notes on abelian varieties [part i]matyd/av1.pdfnotes on abelian varieties [part i] 3 in particular,...

NOTES ON ABELIAN VARIETIES [PART I]

TIM DOKCHITSER

Contents

1. Affine varieties 22. Affine algebraic groups 53. General varieties 94. Complete varieties 105. Algebraic groups 136. Abelian varieties 167. Abelian varieties over C 178. Isogenies and endomorphisms of complex tori 209. Dual abelian variety 2210. Differentials 2411. Divisors 2612. Jacobians over C 27

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1. Affine varieties

We begin with varieties defined over an algebraically closed field k = k.

By Affine space An = Ank we understand the set kn with Zariski topology :V ⊂ An is closed if there are polynomials fi ∈ k[x1, ..., xn] such that

V = x ∈ kn | all fi(x) = 0.Every ideal of k[x1, ..., xn] is finitely generated (it is Noetherian), so it doesnot matter whether we allow infinitely many fi or not. Clearly, arbitraryintersections of closed sets are closed; the same is true for finite unions:fi = 0 ∪ gj = 0 = figj = 0. So this is indeed a topology.

A closed nonempty set V ⊂ An is an affine variety if it is irreducible,that is one cannot write V = V1 ∪ V2 with closed Vi ( V . Equivalently,in the topology on V induced from An, every non-empty open set is dense(Exc 1.1). Any closed set is a finite union of irreducible ones.

Example 1.1. A hypersurface V : f(x1, ..., xn) = 0 in An is irreducibleprecisely when f is an irreducible polynomial.

Example 1.2. The only proper closed subsets of A1 are finite, so A1 andpoints are its affine subvarieties.

Example 1.3. The closed subsets in A2 are ∅, A2 and finite unions of pointsand of irreducible curves f(x, y) = 0.

With topology induced from An, a closed set V becomes a topologicalspace on its own right. In particular, we can talk of its subvarieties (irre-ducible closed subsets). The Zariski topology is very coarse; for example,every two irreducible curves in A2 have cofinite topology, so they are homeo-morphic. So to characterise varieties properly, we put them into a category.A map of closed sets

φ : An ⊃ V −→ W ⊂ Am

is a morphism (also called a regular map) if it can be given by x 7→ (fi(x))with f1, ..., fm ∈ k[x1, ..., xn]. Morphisms are continuous, by definition ofZariski topology. We say that φ is an isomorphism if it has an inverse thatis also a morphism, and we write V ∼= W in this case.

A morphism f : V → A1 is a regular function on V , so it is simply afunction V → k that can be given by a polynomial in n variables. Theregular functions on V ⊂ An form a ring, denoted k[V ], and clearly

k[V ] ∼= k[x1, ..., xn]/I, I = f s.t. f |V = 0.Composing a morphism φ : V → W with a regular function on W gives aregular function on V , so f determines a ring homomorphism φ∗ : k[W ] →k[V ], the pullback of functions. Conversely, it is clear that every k-algebrahomomorphism k[W ] → k[V ] arises from a unique f : V → W . In otherwords, V → k[V ] defines an anti-equivalence of categories

Zariski closed sets −→ finitely generated k-algebras with no nilpotents.

NOTES ON ABELIAN VARIETIES [PART I] 3

In particular, the ring of regular functions determines V uniquely.Now suppose V is a variety. Then k[V ] is an integral domain (Exc 1.3),

and the (anti-)equivalence becomes

affine varieties over k −→ integral finitely generated k-algebras.

The field of fractions of k[V ] is called the field of rational functions k(V ).Generally,

φ : An ⊃ V W ⊂ Am

is a rational map if it can be given by a tuple (f1, ...fm) of rational functionsfi ∈ k(x1, ..., xn) whose denominators do not vanish identically on V . Inother words, the set of points where φ is not defined is a proper closedsubset of V , equivalently φ is defined on a non-empty (hence dense) open.So rational functions f ∈ k(V ) are the same as rational maps V P1.

Example 1.4. The ring of regular functions on An is k[An] = k[x1, ..., xn],and k(An) = k(x1, ..., xn)

It is important to note that the image of a variety under a morphism isnot in general a variety: 1:

Example 1.5. The first projection p : A2 → A1 takes xy = 1 to A1 \ 0,which is not closed in A1.

Example 1.6. The map A2x,y

(xy,y)−→ A2t has image A2 \ x-axis ∪ (0, 0).

The first example can be given a positive twist, in a sense that it actuallygives U = A1 \ 0 a structure of an affine variety. Generally, for a rationalmap φ : V V ′ and U ⊂ V open, say that φ is regular on U if it is definedat every point of U . (For U = V it coincides with the notion of a regularmap as before.) If φ has a regular inverse ψ : V ′ → V with ψ(V ′) ⊂ U , wecan think of U as an affine variety isomorphic to V ′. In the example abovetake V ′ : xy = 1 with φ(t) = (t, t−1) and ψ(x, y) = t.

Example 1.7. If V ⊂ An is a hypersurface f(x1, ...xn) = 0, then the com-plement U = An \ V has a structure of an affine variety with the ring ofregular functions k[x1, ..., xn, 1/f ].

Many properties of V have a ring-theoretic interpretation. Two veryimportant ones are:

The dimension d = dimV is the length of a longest chain of subvarieties

∅ ( V0 ( · · · ( Vd ⊂ V.(For k = C this agrees with the usual dimension of a complex manifold.)With k[Vi] = k[x1, ..., xn]/Pi, this becomes the length of a longest chain ofprime ideals

k[V ] ) P0 ) · · · ) Pd = 0,

1What is true is that the image f(X) ⊂ Y always contains a dense open subset of the

closure f(X) ([?] Exc II.3.19b).

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which is by definition the ring-theoretic dimension of the ring k[V ]. For avariety V it is, equvalently, the transcendence degree of the field k(V ) over k.

Example 1.8. dimAn = dim k[x1, ..., xn] = n.

Example 1.9. A hypersurface H ⊂ An has dimension n− 1.

Let V be a variety and x ∈ V a point. A regular function on V may beevaluated at x, and the kernel of this evaluation map k[V ]→ k is a maximalideal. (Conversely, every maximal ideal of k[V ] is of this form.) The localring Ox = OV,x is the localisation of k[V ] at this ideal. In other words,

Ox =fg ∈ k(V )

∣∣ f, g ∈ k[V ], g(x) 6= 0

This is a local ring, and its maximal ideal mx is the set of rational functionsthat vanish at x.

Write d = dimV . We say that x ∈ V is non-singular if, equivalently,

(1) dimkmxm2x

= n− d. (‘≥’ always holds.)

(2) the completion Ox = lim←−Ox/mjx is isomorphic to k[[t1, ..., td]] over k.

(3) If V is given by f1 = ... = fn = 0, the matrix ( ∂fi∂xj(x))i,j has rank d.

We say that V is regular (or non-singular) if every point of it is non-singular.Generally, a morphism f : V → W is smooth of relative dimension n if forall points f(x) = y the pullback of functions f∗ : Ox ← Oy identifies

Oxf∗←− Oy ∼= Ox[[t1, ..., tn]].

So V is regular if and only if V → pt is smooth and, in general, a smoothmorphism has regular fibres (preimages of points).

Example 1.10. The curves C1 : y = x2 and C2 : y2 + x2 = 1 in A2 arenon-singular, and C3 : y2 = x3 and C4 : y2 = x3 + x2 are singular at (0, 0).

It follows from (3) that the set of non-singular points Vns ⊂ V is open(Exc 1.5), and it turns out it is always non-empty ([?] Thm. I.5.3); inparticular, it is dense in V .

Finally, there are products in the category of varieties, and they corre-spond to tensor products of k-algebras. In other words, if V ⊂ Am and W ⊂An are closed sets (resp. varieties) then so is V ×W ⊂ Am × An = Am+n,and k[V ×W ] ∼= k[V ]⊗k k[W ].

Exc 1.1. A topological space is irreducible if and only if every non-empty open set is

dense.

Exc 1.2. Zariski topology on A2 = A1 × A1 is not the product topology.

Exc 1.3. Prove that the ring of regular functions k[V ] of an affine variety V is an integral

domain.

Exc 1.4. Take the curves C : y2 =x3, D : y2 =x3+x2 and E : y2 = x3 + x in A2, and the

point p = (0, 0) on them. Prove that OC,p ∼= k[[t2, t3]], OD,p ∼= k[[s, t]]/st, OE,p ∼= k[[t]]

and that they are pairwise non-isomorphic.


Exc 1.5. Suppose V is given by f1 = ... = fn = 0. Looking at the determinants of the

minors of the matrix ( ∂fi∂xj

), prove that the set of singular points of V is closed in V .

2. Affine algebraic groups

In the same way as topological groups (Lie groups, ...) are topologicalspaces (manifolds, ...) that happen to have a group structure, affine algebraicgroups are simply closed affine sets with a group structure.

Definition 2.1. A group G is an affine algebraic group over k if it has astructure of a Zariski closed set in some Ank , and multiplication G×G→ Gand inverse G→ G are morphisms.

Equivalently, starting with a closed set V ⊂ An instead, we require

(1) A point e ∈ V (unit element),(2) A morphism m : V × V → V (multiplication),(3) A morphism i : V → V (inverse),

which satisfy the usual group axioms2.

Example 2.2.

(1) The additive group Ga = A1, group operation (x, y) 7→ x+ y.(2) The multiplicative group Gm = A1\0, group operation (x, y) 7→ xy.

