# 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.

294 questions
Filter by
Sorted by
Tagged with
22 views

Let $E$ be a supersingular elliptic curve over $F_q$ where $q=p^n$, then $\operatorname{End}(E)$ is an order in quaterion algebra, hence a non-commutative ring. Question: Is there an endomorphism $\... 0 votes 1 answer 38 views ###$E(\mathbb{F}_q)$is a torsion group where$E$is an elliptic curve? Let$E$be an elliptic curve defined over$\mathbb{F}_q$with$q=p^n$, then how to deduce$E(\mathbb{F}_q)$is a torsion group? In other words, for any$\mathbb{F}_q$-rational point$P$, why does ... 1 vote 1 answer 46 views ### Non-trivial$2$-torsion point on elliptic curves If$E$is an elliptic curve over$\mathbb{F}_p$where$p\geq5$and$\#E(\mathbb{F}_p)$is even, then does$E$have a non-trivial$2$-torsion point defined over$E(\mathbb{F}_p)$? In other words, if we ... 0 votes 0 answers 47 views ### Homogeneous space of elliptic curve in Silverman's AEC I have a question in X.3 proposition 3.2 of Silverman's book AEC. Let$E/K$be an elliptic curve and$C/K$be a homogeneous space for$E/K$. Fix a point$p_0\in C$and define a map$\theta: E\... 