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  • ALGANT Master Thesis - July 2018

    Elliptic Curves and Modular Forms

    Candidate

    Francesco Bruzzesi

    Advisor

    Prof. Dr. Marc N. Levine

    Universitá degli Studi di Milano Universität Duisburg–Essen

  • Introduction

    The thesis has the aim to study the Eichler-Shimura construction associatingelliptic curves to weight-2 modular forms for Γ0(N): this is the perfect topicto combine and develop further results from three courses I took in the firstsemester of the academic year 2017-18 at Universität Duisburg-Essen (namelythe courses of modular forms, abelian varieties and complex multiplication).Chapter 1 gives a brief overview on the algebraic geometry results and toolswe will need along the whole thesis. Chapter 2 introduces and develops thetheory of elliptic curves, firstly as an algebraic curve over a generic field, andthen focusing on the fields of complex and rational numbers, in particular forthe latter case we will be able to define an L-function associated to an ellipticcurve. As we move to Chapter 3 we shift our focus to the theory of modularforms. We first treat elementary results and their consequences, then we see howit is possible to define the canonical model of the modular curve X0(N) overQ, and the integrality property of the j-invariant. Chapter 4 concerns Heckeoperators: Shimura’s book [Shi73 Chapter 3] introduces the Hecke ring and itsproperties in full generality, on the other hand the other two main referencesfor the chapter [DS06 Chapter 5] and [Kna93 Chapters XIII & IX] do notintroduce the Hecke ring at all, and its attributes (such as commutativity forthe case of interests) are proved by explicit computation. We try to take anintermediate approach to the subject and rephrase everything just in terms ofSL2(Z) and congruence subgroups. At the end of the chapter we will be able toassociate an L-function to a cusp form for Γ0(N). Lastly in Chapter 5 we areready to illustrate how to obtain an elliptic curve from a weight-2 cusp form forΓ0(N), making use of the theory of abelian varieties.

    i

  • ii

    Notes and References

    Chapter 1: The organization of Sections 1.1 and 1.2 is based on [Sil09,Chapter 1 & Chapter 2]. The first statement of Proposition 1.10 is taken from[DS06, Proposition 7.2.6]; Proposition 1.14 is from [Kna93, Proposition 11.43];Proposition 1.22 comes from [Was03, Proposition C.2]. The main references forthese two sections are [Har77], [Mir95] and [Sha77]. Section 1.3 treats somestandard results on the Jacobian variety associated to a Riemann surface. Themain references are [Kir92] and [Mir95].

    Chapter 2: Section 2.1 is organized as [Sil09, Chapter 3]. Proposition 2.1 istaken from [Sil09, III.1.4(i)]; Proposition 2.3 and Proposition 2.5 are based onfilling the details of [Kna93, Theorem 11.57 and Theorem 11.58] (respectively);Proposition 2.12 comes from [Sil09, Theorem III.4.10]; Lemma 2.15 is from[Kna93, Lemma 11.63]; Theorem 2.16 and Theorem 2.17 try to fill the detailsof [Kna93, Theorem 11.64 and Theorem 11.66] (respectively). The main refer-ence for Section 2.2 is [Shi73, Chapter 4]. For Section 2.3 we followed [Kna93,Chapter X].

    Chapter 3: Some of the results arise as homeworks and/or are taken fromthe course of modular forms mentioned in the Introduction above and mostof the results are standards: main references are [DS06], [Miy89] and [Lan76].However Section 3.2.2 is taken from [Kna93, Chapter XI, Section 8].

    Chapter 4: The structure of the Chapter is as in [Shi73, Chapter 3]. Propo-sition 4.5 is from [Shi73, Proposition 3.8]; Lemma 4.9 is from[Shi73, Lemma3.12]; Proposition 4.10 is from [Shi73, Proposition 3.14]; Lemma 4.40 and 4.41are respectively from [Shi73, Lemma 3.61 and 3.62]. Results from Theorem 4.33to Proposition 4.39 are taken from [Kna93, Chapter XI, Section 5].