Recall that A1 \0 is a variety via its identification with xy = 1 ⊂ A2.In this notation, multiplication becomes (x1, y1), (x2, y2) 7→ (x1x2, y1y2).This example naturally generalises to GLn (GL1 being Gm):

Example 2.3. Write Mn = An2 for the set of n × n-matrices over k, andIn ∈Mn for the identity matrix. The classical groups

GLn =A ∈Mn

∣∣ detA 6= 0,

SLn =A ∈Mn

∣∣ detA = 1,

On =A ∈Mn

∣∣ AtA = In,

Sp2n =A ∈M2n

∣∣ AtΩA = Ω (

Ω =(

0−In

In0

) ).

are affine algebraic groups. This is clear for G = SLn,On,Sp2n, because thedefining conditions are polynomial in the variables, so G ⊂ An2 is closed.As for GLn, it is the complement to the hypersurface det(aij) = 0, henceaffine by Example 1.7 (cf. also Exc 2.1).

A homomorphism of affine algebraic groups is a morphism that is alsoa group homomorphism, an isomorphism is a homomorphism that has aninverse, and subgroups and normal subgroups will always refer the ones thatare closed in Zariski topology. If H ⊂ G is any ‘abstract’ subgroup, itsZariski closure is a subgroup in this sense (Exc 2.2).

2So V(e,id)−→ V×V m−→V is identity and same for (id, e) (unit), m (m×id) = m (id×m)

as maps V ×V ×V → V (associativity), and Vdiag−→ V ×V (id,i)−→ V ×V m−→ V is constant e

(inverse).

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It is clear that the product of two algebraic groups is an algebraic group,and that the kernel of a homomorphism φ : G1 → G2 is an algebraic group.It is also true that the image φ(G1) is an algebraic group (Exc 2.3).

Example 2.4. The classical groups G = SLn,On,Spn are subgroups ofGLn, and the embeddings G → GLn and the determinant map G → Gm

are homomorphisms.

For g ∈ G, the left translation-by-g map lg : G → G is an isomorphism.Because these maps act transitively on G, every point of G ‘looks the same’.For instance, the irreducible components Gi of G cannot meet. Also, becausethe set of non-singular points Gi,ns ⊂ Gi is non-empty, Gi is non-singular(use left translations again).

In particular, the connected component of identity G0 ⊂ G is a non-singular variety. It is a normal subgroup of G, and its left cosets are theconnected components of G (Exc 2.7). So G/G0 is finite and G = G0 o ∆for some finite group ∆. Thus it suffices to understand connected groups;the classical matrix groups are connected (Exc 2.8).

Example 2.5. Every finite group G is an affine algebraic group, via theregular representation G ⊂ Aut k[G] = GLn.

In particular, every finite affine algebraic group is a closed subgroup ofsome GLn. (In fact, any faithful k-representation of G, not just the regularrepresentation, defines such an embedding.) Somewhat surprisingly, it turnsout that this is true for all affine algebraic groups:

Theorem 2.6. Every affine algebraic group G is a closed subgroup of GLnfor some n.

We will prove this, if only to illustrate that questions about affine va-rieties are really questions about k-algebras. First, some preliminaries arenecessary.

An action of G on a variety V is a group action α : G×V → V which is amorphism. If V = An and the action is linear (i.e αg : V → V is in GLn(k)for all g ∈ G), we say that V is a representation of G. Equivalently, it is ahomomorphism G→ GLn of algebraic groups.

To prove the theorem, we need to find Σ : G→ GL(V ) whose correspond-ing ring map Σ∗ : k[GL(V )] → A is surjective. (Then Σ is an isomorphismof G onto Σ(G).) Putting aside the question of surjectivity, where can wepossibly find a non-trivial representation in the first place?

Let us reformulate this on the level of the ring A = k[G]. Write3

m∗ : A −→ A⊗A, i∗ : A −→ A, e∗ : A→ k

3A k-algebra A with such maps m∗, i∗ and e∗ that satisfy the axioms (e∗⊗ id)m∗= id,(id⊗m∗) m∗=(m∗ ⊗ id) m∗ and (i∗ ⊗ id) m∗= e∗ (dual to the previous footnote) iscalled a Hopf algebra, and the maps comultiplication, counit and coinverse respectively.


for the ring maps corresponding to the multiplication m : G ×G → G, theinverse i : G → G and the identity e : pt → G. To give a representationΣ : G→ GL(V ) is equivalent to specifying a k-linear map

σ : V −→ V ⊗A

such that (id⊗e∗) σ = id and (id⊗m∗) σ = (σ ⊗ id) σ. We say that σmakes V into an A-comodule.

There is only one obvious A-comodule, and that is V = A = k[G] itselfwith σ = m∗, the co-multiplication map. The only problem is that k[G] isan infinite-dimensional k-vector space4, so what we need is the following

Lemma 2.7. Every finite-dimensional k-subspace W ⊂ A is contained in afinite-dimensional subcomodule V ⊂ A (so m∗(V ) ⊂ V ⊗A).

Proof. A sum of subcomodules is again one, so we may assume W = 〈w〉is one-dimensional. Write m∗(w) =

∑ni=1 vi ⊗ ai with a1, ..., an linearly

independent over k and complete them to a k-basis aii∈I of A. We claimthat V = 〈w, v1, ..., vn〉 is a comodule.

Indeed, suppose m∗(ai) =∑cijk aj ⊗ ak. Then∑

m∗(vi)⊗ ai = (m∗ ⊗ id)m∗(w) = (id⊗m∗)m∗(w) =∑

vi ⊗ cijkaj ⊗ ak

in V ⊗A⊗A. Comparing the coefficients of ak, we get m∗(vk) =∑vi⊗cijkaj ,

so m∗(V ) ⊂ V ⊗A.

Proof of Theorem 2.6. Pick some k-algebra generators of A = k[G], and letW ⊂ A be their k-span. Take an A-comodule W ⊂ V ⊂ A as in the lemma,and let v1, ..., vn be a k-basis of V . The image of

Σ∗ : k[GL(V )] = k[x11, ..., xnn, 1/ det] −→ A

contains∑

i e∗(vi)Σ

∗(xij) = (e∗ ⊗ id)m∗(vj) = vj , so Σ∗ is surjective.

Note that the proof is completely explicit.

Example 2.8. Take G = Ga. Then A = k[G] = k[t] is generated by t, sotake W = 〈t〉. The comultiplication

m∗ : k[t] −→ k[t]⊗ k[t] (∼= k[t1, t2])

maps t 7→ t⊗ 1 + 1⊗ t (= t1+t2), so m∗(W ) 6⊂ W ⊗ A. In other words, Wis not a comodule. But V = 〈1, t〉 is one (cf. proof of Lemma 2.7), and thecorresponding embedding Ga → GL2 is

t 7−→(10t1

).

In view of Theorem 2.6, affine algebraic groups are also called linearalgebraic groups. The statement of the theorem can even be refined so thata given H <G can be picked out as a stabiliser of some linear subspace:

4unless G is finite, in which case the construction recovers Example 2.5

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Theorem 2.9 (Chevalley). Let H <G be a (closed) subgroup of an affinealgebraic group G. There is a linear representation G→ GL(V ) and a linearsubspace W ⊂ V whose stabiliser is H.

Proof. Exc 2.10.

Extending this further, one proves that if H <G is normal, it is possibleto find V such that H is precisely the kernel of φ : G → AutV (see [?]§16.3). The image φ(G) is then an algebraic group, so factor groups existfor algebraic groups. (In positive characteristic this factor group is notunique, because there are injective homomorphisms of algebraic groups thatare not isomorphisms, see Exc 2.12; the question whether the quotient G/Hexists in the sense of category theory is subtle, see [?] §15–16.)

Exc 2.1. Write down explicitly the multiplication map and the inverse for GL2 ⊂ A5.

Exc 2.2. Suppose G is an algebraic group and H ⊂ G is a subgroup in the ‘abstact group

sense’. Then its (Zariski) closure H ⊂ G is a subgroup in the algebraic group sense.

Exc 2.3. Show that the image of an algebraic group homomorphism is an algebraic group.

(The image of a variety under a morphism is not always a variety, see Exc 3.5.)

Exc 2.4. (Closed orbit lemma) Suppose G × V → V is an action of G on a variety V ,

and let U = Gv be an orbit. Then the closure U ⊂ V is a variety, U ⊂ U is open and

non-singular, and U \U is a union of orbits (of strictly smaller dimension). In particular,

the orbits of minimal dimension are closed.

Exc 2.5. Let G be an algebraic group and write AutG for the set of isomorphisms G→ G

(as algebraic groups). Determine AutG for G = Ga and G = Gm. Does AutG have a

structure of an algebraic group in these cases, and is it true that its action on G is an

algebraic group action?

Exc 2.6. If G is connected and H /G is finite, then H ⊂ Z(G), in particular H is abelian.

Exc 2.7. Prove that the connected component of identity G0 ⊂ G is a normal subgroup

and its left cosets are the connected components of G.

Exc 2.8. Show that the classical groups GLn, SLn,On and Sp2n are connected.

Exc 2.9. Do Example 2.8 for G = Gm.

Exc 2.10. Prove Theorem 2.9 (Modify the proof of Theorem 2.6.)

Exc 2.11. A character of G is a 1-dimensional representation of G, equivalently a ho-

momorphism G → Gm. Prove that characters are in 1-1 correspondence with invertible

elements x ∈ k[G]× such that m∗(x) = x ⊗ x. What does the product of characters

correspond to? Compute the character group for G = Gm and G = Ga.

