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An abelian variety over a field is a complete, geometrically connected group variety; equivalently it is a connected projective algebraic group in the usual smooth variety setting. Its group operation is necessarily commutative, a consequence of rigidity rather than an extra condition hidden in the name. An elliptic curve with a chosen origin is a one-dimensional abelian variety; products of elliptic curves give higher-dimensional examples. The additive and multiplicative algebraic groups are not complete and hence are not abelian varieties. Their geometry links line bundles, torsion elements and the Jacobian variety of a curve to questions in number theory.^[ref]

A local form of the Mumford rigidity lemma says: if $X$ is a complete integral variety over an algebraically closed field, and a morphism $f:X\times Y\to Z$ sends one fibre $X\times \{y_0\}$ to a point, then on $X\times U$ for some open neighbourhood $U$ of $y_0$, it factors through the projection to $U$. To see the mechanism, choose an affine neighbourhood of that image point. Properness of the projection supplies $U$ whose whole product maps into that neighbourhood. Regular functions on the complete integral $X$ are constant, forcing the factorization there. Pointed global versions add further hypotheses; the local conclusion alone should not be silently replaced by a global one.^[ref]

Over an algebraically closed field $k$, an ample line bundle $L$ on an abelian variety $A$ defines a homomorphismwhere $A^{\vee }$ is the dual abelian variety and $t_x$ is translation by $x$. This isogeny is a polarization of an abelian variety. Over a general field, a polarization is an isogeny that has this form after extending to an algebraic closure; an inducing ample line bundle need not descend to the original field.^[ref]

A principal polarization is an isomorphism $A\to A^{\vee }$. An elliptic curve with its degree-one origin divisor gives a basic example. A polarization supplies positivity data relating the geometry of line bundles to the algebra of isogenies; twisting $L$ by a degree-zero line bundle does not change ${\lambda }_L$.

A principal polarization of an abelian variety $A$ is a polarization of an abelian variety $\lambda :A\to A^{\vee }$ that is an isomorphism. Equivalently, its isogeny has degree one. It identifies $A$ with its dual abelian variety through positivity supplied by an ample line bundle; an arbitrary isomorphism with the dual need not be a polarization of an abelian variety.^[ref]

For an elliptic curve $E$ with origin $0$, the degree-one line bundle ${\mathcal{O}}_E(0)$ gives a principal polarization. Replacing it by ${\mathcal{O}}_E(n\cdot 0)$, for positive $n$, multiplies the corresponding map to the dual by $n$ and gives degree $n^2$. Thus positivity alone does not make a polarization of an abelian variety principal: the size of its kernel matters.

For a line bundle $L$ on an abelian variety $A$ and translations $T_x(a)=a+x$, the Theorem of the square gives an isomorphism of line bundlesConsequently $x\mapsto T_x^{\ast }L\otimes L^{-1}$ is a group homomorphism from $A$ to its dual abelian variety. These are isomorphism classes: no canonical trivialization is supplied without normalization data. On an elliptic curve, translating a degree-one divisor changes its degree-zero difference class additively, illustrating the identity. The Theorem of the Cube controls the additional compatibility when the translation points themselves vary.^[ref]

Let $E/K$ be an elliptic curve and $n>0$ an integer prime to the characteristic. The Weil pairing is the canonical perfect pairingwhere the arguments are geometric $n$-torsion points of an elliptic curve and ${\mu }_n$ is the group of $n$th roots of unity. It is bilinear and alternating, so $e_n(P,P)=1$ and $e_n(P,Q)=e_n(Q,P{)}^{-1}$. Perfection means that every nonzero first argument pairs nontrivially with some second argument.^[ref]

It is compatible with the absolute Galois group: $e_n(\sigma P,\sigma Q)=\sigma (e_n(P,Q))$. For an isogeny $\varphi :E\to E^{\prime }$ and its dual isogeny, $e_n^{E^{\prime }}(\varphi P,Q)=e_n^E(P,\hat{\varphi }Q)$. In a torsion basis, alternation makes the action on the pairing equal to the determinant; hence the determinant of the torsion Galois representation is the cyclotomic character. In characteristic $p$, the pairing on geometric $p$-torsion points cannot have this same prime-to-characteristic interpretation: ${\mu }_p$ has only one geometric point.