Abelian varieties over algebraic number fields are indispensible tools in algebraic geometry and number
theory. In this report, I give a brief introduction of abelian varieties and offer some deeper topics of
interest.
2 Abelian Varieties
Abelian varieties have the special property that their points form a group in such a way that the maps defining the
group structure are given by morphisms. To give a proper definition of an abelian variety, we reintroduce a familiar
definition of a group.
Definition 1.A group consists of a set G together with maps m : G × G → G (the group law),
i : G → G (the inverse), and an identity element e ∈ G such that we have the following eqalities of
maps:
(Associativity) m ∘ (m × idG) = m ∘ (idG× m) : G × G × G → G.
(Identity) m ∘ (e × idG) = j1 : {e}× G → G and m ∘ (idG× e) = j2 : G ×{e}→ G.
j1 and j2 are the canonical identifications {e}× GG and G ×{e}G, respectively. e is the
inclusion map {e}G.
(Left and Right Inverse) e ∘ π = m ∘ (idG× i) ∘ ΔG = m ∘ (i × idG) ∘ ΔG : G → G.
π : G →{e} is the constant map and ΔG : G → G × G is the diagonal map.
Then, we define a group variety in a similar fashion.
Definition 2.A group variety over a field k is a k-variety X together with k-morphisms m : X × X → X (the
group law), i : X → X (the inverse), and a k-rational point e ∈ X(k) (the identity element) such that we have the
following equalities of morphisms.
(Associativity) m ∘ (m × idX) = m ∘ (idXG × m) : X × X × X → X.
(Identity) m ∘ (e × idX) = j1 : Spec(k) × X → X and m ∘ (idX× e) = j2 : X × Spec(k) → X.
j1 and j2 are the canonical identifications Spec(k) × XX and X × Spec(k)X, respectively.
(Left and Right Inverse) e ∘ π = m ∘ (idX× i) ∘ ΔX∕k = m ∘ (i × idX) ∘ ΔX∕k : X → X.
Definition 3.An abelian variety X is a group variety that is complete. In other words, X is a group
variety with the property that for any variety Y , the projection X × Y → Y is a closed morphism.
Remark.A group variety is complete if it is a projective variety.
We are hoping these abelian varieties are commutative. We will prove this by way of the rigidity
lemma.
Lemma 4 (Rigidity Lemma).Let X be a complete variety, Y and Z any varieties, and f : X × Y → Za morphism such that for some y0∈ Y,f(X ×{y0}) is a single point z0of Z. Then, there is a morphismg : Y → Z such that if p2 : X × Y → Y is the projection, f = g ∘ p2. [Mum85]
Proof.For any point x0∈ X, define g : Y → Z by g(y) = f(x0,y). Let U be an affine open neighborhood of
z0 in Z, F = Z -U, and G = p2(f-1(F)). Then, G is closed in Y since X is complete, so p2 is a closed map.
Since y0G, Y -G = V is non-empty open subset of Y . For each y ∈ V , the complete variety X ×{y} gets
mapped by f into the affine variety U, and so it gets mapped into a single point of U. Therefore, for any
x ∈ X,y ∈ V,f(x,y) = f(x0,y) = g ∘ p2(x,y). __
Corollary 5.If X and Y are abelian varieties and f : X → Y is any morphism, f(x) = h(x) + a where his a homomorphism of X into Y and a ∈ Y .
Proof.Let f = f - f(0) and assume that f(0) = 0. Consider the morphism ϕ : X × X → Y defined by
ϕ(x,y) = f(x+y)-f(y)-f(x). Then, ϕ(X ×{0}) = ϕ({0}×X) = 0. By the rigidity lemma, ϕ is equivalent
to 0 on X × X, and so f must be a homomorphism. __
Corollary 6.X is a commutative group.
Proof.The morphism of X into itself mapping each element onto its inverse is a homomorphism by Corollary
1. For x,y ∈ X,(xy)-1 = x-1y-1 = y-1x-1. __
Now, we have the tools to prove the following theorem (although I omit the proof here. See page 45 of
[Mum85].)
Theorem 7.Let X be a complete variety, e ∈ X a point, and
a morphism such taht m(x,e) = m(e,x) = x for all x ∈ X. Then, X is an abelian variety with group lawm and identity e.
From here, it is natural to study divisors on abelian varieties for the purpose of showing that abelian varieties are
indeed projective. Some important tools for this result are the theorem of the square and the see-saw principle (see
chapter 2 of [GM]). Rather than diving deeper into the theory, let’s look at a classic example of an abelian
variety.
3 Elliptic Curves
As Hartshorne puts it, ”the theory of elliptic curves is varied and rich, and provides a good example of the
profound connections between abstract algebraic geometry, complex analysis, and number theory,” [Har77]. In this
section, we show that elliptic curves are abelian varieties by deriving the rational maps (through Sage) that define
the group law.
3.1 The Group Law for Elliptic Curves
The group law for an elliptic curve defined by a Weierstrass equation is given by the following rule
[Sut21]:
For projective curves in Weierstrass form E : y2z = x3 + Axz2 + Bz3, we will take our distinguished point to be
O = (0 : 1 : 0). Then, every point P = (x0 : y0 : 1) that’s not the distinguished point on the curve has a nonzero
z-coordinate (scaled to 1). We find another point Q = (x0 : -y0 : 1) on the curve for which the line through
P and Q is defined by x = x0z. O also lies on the line, we have three points P,Q,O that lie on a
line:
(Identity) O is the identity element. Take any non-distinguished point P. Then, the line between O
and P intersects the curve at -P: O + P + (-P) = O. Thus, O + P = P.
(Inverse) See above. For a point P on the curve, -P is its inverse.
Theorem 8.Every one-dimensional abelian variety is an elliptic curve.
I leave the proof for the reader. One must show that projective curves have genus 1, and the group operation on
the abelian variety is the same as the one induced by the group law, taking the identity element as the distinguished
point.
4 Further Reading
I have left off a portion on the algebraic theoy of abelian varieties concerning cohomology groups and the theorem of
the cube. For this, I would read [Mum85].
There are plenty of directions to go following this material (after studying the above). The following are natural
ideas to explore:
complex tori and their equivalence to abelian varieties
G and G ×{e}
G, respectively. e is the
inclusion map {e}
G.