Exc 2.12. Let G ⊂ An be an affine algebraic group of dimension > 0 over k = Fp.Suppose G is given by the equations fi(x1, ..., xn) = 0. For a polynomial f write f (p)

for the polynomial whose coefficients are those of f raised to the pth power. Prove that

G(p) = x|f (p)i (x) = 0 is also an algebraic group, and that the Frobenius map F (x) = xp

is a homomorphism from G to G(p). Prove that F is bijective but not an isomorphism of

algebraic groups.


3. General varieties

Recall that to define a topological (C∞, analytic, ...) manifold one takesa topological space covered by open sets V =

⋃i Vi, such that

(1) Each Vi is identified with a standard open ball in Rn (or Cn).(2) The transition functions between charts Vi ⊃ Vi ∩ Vj → Vj ∩ Vi ⊂ Vj

are continuous (C∞, analytic,...).(3) V is Hausdorff and second countable.

The Hausdorff condition is necessary to avoid unpleasanties like glueing Rwith R along R− 0: ∪ = cThe two origins cannot be separated by open sets, so the resulting space isnot Hausdorff although both charts are. We do not want this.

We now copy this definition to glue affine varieties together, and we onlyallow finitely many charts.

An algebraic set V is a topological space covered by finitely many opensets V = V1 ∪ ... ∪ Vn (affine charts), such that

(1) Each Vi has a structure of an affine variety.(2) The transition maps Vi ⊃ Vi ∩Vj → Vj ∩Vi ⊂ Vj are isomorphisms5.(3) V is closed in V × V (V is separated).6

An (algebraic) variety is an irreducible algebraic set; in particular, varietiesare connected.

The separatedness condition (3) is equivalent to Hausdorffness for topo-logical spaces, but makes sense for varieties. (Because open sets are usuallydense in Zariski topology, varieties are never Hausdorff). See Excs 3.1–3.4.

Here is the main example:

Example 3.1. Projective space Pn = Pnk is a set of tuples [x0 : · · · : xn] withxi ∈ k and not all 0, modulo the relation that [ax0 : · · · : axn] defines thesame point for all a ∈ k×.

A subset V ⊂ Pn is closed if there are homogeneous polynomials fi inx0, ..., xn such that

V = x ∈ Pn | all fi(x) = 0.

As fi are homogeneous, the condition fi(x) = 0 is independent of the choiceof a tuple representing x.

To give Pn a structure of a variety, cover it Pn = An(0) ∪ · · · ∪ An(n) with

An(j) = [x0 : . . . xj−1 : 1 : xj+1 : . . . xn] ⊂ Pn

5That is, rational functions φij : Vi → Vj , defined everywhere on Vi ∩ Vj ⊂ Vi andmapping it to Vj ∩ Vi ⊂ Vj .

6The topology on V × V comes from Zariski topology on affine varieties Vi × Vj thatcover it.

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The transition maps between charts are indeed morphisms

An(j)−xk=0 −→ An(k)−xj =0, (xi) 7→ (xixjxk

),

so Pn becomes an algebraic variety. Closed irreducible subsets of Pn arecalled projective varieties; see Exc 3.7.

Example 3.2. If X is an affine variety and V ⊂ X an open set, then V isgeneral not an affine variety. However, it can always be covered by finitelymany affine subvarieties of X, so it is a variety. Generally, both open andirreducible closed subsets of a variety are varieties (Exc 3.6).

A morphism X → Y of algebraic sets is a continuous map which is amorphism when restricted to affine charts. A rational map X Y is amorphism from a dense open set. As before, regular and rational functionson X as morphisms X → A1 and rational maps X P1, respectively.The former form a ring k[X], and the latter a field k(X) if X is a variety.However, unless X is affine, k(X) is usually much larger than the field offractions of k[X] (Exc 3.8).

We refer to 1-dimensional varieties as curves and 2-dimensional varietiesas surfaces. A 0-dimensional variety is a point.

Exc 3.1. Every variety X is compact, and X is not Hausdorff unless X is a point.

Exc 3.2. A topological space X is Hausdorff if and only if the diagonal X ⊂ X × X is

closed in the product topology.

Exc 3.3. (‘Valuative criterion of separatedness’) Suppose V satisfies (1) and (2) in the

definition of an algebraic set. Then it satisfies (3) if and only if for every curve C and a

non-singular point P ∈ C, any morphism C \ P → X has at most one extension to a

morphism C → X.

Exc 3.4. If f, g : X → V are morphisms of varieties, the set of points x ∈ X where

f(x) = g(x) is closed in X. In fact, for a fixed V this holds for all X, f, g if and only if V

is separated.

Exc 3.5. Show that the image of A2 under (xy, x) : A2 → A2 is not a variety.Exc 3.6.

(a) Prove that A2 − 0 is not isomorphic to an affine variety.(b) Suppose X is an affine variety and ∅ 6= V ⊂ X open. Then V can be covered by

finitely many affine subvarieties of X, so it is a variety.(c) Prove that both open and irreducible closed subsets of a variety are varieties.

Exc 3.7. Prove that An and Pn are varieties, that is satisfy the separatedness condition.

In particular, affine varieties and projective varieties are actually varieties.

Exc 3.8. Show that k[P1] = k and k(P1) ∼= k(t).

Exc 3.9. Prove that every curve has cofinite topology.

4. Complete varieties

The nicest manifolds are the compact ones, but Zariski compactness is notthe right notion for varieties — from finite-dimensionality it follows easilythat every affine variety is compact (Exc 3.1).


A variety X is complete if for every variety Y , the projection X×Y p2−→Ymaps closed sets to closed sets7.

For k = C this is the same as requiring X to be compact in the usualtopology. Also, in the same way as separatedness is a reformulation of beingHausdorff, completeness is equivalent to compactness for topological spaces(Exc 4.1).

Any topological space can be compactified, and the same is true in oursetting:

Theorem 4.1 (Nagata). Every variety can be embedded in a complete va-riety as a dense open subset.

Here are two alternative definitions of completeness that are a bit morenatural. Both say that X is as large as possible in some sense, and has nomissing points. For the proof of equivalence, see Exc 4.2.

Lemma 4.2. For a variety X the following conditions are equivalent:

(1) (‘Universally closed’) X is complete.(2) (’Maximality’) If X ⊂ Y is open with Y a variety, then X = Y .(3) (’Valuative criterion’) For every curve C and a non-singular point

P ∈ C, any morphism C\P → X extends to a morphism8 C → X.

Example 4.3. A1 is not complete:

(1) fails because under A1×A1 p2−→A1, the set xy = 1 projects to A1\0.(2) fails because A1 may be embedded in P1 as a dense open subset.

(3) fails because A1\0 x 7→x−1

−→ A1 does not extend to A1 → A1.

Here are some important consequences of completeness:

Lemma 4.4. Suppose X is a complete variety.

(a) Every closed subvariety Z ⊂ X is complete.(b) For any morphism f : X → Y the image f(X) is complete and

closed in Y .(c) k[X] = k, in other words X has no non-constant regular functions.(d) If X is affine, then X is a point.

Proof. (a) Clear from the definition.(b) f(X) is the same as p2 of the “graph of f” (x, f(x)) ⊂ X × Y .

(c) The image of X under the composition Xf→ A1 → P1 is connected,

closed and misses ∞, so it must be a point.(d) Affine varieties are characterised by k[X].

The main example of a complete variety is Pn (Exc 4.4). By (a), allprojective varieties are therefore complete, and these are the only complete

7One says that π : X → pt is ‘universally closed’ (π× idY is a closed map for all Y ).8necessarily unique as X is a variety and is therefore separated

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varieties we will ever encounter.9 Incidentally, note that this proves Nagata’stheorem for affine varieties: embed V ⊂ An ⊂ Pn and take the closure Vof V in Pn (the intersection of all closed subsets of Pn containing V ). ThenV ⊂ V is dense open and V is a projective variety, hence complete.10

Here is one standard illustration of the power of completeness:

Lemma 4.5. Suppose C1 and C2 are complete non-singular curves.

(i) Any morphism C1 \ P1, ..., Pn → C2 extends uniquely to C1 → C2.(ii) Every non-constant map f : C1 → C2 is surjective.

Proof. (i) use 4.2 (3); uniqueness follows from separatedness of C2. (ii) Im fis closed, so it is either a point or the whole of C.

What (i) says is that every rational map C1 → C2 extends to a uniquemorphism, so for non-singular complete curves there is no real distinctionbetween rational maps and morphisms. From here it is not hard to get thatC 7→ k(C) defines an equivalence of categories

complete non-singularcurves over k

−→ Finitely generated field extensionsK/k of transcendence degree 1.

In higher dimensions this is not true — for instance P2 and P1 × P1 havethe same field of rational functions, but are not isomorphic (Exc 4.8).

When k = C, complete non-singular curves are the same as (compact)Riemann surfaces. In particular, every Riemann surface is algebraizable(arises from such a curve), but this is not true for higher-dimensional com-pact complex manifolds. Nevertheless, for complete varieties there is a veryclose connection between the usual complex topology and Zariski topology:

Theorem 4.6 (Chow). Let X,Y be complete varieties over C.

(1) Every analytic subset11 of X is closed in Zariski topology.(2) Every holomorphic map f : X → Y is induced by a morphism of

varieties.

Chow proved (1) for X = Pn and the rest of the assertions follow relativelyeasily (using Chow’s lemma for (1) and applying (1) to the graph of f in(2); see [?] §1.3).

In particuar, the only meromorphic functions on a complete variety overC are rational functions (take Y = P1).

