the rational number field $\mathbb{Q}$, and I have seen $\text{End}_{\mathbb{Q}}(E)=\mathbb{Z}$ in many places ,but no proof is given. So I want to ask for a proof or ...
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Group action on pullback sheaf.

I want to prove the following fact: If $G$ is a finite group scheme acting freely by $\mu$ on an abelian variety $X$ and $\pi \colon X \rightarrow X/G$ is the quotient map then for any coherent sheaf ...
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Seesaw theorem for the pullback of a line bundle via projection map.

I am trying to understand Mumford's statement of the Seesaw theorem given at the start of Chapter 3 in his book Abelian Varieties. The statement is: Let $X$ be a complete variety, $Y$ a scheme (both ...

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