    Chapter 5: The organization of the whole chapter is as in [Kna93, Chap-ter XI, Sections 10 & 11]. For Section 5.1 we used as references [Lan59] and[Swi74]. Section 5.2 is from the last part of [Kna93, Chapter XI, Section 10];Proposition 5.15 is taken from [Kna93, Theorem 11.74].

  • iii

    Notation

    Z,Q,R,C integers, rationals, reals, complex numbersFp the field with p elementsH complex upper half planeRe(z), Im(z) real, imaginary part of zK algebraic closure of the field KAut(K/L) automorphism group of K fixing the subfield L ⊂ KMn(K) n× n matrices with coefficients in KGLn(K) invertible n× n matrices with coefficients in KSLn(K) n× n matrices with coefficients in K and determinant 1[A : B] index of B in A or degree of A in BW∨ dual space of the vector space WA× group of invertible elements of the ring A#S cardinality of the set S

  • iv

    Acknowledgements

    First and foremost, I would like to express my gratitude to Professor Dr.Marc N. Levine, who accepted to supervise my study in this topic and patientlyspent time to enlight me with his deep mathematical insights. He motivated meto do always better.

    I want to thank all the people I met in the last two year in the Algant Master,both Professors and students. In particular Bob, John and Francesco: we sharedevery moment, all the laughs and all the dissapointments. We constantly helpedeach other, both in life and in university. We all know how frustating it feels tostudy math sometimes, and I am thankful that we overcome the difficulties ofthese two years together.

    A special thanks goes to Federica, who had to bear me every single day ofthis stressful period. Even though she had to listen to my complaints, she mademe smile in every situation. Thank you for having coped with all my strugglesand problems.

    Also I would like to mention my dear friend Antonio, the person who helpedme to develop a taste for number theory, I couldn’t ask for a better mentor andfriend. No matter how long we don’t see each other, we always have a great timeand connection.

    I am thankful to all my hometown friends and relatives. You are the reasonwhy I keep coming back home and on all occasions it feels like time didn’t pass.

    Last, but certainly most important, I want to thank my parents Lanfrancoand Laura for providing me with unfailing support and continuous encourage-ment throughout my years of study. This accomplishment would not have beenpossible without them: words cannot really describe how grateful I am.

    Francesco

  • Contents

    Introduction iNotes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiAckowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

    1 Algebraic Geometry 11.1 Algebraic Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Algebraic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    1.2.1 Fields of positive characteristic . . . . . . . . . . . . . . . 101.2.2 Riemann-Roch theorem . . . . . . . . . . . . . . . . . . . 13

    1.3 Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

    2 Elliptic Curves 192.1 Elliptic curves over an arbitrary field . . . . . . . . . . . . . . . . 19

    2.1.1 Weierstrass form & abstract elliptic curves . . . . . . . . 192.1.2 Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    2.2 Elliptic curves over C . . . . . . . . . . . . . . . . . . . . . . . . 272.2.1 The Weierstrass ℘-function . . . . . . . . . . . . . . . . . 292.2.2 Isogenies over C . . . . . . . . . . . . . . . . . . . . . . . 332.2.3 Automorphisms of an elliptic curve . . . . . . . . . . . . . 35

    2.3 Elliptic curves over Q . . . . . . . . . . . . . . . . . . . . . . . . 362.3.1 L-function associated to an elliptic curve . . . . . . . . . . 372.3.2 Hasse theorem . . . . . . . . . . . . . . . . . . . . . . . . 38

    3 Modular Forms 403.1 Modular forms for SL2(Z) . . . . . . . . . . . . . . . . . . . . . . 40

    3.1.1 Functions of lattices . . . . . . . . . . . . . . . . . . . . . 403.1.2 The action of SL2(Z) on H . . . . . . . . . . . . . . . . . 423.1.3 Divisors of modular functions . . . . . . . . . . . . . . . . 453.1.4 The space of modular forms . . . . . . . . . . . . . . . . . 493.1.5 The modular curve X0(1) . . . . . . . . . . . . . . . . . . 52