∗Exc 4.1. A topological space X is compact if and only if X × Y p2−→ Y takes closed sets

to closed sets for every topological space Y .

Exc 4.2. Prove Lemma 4.2.

9In fact, complete curves and surfaces are always projective, and it is quite non-trivialto construct a non-projective complete 3-dimensional variety [?].

10Thus Nagata’s theorem for arbitrary varieties becomes a question of arranging thecompletions of affine charts so that they can be glued together. This may be done using‘blowing-ups’ and ‘blowing-downs’, see [?].

11Locally (in the usual complex topology) a zero set of holomorphic functions


Exc 4.3. Let C be a curve and P ∈ C a non-singular point. Show that the local ring

OC,P ⊂ k(C) of functions defined at P is a discrete valuation ring.

Exc 4.4. Use Exc 4.3 and the ‘valuative criterion’ to show that Pn is complete.

Exc 4.5. Determine k[Pn] and k(Pn).Exc 4.6. Suppose C is a curve.

(a) Show that there is a non-singular complete curve C birationally isomorphic to C.

In other words, there are rational maps φ : C C and ψ : C C with φψ = idand ψφ = id. Show that any two such C are isomorphic.

(b) If C is complete, then ψ is a surjective morphism.(c) If C is non-singular, then φ is an injective morphism identifying C with an open

set of the form C \ P1, ..., Pn.Exc 4.7. Explain why Lemma 4.5 fails if one of the words ‘complete’, ‘non-singular’ or

‘curves’ is omitted.

Exc 4.8. Show that P2 and P1 × P1 have the same field of rational functions but are

not isomorphic. (You may want to use that a non-singular curve C ⊂ P2 defined by a

homogenous equation f(x0, x1, x2) = 0 of degree d has genus g = (d− 1)(d− 2)/2.)

Exc 4.9. Prove that the complete variety in Nagata’s theorem is not necessarily unique.

Exc 4.10. Show that for complex varieties that are not complete Chow’s theorem may fail.

Exc 4.11. Prove that a compact complex manifold has at most one algebraic structure.

5. Algebraic groups

As before, k = k is an algebraically closed base field. We define algebraicgroups from varieties exactly as in the affine case (Def. 2.1).

Definition 5.1. A group G is an algebraic group if it has a structure ofan algebraic set, and multiplication G × G → G and inverse G → G aremorphisms.

As before, homomorphisms refer to morphisms that are group homomor-phisms, isomorphisms are isomorphisms of both groups and varieties, andsubgroups and normal subgroups always refer to closed ones.

Example 5.2. Affine algebraic groups Gm,Ga,GLn, ... are algebraic groups.

Example 5.3. Elliptic curves and their products are algebraic groups.

Example 5.4. The multiplication-by-m map [m] : G → G is a homomor-phism for any commutative algebraic group G and m ∈ Z (Exc 5.1).

Basic properties of algebraic groups carry over immediately from the affinecase: an algebraic group G is a semidirect product G = G0 o ∆ of its con-nected component of identity G0 and a finite discrete group ∆. Again, G0

is a non-singular variety. Kernels and images of algebraic group homomor-phisms exist, and so do factor groups in the same ‘naıve’ sense as before.

Example 5.5. Algebraic groups often occur naturally as automorphismgroups of varieties (see Exc 5.2 though). For example, suppose C is a com-plete non-singular curve of genus g.

(g = 0) C ∼= P1, and AutC ∼= PGL2 = GL2 /Gm is a Mobius group (Exc 5.3).

14 TIM DOKCHITSER

(g = 1) Choosing a point O ∈ C makes C into an elliptic curve, and fromthe theory of elliptic curves it follows that AutC ∼= C n Aut(C,O)with Aut(C,O) finite of order ≤ 24 (usually just id, [−1].)

(g ≥ 2) AutC is finite.12

Proposition 5.6. The only one-dimensional connected algebraic groups areGa, Gm and elliptic curves.

Proof. Write G = C \ P1, ..., Pn with C a non-singular complete curve(Exc 4.6), and take x ∈ G. The left translation map lx : y 7→ xy on Gextends to an automorphism

lx : C −→ C,

because C is non-singular and complete. So C has infinitely many auto-morphisms that are (a) fixed point free on G, and (b) preserve the set of‘missing points’ P1, ..., Pn.

Write g for the genus of C, and e ∈ G for the identity element.g ≥ 2: AutC is finite, so this is impossible.g = 1: If n ≥ 1 then |Aut(C,P1)| ≤ 24, so n = 0 and G is complete. For

x ∈ G there is a unique fixed point free map taking the identity element eto x, which must be lx in every group law on G which has e as the identityelement. So the group law must be the same one as the standard one on anelliptic one.g = 0: Now C ∼= P1 and AutC ∼= PGL2(k) is the group of Mobius

transformations. These are uniquely determined by what they do to 3 givenpoints, in particular AutG 6= 1 implies n ≤ 2.n = 0: Then lx : P1 → P1 has no fixed points, which is impossible.n = 1: Change the coordinate on P1 to move P1 to ∞ and e to 0. The

fixed point free automorphisms of P1−∞ are translations

z 7→ z + a,

again there is a unique one taking 0 to a given a ∈ G, and G = Ga.n = 2: Similarly, move P1 7→ 0, P2 7→ ∞ and e 7→ 1. The automorphisms

of P1\0,∞ are z 7→ az and z 7→ a/z. Only the former ones are fixed pointfree, and there is a unique one taking e→ x for a given x. So the group lawis unique, G = Gm.

Suppose G is an algebraic group over k. Recall that if G is an affinevariety, G is also called a linear algebraic group.

Definition 5.7. An abelian variety is a complete connected algebraic group.

12Hurwitz (1893) showed that |Aut(C)| ≤ 84(g−1) over C, and the bound is sharp forinfinitely many g (Macbeath 1961). Schmid (1938) proved finiteness when char k = p > 0and noted that Hurwitz’ bound fails for small p. It still holds when p > g + 1, except foryp − y = x2 which has g = p−1

2and |Aut(C)| = 8g(g + 1)(2g + 1) (Roquette 1970).


We have seen that 1-dimensional algebraic groups are either linear (Ga,Gm) or abelian varieties (elliptic curves). Much more generally, these twoextremes build all algebraic groups:

Theorem 5.8 (Barsotti-Chevalley). Every connected algebraic group G fitsinto an exact sequence

1 −→ H −→ G −→ A −→ 1

with H / G the unique largest linear connected subgroup of G, and A anabelian variety.

Remark 5.9. With the theory of linear groups thrown in, the classificationcan be extended. There is a unique filtration of G of the form

finite AV semisimple torus

G G0 G1 G2 G3 1connected linear solvable unipotent

with Gi connected and normal in Gi−1. Here:A torus is an algebraic group isomorphic to Gm × · · · ×Gm;A unipotent group is a subgroup of upper-triangular matrices with ones onthe diagonal;A solvable group is one admitting a filtration 1 = H0 / H1 / · · · / Hk = Gwith Hi/Hi−1 commutative;A semisimple group is one whose radical G2 (the unique maximal connectedlinear solvable normal subgroup) is trivial; a semisimple group admits afinite covering G1× · · ·×Gn → G with Gi almost simple (finite centre C andG/C simple). Every almost simple group is isomorphic to either SLn+1 (typeAn), Sp2n (type Cn), E6, E7, E8, F4, G2 (exceptional groups) or isogenous toan orthogonal group (types Bn, Dn).

See [?] Ch. X for a more extended summary and references.

Example 5.10. G = GLn. Then G = G1, G2 = Z(G) = Gm and G/G2 =PGLn is simple.

Example 5.11. Is it not hard to deduce that the only connected commuta-tive linear groups are U ∼= (Gm)m×(Ga)

n. So every commutative connectedalgebraic group G has the form G = A o U with A an abelian variety, U asabove and acting trivially on A.

Exc 5.1. Prove that the multiplication-by-m map [m] : G → G is a homomorphism for

any commutative algebraic group G and m ∈ Z.

Exc 5.2. Give an example of a variety V such that AutV has no natural structure of an

algebraic group.

Exc 5.3. Show that every automorphism of P1k is of the form t 7→ at+b

ct+d. Deduce that

AutP1 ∼= PGL2(k) (= GL2(k)/k∗).

Exc 5.4. Show that there are no non-constant algebraic group homomorphisms from an

abelian variety to a linear algebraic group.

16 TIM DOKCHITSER

6. Abelian varieties

Suppose k = k as before. Recall that an abelian variety A/k is an alge-braic group over k, which is a complete variety. Let us validate ‘abelian’and prove that abelian varieties are always commutative. This must clearlyrely on completeness, and we follow Mumford’s approach using rigidity:

Lemma 6.1 (Rigidity). Suppose f : V ×W → U is a map of varieties, V iscomplete, and

f(v0 ×W ) = f(V × w0) = u0for some points v0, w0 and u0. Then f is constant, f(V ×W ) = u0.

Proof. 13 Let U0 be an open affine neighbourhood of u0 and Z = f−1(U−U0).This is a closed set, and so is its image under the projection p2 : V ×W →W ,as V is complete.

As w0 /∈ p2(Z), the complement W0 = W − p2(Z) is open dense in W .But for all w ∈W0 the image f(V ×w) ⊂ U0 must be a point, as V ×wis complete and U0 is affine. In other words, f(V × w) = f((v0, w)) =u0.So f−1(u0) contains a dense open V ×W0; as f−1(u0) is also closed, it mustbe the whole space, so f is constant.

Corollary 6.2. If U, V,W are varieties, V is complete, U is an algebraicgroup, and f1, f2 : V ×W → U are morphisms that agree on v0 ×W andon V × w0, then they agree everywhere.

Proof. The map x 7→ f1(x)f2(x)−1 is constant by the rigidity lemma.

Corollary 6.3. Abelian varieties are commutative.

Proof. The maps xy and yx from X ×X to X agree on X ×e and e×X,so they must agree everywhere by the previous corollary.

From now on let us write the group operation on abelian varieties asaddition, and denote the identity element by 0.

Corollary 6.4. If A,B are abelian varieties, then every morphism of vari-eties f : A→ B that takes 0 to 0 is a homomorphism of abelian varieties.

Proof. The morphisms f(x) + f(y) and f(x+ y) from A×A to B agree on0 ×A and on A× 0, so they are equal.

Corollary 6.5. Every morphism f : A → B between abelian varieties is acomposition of a translation on B and a homomorphism A→ B.

13Over k = C, this works as follows: if w is close to w0, then f(V × w) is close tou0, by the compactness of V and continuity of f . So f(V × w) is contained in someopen ball around u0. But there are no non-constant analytic maps from V to an open ball(maximum principle), so f(V × w) is a point for such w, namely f((v0, w)) = u0. Thisproves that the set of such w is open; but it is also closed, so f is constant.


In fact, in the two preceding corollaries B could be any algebraic group.Rigidity has another curious consequence: in defining an abelian variety

we could have dropped the associativity condition, as it also follows auto-matically from rigidity! (Exc 6.1) For instance, for elliptic curves this givesa quick proof of the associativity of the group law, that only relies on Ebeing complete.

Remark 6.6. It is possible to extend Corollary 6.5 slightly: any rationalmap φ : G A from a connected algebraic group to an abelian variety is acomposition of a translation with a homomorphism G → A (in particular,φ a morphism).

Exc 6.1. Suppose V is a complete variety, e ∈ V (k) a point, and we have a morphism

∗ : V ×V → V and an isomorphism i : V → V . If x∗e = e∗x = x and x∗i(x) = i(x)∗x = e,

then ∗ is associative, so V is an abelian variety.

Exc 6.2. Prove that for an abelian variety A, every rational map P1 A is a constant

morphism (hint: 6.6). Deduce that every rational map Pn A is also constant.

7. Abelian varieties over C

If the base field k = k has characteristic 0, by Lefschetz principle (e.g. [?]§VI.6) understanding varieties over k essentially reduces to understandingthose over C.

Suppose k = C and X/C is a g-dimensional abelian variety. As X isregular and complete, in classical topology (also referred to as Euclidean orstrong topology) it is a compact complex manifold with a group structure.In other words, it is a compact complex-analytic Lie group of C-dimension g.

Theorem 7.1. Let X/C be a g-dimensional abelian variety. As complexLie groups, X ∼= Cg/Λ for some lattice Λ ⊂ Cg of rank 2g.

Proof. Suppose X is any complex Lie group, and write V = TeX for thetangent space at identity. From the theory of Lie groups, for every v ∈ Vthere is a unique analytic group homomorphism (“one-parameter subgroup”)

φv : C −→ X

with dφv(0) = v, and φ·(·) : V × C → X analytic. Define the exponentialmap exp:V →X by exp v = φv(1). It is analytic, and we have

(1) φv(t) = exp(tv). [φv(st) = φtv(s) by uniqueness of φv; now put s = 1.](2) If X is commutative then exp : V → X is a group homomorphism.

[As X is abelian, ψ : t 7→ exp(tx) exp(ty) is a homomorphism C → X. Butdψ(1) = x+ y, so ψ = φx+y again by uniqueness; now put t = 1.]

Now if X is an abelian variety, then exp : V → X is surjective becauseexp(V ) ⊂ X is an open subgroup (Exc 7.1). Finally, ker exp ⊂ V is a discretesubgroup with compact quotient, so it is a lattice of maximal rank.

18 TIM DOKCHITSER

Corollary 7.2. Let X be a g-dimensional abelian variety over C, and writeX[n] for the n-torsion subgroup of X (all x with nx = 0). Then14

X[n] ∼= (Z/nZ)2g (n ≥ 1).

Proof. X ∼= C2g/Λ, so X[n] ∼= 1nΛ/Λ ∼= ( 1

nZ/Z)2g as a group.

The proof of Theorem 7.1 does not use compactness of X until the lastline, so all conclusions except Λ = ker exp having maximal rank remain validfor all commutative Lie groups over C:

Example 7.3. For X = Ga, exp : C→ C = Ga is identity, and Λ = 1.

Example 7.4. For X = Gm, exp : C → C∗ = Gm is the usual exponentialfunction z 7→ ez, and Λ = 2πiZ is a lattice of rank one; in other words,Gm∼= C/Z as an analytic Lie group.

Example 7.5. (Elliptic curves) If Λ ⊂ C is a lattice, the parametrisation

C/Λ (℘,℘′,1)−−−−−→ E : y2 = 4x3−g2x−g3 ⊂ P2

is given by the elliptic functions g2 = g2(Λ), g3 = g3(Λ) and ℘(z) = ℘(Λ, z):

g2 = 60∑w∈Λw 6=0

1w4 , g3 = 140

∑w∈Λw 6=0

1w6 , ℘(z) = 1

z2 +∑w∈Λw 6=0

(1

(z−w)2 −1w2

).

Note that ℘(z) is a transcendental function, in other words the morphismA1C = C→ E is by no means algebraic.15

The three examples combine to an analytic analogue of the classificationof 1-dimensional algebraic groups (Prop. 5.6): a connected 1-dimensionalcommutative complex Lie group is either Ga = C, Gm = C/Z or an ellipticcurve E = C/Zτ1 ⊕ Zτ2.

A compact analytic Lie group of the form Cg/Λ is called a (g-dimensional)complex torus. By the example above, every 1-dimensional complex torus isan elliptic curve. Unfortunately, this fails for g > 1: not every complex torusis an abelian variety. The ones that are varieties are called algebraizable, andwe give a criterion below.

Everything else goes well though:

Theorem 7.6. Suppose A = Cg/Λ and A′ = Cg′/Λ′ are abelian varieties.

(1) Abelian subvarieties U ⊂ A are in one-to-one correspondence withthe C-subspaces V ⊂ Cg for which rkZ(V ∩ Λ) = 2 dimV .

(2) HomAV(A,A′) = M ∈ Matg′×g(C) |MΛ ⊂ Λ′.(3) A ∼= A′ if and only if MΛ = Λ′ for some M ∈ GLg(C).

Proof. (3) is a special case of (2). Half of (2) is also easy: a homomorphism

14By Lefschetz principle, the same holds over any k = k of characteristic 0 (Exc 7.2)15E.g. its kernel Λ is not Zariski closed; in fact, there are no non-constant morphisms

A1 → E by Exc 6.2.


φ : A→ A′ induces a C-linear map on the tangent spaces at e,

dφ : Cg → Cg′

and exp dφ = φ exp by the uniqueness of exp, so φ(Λ) ⊂ Λ′. Applyingthis to the inclusion U ⊂ A gives half of (1) as well.

The other half of (1) and (2) is a special case of Chow’s theorem 4.6.

Corollary 7.7. If A = Cg/Λ and A′ = Cg′/Λ′ are abelian varieties, thenHomAV(A,A′) is a free Z-module of rank ≤ 2gg′.

Proof. Let L ⊂ Λ be a rank g sublattice which spans Cg over C. ThenHomAV(A,A′) → HomZ(L,Λ′) from 7.6 (2).

This bound is best possible for all g and g′ (Exc 8.3).

Example 7.8. If E = C/Λ is an elliptic curve, then EndE = Hom(E,E)is either Z or has rank 2. In the latter case, it is an order in an imaginaryquadratic field (Exc 7.3) and we say that E has complex multiplication.

Another immediate corollary of Chow’s theorem is a description of ratio-nal functions on an abelian variety A = Cg/Λ:

Rational functions on A = Meromorphic functions on A =Λ-periodic meromorphic functions on Cg.

It turns out that algebraicity of a complex torus amounts to requiringthat it has ‘enough’ meromorphic functions, and has the following explicitdescription:

Theorem 7.9. (Lefschetz) Let Λ ⊂ Cg be a Z-lattice of rank 2g. For X =Cg/Λ the following conditions are equivalent:

(1) X is an abelian variety.(2) X is a projective variety.(3) X has g algebraically independent meromorphic functions.(4) There is a positive-definite Hermitian form H : Cg × Cg → C such

that E = ImH takes values in Z on Λ× Λ.

A form H as in (4) is called a Hermitian Riemann form, and E = ImHan alternating Riemann form. Note that

E : Λ× Λ→ Zis alternating and E(iu, iv) = E(u, v) (use that H is Hermitian). Conversely,such a E recovers H by H(u, v) = E(iu, v) + iE(u, v), see Exc 7.4.

Example 7.10. (Elliptic curves) For 1-dimensional complex tori X = C/Λthe conditions are always satisfied:

(1) Every compact Riemann surface in algebraizable.(2) The field of meromorphic functions is generated by ℘(z) and ℘′(z)

with one relation (Example 7.5).(3) Rescale the lattice so that Λ = Z⊕Zτ . Then H(z, w) = 1

Im τ zw is aHermitian Riemann form.