    3.2 Congruence subgroups . . . . . . . . . . . . . . . . . . . . . . . . 583.2.1 Modular functions of higher level . . . . . . . . . . . . . . 633.2.2 The canonical model of X0(N) over Q . . . . . . . . . . . 67

    3.3 Integrality of the j-invariant . . . . . . . . . . . . . . . . . . . . . 693.3.1 j(z) is an algebraic number . . . . . . . . . . . . . . . . . 703.3.2 j(z) is integral . . . . . . . . . . . . . . . . . . . . . . . . 71

    v

  • CONTENTS vi

    4 Hecke Operators 744.1 The Hecke ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

    4.1.1 The structure of R(Γ,∆) . . . . . . . . . . . . . . . . . . 764.2 Action on modular functions . . . . . . . . . . . . . . . . . . . . 83

    4.2.1 Hecke operators on SL2(Z) . . . . . . . . . . . . . . . . . 844.2.2 Hecke operators on congruence subgroups . . . . . . . . . 86

    4.3 L-function of a cusp form . . . . . . . . . . . . . . . . . . . . . . 93

    5 Eichler-Shimura Theory 975.1 Complex abelian varieties and Jacobian varieties . . . . . . . . . 975.2 Technical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.3 Elliptic curves associated to weight-2 cusp forms . . . . . . . . . 104

    5.3.1 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . 108

    Bibliography 109

  • Chapter 1

    Algebraic Geometry

    Throughout this whole chapter, let K0 denote a field and K be an alge-braically closed field containing K0

    1.1 Algebraic Varieties

    Definition. Define the affine n-dimensional space over K as

    An = An(K) = {P = (x1, ..., xn) | xi ∈ K}

    Similarly, define the set of K0-points (or K0-rational points) of An as the set

    An(K0) = {P = (x1, ..., xn) ∈ An | xi ∈ K0}

    Remark 1.1. We have an action of the Galois group Gal(K/K0) on An: letσ ∈ Gal(K/K) and P ∈ An, then Pσ = (xσ1 , ..., xσn). It follows that

    An(K0) = {P ∈ An | Pσ = P ∀σ ∈ Gal(K/K0)}

    Recall that by the Hilbert basis theorem any polynomial ring over a field isa Noetherian ring, thus every ideal is finitely generated.

    Definition. Consider the polynomial ring in n variables K[X1, .., Xn], to anyideal J ⊂ K[X1, .., Xn] we can associate a subset of An, called affine algebraicset,

    VJ = {P ∈ An | f(P ) = 0 ∀f ∈ J}

    Viceversa, to any algebraic set V ⊂ An we can associate an ideal of polynomialsvanishing on V , called ideal of V ,

    I(V ) = {f ∈ K[X1, .., Xn] | f(P ) = 0 ∀P ∈ V }

    In particular we say that an algebraic set is defined over K0 if its ideal I(V ) canbe generated by polynomials in K0[X1, .., Xn] and we write V/K0. Then if V isdefined over K0, the set of K0-rational points of V is

    V (K0) = V ∩ An(K0)

    1

  • 1.1. ALGEBRAIC VARIETIES 2

    Remark 1.2. Assume that V is an algebraic set defined over K0 and let

    f1, ..., fr ∈ K0[X1, .., Xn] be the generators for I(V/K0)def= I(V )∩K0[X1, .., Xn],

    then

    V (K0) = {P = (x1, .., xn) ∈ An(K0) | f1(P ) = ... = fr(P ) = 0}

    Noice that Gal(K/K0) acts on K[X1, .., Xn] by acting on the coefficients of anelement f ∈ K[X1, .., Xn]. Then for f ∈ K[X1, .., Xn], P ∈ An, σ ∈ Gal(K/K0)