20 TIM DOKCHITSER

For g > 1, a general lattice Λ ⊂ Cg does not admit any alternatingRiemann forms: if ei is a Z-basis of Λ, an alternating form E : Λ×Λ→ Zis determined by E(ei, ej) ∈ Z. The condition E(iu, iv) = E(u, v) forceslinear relations with real coefficients upon them, and these will have no non-zero integer solutions in general. Thus, ‘most’ tori are not algebraizable.

Exc 7.1. If G is a connected topological group and U ⊂ G an open subgroup, then U = G.

Exc 7.2. Use Lefschetz principle to show that A[n] ∼= (Z/nZ)2g for a g-dimensional abelian

variety A over a field k = k of characteristic 0.

Exc 7.3. Suppose E = C/(Z + τZ) is an elliptic curve with EndE 6= Z. Show that

K = Q(τ) is an imaginary quadratic field, and EndE → K is naturally a subring which

has rank 2 over Z (so it is an order of K).

Exc 7.4. Show that H 7→ ReH defines a 1-1 correspondence between Hermitian forms H

on Cg and real-valued alternating forms E on Cg satisfying E(iu, iv) = E(u, v).

Exc 7.5. Let E : Z2g × Z2g → Z be an alternating non-degenerate form. There is a basis

for Z2g in which E has the shape(

0−M

M0

)with M a diagonal matrix with positive integer

entries on the diagonal.

Exc 7.6. Write down an explicit example of a non-algebraizable complex torus.

8. Isogenies and endomorphisms of complex tori

Recall that by 7.6(2), homomorphisms φ : Cg/Λ → Cg′/Λ′ of abelian

varieties are just C-linear maps Cg → Cg′ mapping Λ to Λ′. Let us call thelatter homomorphisms of complex tori even when they are not algebraizable.(Equivalently, it is a homomorphism as analytic Lie groups.)

Definition 8.1. A homomorphism φ : X → X ′ of complex tori is an isogenyof degree m if it is surjective with kernel of size m. We write deg φ = m,and we say that the two tori are isogenous (denoted X ∼ X ′).

Equivalently, φ is a homomorphism which is a covering map of degree min the topological sense. If φ : Cg/Λ→ Cg′/Λ′ is such an isogeny, then g = g′

and we identify Cg = Cg′ via φ, it gives inclusion of lattices φ : Λ → Λ′,identifying Λ with a sublattice of index m in Λ′.

Here are some immediate properties of isogenies:

• For isogeniesXφ−→X ′ ψ−→X ′′ of complex tori, deg(ψφ) = deg φ degψ.

[Consider Λ → Λ′ → Λ′′].• The multiplication-by-m map [m] : Cg/Λ → Cg/Λ is an isogeny of

degree m2g [cf. 7.2].• There is an isogeny X → X ′ if and only if there is one X ′ → X.

[If Λ ⊂ Λ′ has index m, then mΛ′ ⊂ Λ.]

Thus, being isogenous is an equivalence relation. Specifically, for a degree misogeny φ : X → X ′ we constructed a unique isogeny φ : X ′ → X (of degree

m2g−1) with φφ = [m], called the conjugate isogeny. For elliptic curvesφ˜= φ, but this fails in higher dimensions (wrong degree).

• For isogenies Xφ−→X ′

ψ−→X ′′ of complex tori, ψφ = φ ψ.


• A complex torus isogenous to an abelian variety is an abelian variety.[Restrict the Riemann form to Λ ⊂ Λ′ 16].

The endomorphism ring EndA of a complex abelian variety A is finitelygenerated and free as a Z-module (7.7), and we call End0A = EndA⊗Z Qthe endomorphism algebra of A. Its units are isogenies A→ A.

We call A simple if it has no abelian subvarieties apart from 0 and A.

Lemma 8.2.

(1) A is simple if and only if End0A is a division algebra.(2) If A and A′ are simple, then either A ∼ A′ or Hom(A,A′) = 0.(3) If A ∼ A′ then End0A ∼= End0A′.

Proof. (1, 2) Exc 8.5. (3) If φ : A→ A′ is an isogeny of degree m, the maps

α : f 7→ φfφ : EndA→ EndA′ and β : f 7→ φfφ : EndA′ → EndA

compose to αβ= [m]∈EndA and βα= [m]∈EndA′. Because [m] becomesa unit after ⊗Q, α and β are isomorphisms on the level of End0.

Theorem 8.3 (Poincare reducibility). Suppose A′ ⊂ A is an abelian sub-variety of a complex abelian variety. Then there is an abelian subvarietyA′′⊂A such that A=A′+A′′ and A′∩A′′ is finite; A is isogenous to A′×A′′.

Proof. Exc 8.6.

Thus, ever non-simple abelian variety is isogenous to a product A′ × A′′of abelian varieties of smaller dimension. Continuing this process, we get

Corollary 8.4 (Complete reducibility). Every complex abelian variety isisogenous to a product simple abelian varieties,

A ∼ An11 × · · · ×A

nkk .

For such an A,

End0A ∼= Matn1×n1(D1)× · · · ×Matn1×nk(Dk), Di = End0(Ai),

in particular it is a semisimple Q-algebra (product of matrix algebras overdivision algebras).

A formal way of saying this is that the category of ‘abelian varieties upto isogeny’ (objects = abelian varieties, morphisms = HomAV⊗ZQ) is asemisimple Q-linear category (Homs are Q-vector spaces and every object isa direct sum of simple objects).

It remains to understand algebraically which division rings may actuallyoccur as endomorphism algebras of simple abelian varieties. For this we firstneed to reformulate the algebraizability condition (existence of a Riemannform).

16The same argument also gives an alternative proof that subtori of abelian varietiesare abelian varieties (half of Thm. 7.6(2)), avoiding the use of Chow’s theorem.

22 TIM DOKCHITSER

Exc 8.1. Show that the elliptic curves En = C/(Z+niZ) for n ≥ 1 are pairwise isogenous.

Determine EndEn and End0 En.

Exc 8.2. ‘A generic complex torus has endomorphism ring Z’. Explain.

Exc 8.3. For any g, g′ ≥ 1 give an example of complex abelian varieties with dimA = g,

dimA′ = g′ and dim Hom(A,A′) = 2gg′. Give also an example with Hom(A,A′) = 0.Exc 8.4. Give an example of an abelian variety which is isogenous to a product of two

(positive-dimensional) abelian varieties, but is itself not such a product.

Exc 8.5. Prove that A is a simple abelian variety if and only if End0 A is a division algebra.

Show that if A and A′ are simple abelian varieties, then either A ∼ A′ or Hom(A,A′) = 0.

Exc 8.6. Prove the Poincare reducibility theorem (8.3). Hint: Take the orthogonal com-

plement V of Λ′ ⊂ Λ with respect to the Hermitian Riemann form H, and let Λ′′ = V ∩Λ.

9. Dual abelian variety

Let X = V/Λ be a complex torus of dimension g (so V ∼= Cg). Define

V ∗ = f ∈ HomR(V,C) | f(αv) = αf(v), all α ∈ C, v ∈ V ,Λ∗ = f ∈ V ∗ | Im f(v) ∈ Z all v ∈ Λ.

Because (v, f) 7→ Im f(v) is a non-degenerate R-linear pairing V ×V ∗ → R,Λ∗⊂V ∗ is again a lattice of rank 2g. Call X∗=V ∗/Λ∗ the dual torus of V/Λ.Clearly,

• X∗∗ = X canonically;• a homomorphism f : X → Y induces f∗ : Y ∗ → X∗. Moreover, f is

an isogeny if and only if f∗ is; if they are, then deg f = deg f∗;

• (ψ φ)∗ = φ∗ ψ∗ for homomorphisms Xφ−→X ′

ψ−→X ′′;• [m]∗ = [m] for the multiplication-by-m map [m] : X → X,

the properties expected from a decent duality.If X = V/Λ is an abelian variety, then a Hermitian Riemann form on X

defines a C-linear isomorphism λH : v 7→ H(v, ·) from V to V ∗ which mapsΛ to Λ∗, because ImH is integer-valued on Λ× Λ by definition. So

λH : X → X∗

is an isogeny17. Such as isogeny λ : X → X∗ induced by a positive-definiteHermitian form H is called a polarisation on X, and Lefschetz’ Theorem 7.9asserts that abelian varieties are precisely those tori that admit a polarisa-tion. Note also that λ∗ = λ (identifying X∗∗ with X).

We call λ a principal polarisation if it has degree 1, in other words if it isan isomorphism X ∼−→X∗. By Exc 7.5, a principal polarisation comes froma Riemann form

(0−I

I0

)in some basis of Λ (I is the g × g identity matrix).

Not every abelian variety admits a principal polarisation, but it is alwaysisogenous to one that does (Exc 9.2).

Example 9.1. On an elliptic curve X = C/(Z+Zτ), any Riemann form isas integer multiple of the standard one H(z, w) = 1

Im τ zw. An elliptic curve

17whose degree is√| detE|, where E = ImH (Exc 9.1)


is principally polarised via λH : X ∼−→X∗, and every other polarisation hasthe form [n]λH for some non-zero n.

Now we return to the question of understanding the endomorphism al-gebra End0A of an abelian variety A. Fix a polarisation λ : A → A∗.As it is an isogeny, it induces a ring isomorphism End0A → End0A∗ viaφ 7→ λφλ−1. But also the duality map

f 7→ f∗ : EndA ∼−→ End(A∗),

induces an anti18-isomorphism End0A→ End0A∗. Combining the two givesan anti-involution on End0A:

Definition 9.2. The anti-involution ι : φ 7→ φι = λφ∗λ−1 on End0A iscalled the Rosati involution induced by the polarisation λ.