    (f(P ))σ = fσ(Pσ)

    Remark 1.3. If f ∈ K0[X1, .., Xn] and P ∈ An, then for σ ∈ Gal(K/K0) wehave f(Pσ) = f(P )σ. Therefore if V is defined over K0 we can characterizeV (K0) as

    V (K0) = {P ∈ V | Pσ = P ∀σ ∈ Gal(K/K0)}

    Definition. An affine algebraic set is irreducible if it is not union of two properaffine algebraic sets. An irreducible affine algebraic set is called affine variety

    Proposition 1.4. Let V ̸= ∅ be an affine algebraic set, then V is irreducible ifand only if I(V ) is a prime ideal.

    Proof. Assume V is reducible, so that V = V1 ∪ V2 for some proper algebraicsubsets V1, V2 $ V . Since the containment is proper, there exist f ∈ I(V1)rI(V )and g ∈ I(V2)rI(V ). Therefore fg vanishes on V1∪V2 = V and thus fg ∈ I(V ).We conclude that I(V ) is not a prime ideal.Viceversa, suppose that I(V ) is not prime, then there exist f, g ∈ K0[X1, .., Xn]rI(V ) but fg ∈ I(V ). Let us define J1 = (I(V ), f) and J2 = (I(V ), g), thenV1 = V (J1), V2 = V (J2) are both strictly contained in V . On the other handV ⊂ V1∪V2 since if P ∈ V then fg(P ) = 0, so that f(P ) = 0 or g(P ) = 0 whichyields that P ∈ V1 or P ∈ V2. This shows that V is reducible.

    Definition. Given an affine variety V , define its affine coordinate ring as

    K[V ] = K[X1, .., Xn]/I(V )

    Since I(V ) is a prime ideal, K[V ] is an integral domain, thus we can form itsquotient field K(V ) called function field of V .In case that V is defined over K0 we have K0[V ] = K0[X1, .., Xn]/I(V/K0).Since I(V ) is prime, so is I(V/K0): therefore K0[V ] is an integral domain aswell.

    Remark 1.5. Recall that f is a polynomial function on V if there existsF ∈ K[X1, .., Xn] such that F (P ) = f(P ) ∀P ∈ V .Then K[V ] can be identified with the set {f : V −→ K | f is a polynomial function}

    Definition. The dimension dim(V ) of an affine variety V is the transcendencedegree of K(V ) over K. In particular an affine curve is an affine variety ofdimension one.

    Definition. Let V ⊂ An be an affine variety and let I(V ) = ⟨f1, ..., fr⟩. A pointP ∈ V is a nonsingular point if the matrix⎛⎜⎝

    ∂f1∂X1

    (P ) . . . ∂fr∂X1 (P )...

    ...∂f1∂Xn

    (P ) . . . ∂fr∂Xn (P )

    ⎞⎟⎠ (1.1.1)

  • 1.1. ALGEBRAIC VARIETIES 3

    has rank n− dim(V ). V is nonsingular if each point P ∈ V is nonsingular.

    Remark 1.6. These definitions do not depend upon the choice of the generatorsf1, .., fr of I(V ).

    To see this, let P ∈ V and define

    mP = {f ∈ K[V ] | f(P ) = 0}

    mP is maximal in K[V ] since evaluation at P is an isomorphism from K[V ]/mPto K and mP /m

    2P is a finite dimensional vector space over K.

    Fact 1.7. P ∈ V is nonsingular if and only if dim(mP /m2P ) = dim(V )

    Proof. See [Har77] I.5.1

    Definition. Define the local ring of V at P as

    K[V ]P = {f/g ∈ K(V ) | g(P ) ̸= 0}

    Its elements are said to be regular (or defined) at P .

    K[V ]P is a local ring with maximal ideal

    MP = {f/g ∈ K[V ]P | f(P ) = 0} = mP ·K[V ]P

    MP is maximal and consists of all non-invertible elements of K[V ]P , thereforeit is the unique maximal ideal.Let us now consider the natural map ι : mP −→MP /M2P given by f ↦→ f +M2P

    • ker(ι) = mP ∩M2P = m2P

    • ι is surjective since for all f/g ∈MP we have f/g(P ) ∈ mP and

    f

    g(P )− fg=f · (g − g(P ))g · g(P )

    ∈M2P

    which means that ι(f/g(P )) = f/g +M2P .

    We can conclude that ι induces an isomorphism mP /m2P

    ∼−→MP /M2P

    Definition. The projective n-space over K is

    Pn = Pn(K) = {0 ̸= P = (x0, .., xn) ∈ An+1}/ ∼

    where (x0, .., xn) ∼ (y0, .., yn) if there exists λ ∈ K×

    such that yi = λxi for alli = 0, .., n. We denote [x0, .., xn] such equivalence class, thus we can write

    Pn(K) = {[x0, .., xn] | xi ∈ K,∃i : xi ̸= 0}

    x0, .., xn are called homogeneous coordinates of the corresponding point in Pnand the set of K0-rational points in Pn is

    Pn(K0) = {[x0, .., xn] ∈ Pn | xi ∈ K0 ∀i}

  • 1.1. ALGEBRAIC VARIETIES 4

    As for the affine case, the Galois group Gal(K/K0) acts on Pn by acting onhomogeneous coordinates:

    [x0, .., xn]σ = [xσ0 , .., x

    σn]

    Definition. A polynomial f ∈ K[X0, .., Xn] is homogeneous of degree d iff(λX0, .., λXn) = λ

    df(X0, .., Xn) for all λ ∈ K×.

    An ideal J ⊂ K[X0, .., Xn] is homogeneous if it is generated by homogeneouspolynomials.

    Definition. To any homogeneous ideal J ⊂ K[X0, .., Xn] we cab associate itszero set in Pn

    WJ = {P = [x0, .., xn] ∈ Pn | f(P ) = 0 ∀f ∈ J}

    called projective algebraic set.Viceversa, if W ⊂ Pn is a projective algebraic set, its homogeneous ideal ofpolynomials vanishing on it is the ideal I(W ) generated by all homogeneouspolynomials f ∈ K[X0, .., Xn] such that f(x0, .., xn) = 0 for all [x0, .., xn] ∈W .

    As for the affine case, W is defined over K0 if I(W ) can be generated byelements of K0[X0, .., Xn] and we write W/K0. The set of K0-rational points ofW is W (K0) =W ∩ Pn(K0).

    Definition. A projective algebraic set is irreducible if it is not union of twoproper projective algebraic sets. A irreducible projective algebraic set W iscalled projective algebraic variety.

    Now notice that Pn(K) = U0 ∪ · · · ∪ Un where Ui = {[x0, .., xn] ∈ Pn(K) |xi ̸= 0}.Then for each i ∈ {0, .., n} the map Ui

    ϕi−→ An(K) given by

    [x0, .., xn] ↦→(x0xi, ..,

    xi−1xi

    ,xi+1xi

    , ..xnxi

    )is a bijection (actually it is a homeomorphism with respect to the Zariski topol-ogy).It follows that any projective variety W has a standard covering

    W =

    n⋃i=0

    Wi where Wi =W ∩ Ui

    And by mean of ϕi, each Wi can be regarded as an affine variety.

    Remark 1.8. The map W ↦→W0 =W ∩ U0 yields a bijection{projective varietiesW ⊂ Pn |W * {x0 = 0}

    }←→

    {affine varietiesW0 ⊂ An

    }which inverse map is given by the projective closure. Namely if V is an affinealgebraic set with ideal I(V ), its projective closure is the projective set W = Vwhose homogeneous ideal is generated by {Xd0f(X1X0 , ..,

    XnX0

    ) | f ∈ I(V )}

  • 1.1. ALGEBRAIC VARIETIES 5

    Definition. If W is a projective variety its function field is

    K(W ) = {f/g | f, g ∈ K[X0, .., Xn],homogeneous of same degree d, g /∈ I(W )}/ ∼

    where f/g ∼ f ′/g′ if and only if fg′ − gf ′ ∈ I(W ).The elements of K(W ) are called rational functions on W .