Note that ι([m]) = λ[m]∗λ−1 = λ[m]λ−1 = [m]λλ−1 = [m] (isogeniescommute with addition), so ι acts trivially on Q ⊂ End0A.

Example 9.3. If E is a non-CM elliptic curve, then End0E = Q and theRosati involution induced by any λ is the identity map ι : Q→ Q.

Example 9.4. If E is a CM elliptic curve, then the Rosati involution in-duced by any λ is the complex conjugation on End0E ∼= Q(

√−d) (check).

Now suppose A is simple, so D = End0A is a finite-dimensional divisionalgebra over Q. Denote by Tr = TrD/Q the trace19 from D to Q. It is easyto see what the positive-definiteness of the Riemann form corresponds to:

Lemma 9.5. The quadratic form φ 7→ Tr(φφι) is positive definite on End0A.

Division algebras with these properties can be classified completely:

Theorem 9.6 (Albert). Let D be a finite-dimensional division algebra overQ with a positive-definite anti-involution ι. Write K for the centre of D,and K0 for the subfield of K of ι-invariants. Then K is a field, K0 is atotally real field, and the pair (D, ι) belongs to one of the following types:

Type I: D = K = K0 and ι is identity.Type II: K = K0, D is a quaternion algebra over K, and there is an

isomorphism

D ⊗Q R ∼= Mat2×2(R)× · · · ×Mat2×2(R),

taking ι to the involution (x1, ..., xe) 7→ (xt1, ..., xte).

Type III: K = K0, D is a quaternion algebra over K, and there is anisomorphism

D ⊗Q R ∼= H× · · · ×Htaking ι to (ιH, ..., ιH) with ιH(x) = TrH/R(x) − x, the usual involution onthe standard (Hamiltonian) quaternions H.

18i.e. it preserves addition and changes the order in multiplication19=trace of the left multiplication by α ∈ D as a Q-linear endomorphism of D(∼= Qn)

24 TIM DOKCHITSER

Type IV: K/K0 is a totally imaginary quadratic extension, D is a divisionalgebra over K (of some rank d2) whose Brauer invariants above any fixedfinite place of K0 sum to 0, and there is an isomorphism

D ⊗Q R ∼= Matd×d(C)× · · · ×Matd×d(C),

taking ι to the involution (x1, ..., xe0) 7→ (xt1, ..., xte0).

Let D be a division algebra as in the theorem, and denote e0 = [K0 : Q],e = [K : Q], d2 = dimK D. Can D actually occur as the endomorphismalgebra of some abelian variety A = Cg/Λ? One restriction is that d2e =dimQD|2g, since D should act Q-linearly on Λ⊗Q ∼= Q2g.

Conversely, suppose g ≥ 1 an integer with dimQD|2g. It turns out thatexcept when D has Type III with g/2e ∈ 1, 2 or Type IV with g/e0d

2 ∈1, 2 there always is a simple complex abelian variety of dimension g withendomorphism algebra D (see [?]). In the exceptional cases there are explicitadditional conditions guaranteeing the existence (see [?]).

Exc 9.1. Show that the degree of the polarisation induced by a Hermitian Riemann form

H satisfies (deg λH)2 = | detE| (the 2g × 2g-determinant of E = ImH on Λ× Λ).

Exc 9.2. Give an example of an abelian variety that has no principal polarisation. Show,

however, that every abelian variety over C is isogenous to a principally polarized one.

10. Differentials

Let X be a variety over a field k = k. The rational k-differentials on Xare formal finite sums ω =

∑i fidgi with fi, gi ∈ k(X), modulo the relations

d(f + g) = df + dg, d(fg) = fdg + gdf, da = 0 (a ∈ k).

If char k = 0, dimX = n and t1, ..., tn ∈ k(X) are algebraically independent,every differential can be written uniquely as g1dt1+. . .+gndtn with gi ∈ k(X)(Exc 10.1), so their space is ∼= k(X)n as a k-vector space.

Example 10.1. X = P1 has differentials f(x)dx for f ∈ k(X) = k(x).

Example 10.2. On the curve C : y2 = x3 + 1 every differential can bewritten uniquely as f(x, y)dx, and also as h(x, y)dy with f, h ∈ k(C).(use that 0 = d(y2−x3−1) = 2ydy − 3x2dx to transform between the two).

A differential ω is regular at P ∈ X if it has a representation ω =∑

i fidgiwith fi, gi regular at P .20 If ω is regular everywhere, we call it a regulardifferential, and we write ΩX for the k-vector space of those. A morphismφ : X → Y naturally induces a k-linear map

φ∗ : ΩY → ΩX ,∑

i fidgi 7→∑

i(φ∗fi)d(φ∗gi),

20If P ∈ X is a non-singular point, there is an easy test: pick the algebraically in-dependent functions g1, ..., gd ∈ k(X) so that gi − gi(P ) generate mP /m

2P , and write

ω =∑i fidgi. Then ω is regular at P if and only if all the fi are (Exc 10.2)


the pullback of differentials. For complete varieties dimk ΩX is finite; if,moreover, X is projective then regular differentials are the same as holo-morphic differentials21.

Example 10.3. P1 (and, generally, Pn) has no non-zero regular differentials.

Example 10.4. The curve C : y2 = x3 +1 has ΩC = k dxy , of k-dimension 1.

Example 10.5. An abelian variety A = Cg/Λ has ΩA = 〈dz1, ..., dzg〉,where zi are the standard coordinates. In fact, dimk ΩA = dimA for anabelian variety over any k, and all ω ∈ ΩA are invariant under translationson A.22

Definition 10.6. The genus g of a complete non-singular curveX is dimk ΩX .

Thus, P1 has genus 0 and y2 = x3 + 1 ⊂ P2 genus 1. Generally:

Example 10.7. (Plane curves) A curve given by a non-singular homoge-

neous equation f(x, y, z) ⊂ P2 has genus g = (d−1)(d−2)2 , d = deg f .

Example 10.8. (Hyperelliptic curves, char k 6= 2) Let f(x) be a polynomialof degree 2g + 1 or 2g + 2 with no multiple roots, for some g ≥ 0. The twoaffine charts

y2 = f(x) and Y 2 = X2g+2f( 1X )

glue via Y = yxg+1 , X = 1

x to a curve C. The curve is complete, non-singular

and admits a degree 2 map C → P1 (via (x, y) 7→ x). Such a curve is calledhyperelliptic, and every hyperelliptic curve has a model as above (use that[k(C) : k(x)] = 2). The reflection automorphism

(x, y) 7→ (x,−y) : C → C

is called the hyperelliptic involution. Here

ΩC =⟨dxy,xdx

y, . . . ,

xg−1dx

y

⟩,

so C has genus g. (See Exc 10.4).

It is consequence of Riemann-Roch theorem that for g ≤ 2 every completenon-singular curve is hyperelliptic, and genus 3 curves are either hyperellpticor non-singular plane quartics.

Exc 10.1. Suppose X is an n-dimensional variety, t1, ..., tn ∈ k(X) are algebraically

independent and the (finite) extension k(X)/k(t1, ..., tn) is separable (e.g. char k = 0).

Then every rational differential on X can be written uniquely as g1dt1 + . . .+ gndtn with

gi ∈ k(X).

21These are special cases of finite-dimensionality of the space of global sections of acoherent sheaf on a complete variety, and of Serre’s ‘Geometrie Algebrique GeometrieAnalytique’ (GAGA) principle for complex projective varieties.

22Otherwise the action by translations would give a non-constant morphism A→GL(ΩA).

26 TIM DOKCHITSER

Exc 10.2. Suppose X is a variety, P ∈ X a non-singular point, and g1, ..., gdimX ∈ k(X)

are algebraically independent functions so that gi − gi(P ) generate the k-vector space

mP /m2P . (mP ⊂ OX,P is the maximal ideal.) Then ω =

∑i fidgi is regular at P if and

only if all the fi are.

Exc 10.3. Prove that Pn has no non-zero regular differentials.

Exc 10.4. Suppose char k 6= 2. Let C : y2 = f(x) with f(x) ∈ k[x] of degree 2g + 1 or

2g + 2 with no multiple roots. Prove that ΩC = 〈 dxy, xdxy, . . . , x

g−1dxy〉.

11. Divisors

Let C be a complete non-singular curve over k = k. A divisor on C is afinite formal linear combination of points,

D =

r∑i=1

ni(Pi), ni ∈ Z, Pi ∈ C.

The degree of D is∑ni.

If f ∈ k(C)× is a non-zero rational function, define the divisor of f

(f) =∑P∈C

ordP (f) (P ).

(Recall that the local rings OC,P are discrete valuation rings, and ordPdenotes the corresponding valuation on k(X).) Divisors of the form (f) arecalled principal, and it is not hard to see that they have degree 0. Clearlythey form a subgroup, and the quotient groups

PicC =divisors on C

principal divisors, Pic0C =

divisors of degree 0 on C

principal divisors

are called the Picard group of C and the degree 0 Picard group of C. Twodivisors are called linearly equivalent if they have the same class in PicC.There is an obvious (split) exact sequence

0 −→ Pic0C −→ PicCdeg−→ Z −→ 0.

Example 11.1. On C=P1 any divisor D=∑a∈P1

na(a) of degree 0 is principal,

D = (f), f =∏a6=∞

(x− a)na .

So Pic0 P1 = 0 and PicP1 = Z.

Example 11.2. On an elliptic curve E with origin O every divisor of degree0 is linearly equivalent to a divisor of the form (P )−(O) for a unique P ∈ E.Thus Pic0E ∼= E and PicE ∼= E × Z as abelian groups.

A morphism of non-singular complete curves φ : C → C ′ induces a map

φ∗ : divisors on C ′ −→ divisors on C,


the pullback of divisors. For a point D = (P ),

φ∗(P ) =∑

f(Q)=P

eQ(Q),

the sum taken over the pre-images Q of P , and the coefficient eQ the rami-fication index23 of f at Q. It is easy to check that

deg φ∗D = deg φ degD and φ∗(f) = (f φ) for f ∈ k(D)×,

in particular φ∗ takes divisors of degree 0 to divisors of degree 0 and principaldivisors to principal divisors. So it induces maps

φ∗ : PicC ′ → PicC and Pic0C ′ → Pic0C.

Example 11.3. Let C : y = x2 ⊂ P2 and D = P1, with φ : (x, y) 7→ y.Then φ∗(a) = (

√a, a) + (−

√a, a) for non-zero a, and φ∗(0) = 2(0, 0).

12. Jacobians over C

Generalising Examples 11.1 and 11.2, it turns out that for any completenon-singular curve C of genus g the group Pic0C has a natural structure ofa principally polarized abelian variety of dimension g, the Jacobian of C.

Over the complex numbers this is classical theory of Abel and Jacobi.Thus, suppose k = C (so C is a compact Riemann surface), and choose

(1) a basis ω1, . . . , ωg of the space of regular differentials ΩC and(2) a basis γ1, . . . , γ2g of the first homology group H1(C,Z).

Fix a point P0 ∈ C and look at the map

C → Cg, P 7→(∫ P

P0

ω1, . . . ,

∫ P

P0

ωg

).

This is not well-defined because the integrals are path-dependent; changinga path by a loop with a class in

∑niγi ∈ H1(C,Z) modifies the integral by∑

niλi, where

λi =(∫

γi

ω1, ...,

∫γi

ωg

), (i = 1, . . . , 2g).

(The integrals∫γiωj are called the periods of C). Letting

Λ = Zλ1 + . . .+ Zλ2g ⊂ Cg (“period lattice”),

the integration map C → Cg/Λ is now well-defined. Extending it by linearitydefines the Abel-Jacobi map

(12.1) divisors on C −→ Cg/Λ.

It turns out that (H1(C,Z) =) Λ → Cg (= ΩC) is a lattice, and we have

23Defined by eQ =ordQ φ∗f

ordP ffor any f ∈ k(C′) with ordP f 6= 0.

28 TIM DOKCHITSER

Theorem 12.2 (Abel–Jacobi). The map (12.1) a bijection Pic0C → Cg/Λ.24

The lattice Λ = H1(C,Z) carries a natural non-degenerate alternatingform E : Λ×Λ→ Z of determinant 1, the intersection pairing for homol-ogy classes25; one checks that after embedding Λ → Cg, the form satisfiesE(iu, iv)=E(u, v) for u, v ∈ Cg. So E is a Riemann form and gives Cg/Λ thestructure of a principally polarized abelian variety, the Jacobian JacC of C.

Example 12.3. Let C be the genus 2 hyperelliptic curve y2 = x6−1 over C.Here ΩC = 〈dxy ,

xdxy 〉, and we let α1, α2, β1, β2 be the standard ‘symplectic’

basis of H1(C,Z) as in Figure 1; thus E(αj , βj) = 1 = −E(βj , αj), and therest are 0.

Figure 1

To represent C as a Riemann surface, make 3 cuts in C between the rootsof x6−1 (6th roots of unity), see Figure 2. There are two continuous branches

of√x6 − 1 on the complement, in other words every point (x, y) ∈ C with

x /∈ cuts lies on one of the two sheets. Glueing them together gives C,which is topologically a surface as in Figure 3 (the cuts become joiningcircles), visibly of genus 2.

Figure 2 Figure 3 Figure 4

The period lattice Λ ⊂ C2 is spanned by 4 vectors(∫γdxy ,∫γxdxy

)=(∫γ

dz√z6−1 ,

∫γ

zdz√z6−1

), γ ∈ α1, α2, β1, β2,

24Abel showed that the degree 0 divisors in the kernel of (12.1) are precisely the divisorsof meromorphic functions, and Jacobi proved surjectivity.

25To define E(γ, γ′) pick closed loops representing γ and γ′ that intersect each othertransversally in a finite set of points. Then follow γ, and count the number of times γ′

meets it from the left minus the ones from the right (C is oriented). This is bilinearand alternating by construction, and also non-degenerate of determinant 1 (look at thestandard generators of H1(C,Z), loops going ‘once around the holes’ as in Figure 1).


the plane integrals taken over the four loops in Figure 4, and the dotted linesindicate ‘the other sheet’, i.e. −

√z6 − 1 in the denominator instead. This

is the same as changing the sign and inegrating in the opposite direction.By continuity we can deform all 4 integrals to twice the integrals along theline segments as in Figure 5 (the third one with minus).

Figure 5

The 2 components for the 4 basis vectors of Λ are therefore∫ (j+1)/6

j/62πi

e2πitk√e12πit − 1

, j ∈ 0, 4, 1, 3, k ∈ 1, 2,

again with minus for the third integral. Evaluated numerically, they are

constant×(ζ 1 1 −ζζ2i i −i ζ2i

), ζ = e2πi/3.

Move the first 2 vectors to(10

)and

(01

)with an element of GL2(C), we get

JacC ∼= C2/Λ, Λ =⟨(

10

),(01

), i√

3

(2−1), i√

3

(−12

)⟩.

Note that Λ contains an index 12 sublattice

2⟨(

10

),(01

),(ζ0

),(0ζ

)⟩, so

JacC ∼ E ×E (via the corresponding 12-isogeny), where E = C/Z + ζZ isthe unique elliptic curve over C with an automorphism of order 6,

E : y2 = x3 + 1.

In particular, End0(JacC) ∼= GL2(Q(ζ)).

Now we go back to an arbitrary complete non-singular curve C over Cwith a fixed base point P0 ∈ C. We have defined a map

P 7→(∫ P

P0

ω1, . . . ,

∫ P

P0

ωg) : C −→ Cg/Λ = JacC,

which is clearly holomorphic, and therefore a morphism by Chow’s theorem.(Recall that both compact Riemann surfaces and abelian varieties over Care projective varieties.) Seeing the Jacobian as Pic0, the map is

C −→ Pic0C, P 7→ (P )− (P0),

and it is an embedding unless C ∼= P1.

30 TIM DOKCHITSER

The Jacobian of C is functorial in two ways. A non-constant map ofcurves

φ : C 7→ C ′

induces maps φ∗ : JacC −→ JacC ′ and φ∗ : JacC ′ −→ JacC, given on thelevel of divisors by the usual pushforward and pullback of divisors

φ∗(Q) = (φQ), φ∗(P ) =∑

f(Q)=P

eQ(Q),

restricted to Pic0 = Jac. Clearly φ∗ φ∗ = [deg φ], in particular φ∗ is onto,and kerφ∗ is finite. On the level of ambient spaces,

φ∗ : ΩC′/H1(C′,Z) = Cg

′/Λ′ −→ Cg/Λ = ΩC/H1(C,Z)

is induced by the usual pullback of differentials φ∗ : ΩC′ → ΩC .

Example 12.4. Let us use these purely functorial properties to compute(up to isogeny) the Jacobian of

C : y2 = ax6 + bx4 + cx2 + d.

Supposing the right-hand side has no multiple roots, C is a genus 2 hyper-elliptic curve. It admits two non-constant maps to elliptic curves,

φ : C E1 : y2 = ax3 + bx2 + cx+ dψ : C E2 : y2 = dx3 + cx2 + bx+ a

given by (x, y) 7→ (x2, y) and (x, y) 7→ ( 1x2 ,

yx3 ) respectively. The pullback

gives morphisms of abelian varieties with finite kernels

φ∗ : E1 −→ JacC, ψ∗ : E2 −→ JacC.

By Poincare reducibility theorem, JacC ∼ E1 ×E2 if E1 6∼ E2. If E1 ∼ E2,the claim is still true, because

φ∗ dxy = d(x2)y = 2xdx

y

ψ∗ dxy = d(1/x2)y/x3 = −2dx

y

are linearly independent, so φ∗ and ψ∗ map the lattices of E1 and E2 intodifferent complex subspaces of C2, and the two image lattices generate thatof JacC up to finite index.

Conversely, any genus 2 curve whose Jacobian is not simple is isomorphicto one of the form y2 = ax6 + bx4 + cx2 + d (Exc 12.2).

Exc 12.1. Suppose C : y2 = f(x) with f ∈ C[x] of even degree 2g + 2 with distinct rootsαi. (Any genus g hyperelliptic curve over C has such an equation.) For j = 2, . . . , 2g + 2fix any path in C from α1 to αj that does not pass through the αk and any continuous

branch of√f(z) along the path. Generalising Example 12.3, show that the vectors( ∫ αj

α1

dz√f(z)

,∫ αj

α1

zdz√f(z)

, . . .∫ αj

α1

zg−1dz√f(z)

), j = 2, . . . , 2g + 2

generate the period lattice of C.

Exc 12.2. Suppose C is a hyperelliptic genus 2 curve whose Jacobian is not simple. Then

C ∼= y2 = ax6 + bx4 + cx2 + d. (You may use Torelli’s theorem — see below.)

notes on abelian varieties [part i]matyd/av1.pdfnotes on abelian varieties [part i] 3 in particular,...

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