    Lemma 1.9. Let W be a projective variety such that W * V (x0), with stan-dard covering W =W0 ∩ .. ∩Wn. We have an isomorphism

    K(W )∼−→ K(W0)

    Proof. We have the following maps are mutually inverse:

    K(W ) −→ K(W0) given by f(X0, .., Xn)/g(X0, .., Xn) ↦→ f(1, X1, .., Xn)/g(1, X1, .., Xn)

    K(W0) −→ K(W ) given by f(X1, .., Xn)/g(X1, .., Xn) ↦→ f(X1X0

    , ..,XnX0

    )/g(X1X0

    , ..,XnX0

    )

    Definition. The dimension dim(W ) of a projective variety W is the transcen-dence degree of K(W ).

    Definition. A rational function F ∈ K(W ) is regular at P ∈ W if there is arepresentation F = f/g such that g(P ) ̸= 0. The local ring of W at P ∈W is

    K[W ]P = {F ∈ K(W ) | F is regular at P}

    with maximal ideal MP = {F ∈ K[W ]P | F (P ) = 0}.A point P ∈ W is a nonsingular point if dim(MP /M2P ) = dim(W ); W is non-singular if each point P is nonsingular.

    Now let W1 ⊂ Pm and W2 ⊂ Pn be projective varieties in the respectiveprojective spaces.

    Definition. A rational map F :W1 −→W2 is a tuple F = [f0, .., fn] of elementsof K(W1) such that ∀P ∈ W1 where f0, .., fn are all defined, then F (P ) =[f0(P ), .., fn(P )] belongs to W2.F is defined over K0 if W1/K0, W2/K0 and f0, .., fn can be multiplied by thesame invertible elements of K so that f0, .., fn ∈ K0(W1)

    Definition. A rational map F = [f0, .., fn] : W1 −→ W2 is regular at P ∈ W1if there exists g ∈ K(W1) such that gfi is regular at P and [gf0(P ), .., gfn(P )]is not the 0-tuple.A rational map regular at all points of W1 is called morphism.

    Definition. We say that W1 and W2 are isomorphic if ∃F : W1 −→ W2 and∃G : W2 −→ W1 morphisms such that F ◦ G and G ◦ F are the respectiveidentity maps.W1/K0 and W2/K0 are isomorphic over K0 is F and G as above are definedover K0.

    Definition. A rational map F :W1 −→ W2 is dominant if its image F (W1) isa Zariski dense subset of W2.

  • 1.2. ALGEBRAIC CURVES 6

    Now if F :W1 −→W2 is a rational map and F is dominant, then we have awell-defined K-algebra map

    F ∗ : K(W2) −→ K(W1)

    such that F ∗(g/h) = F ∗(g)/F ∗(h) = g ◦ F/h ◦ F .In particular if F is an isomorphism then F ∗ is an isomorphism of function fieldsand an isomorphism of local rings at each point.

    1.2 Algebraic Curves

    Proposition 1.10. Let C be a projective curve and P be a nonsingular point

    on the curve. Then MP = (t) for some t ∈MP rM2P and∞⋂h=1

    MhP = {0}

    Proof. P ∈ C nonsingular implies that mP /m2P∼−→ MP /M2P has dimension 1

    as K-vector space, therefore mP /m2P = ⟨t⟩ for some t ∈ K[C]. We claim that

    MP = t ·K[C]P .Let N

    def= t ·K[C]P ⊆MP , and let us prove that the quotient MP /N is trivial,

    so that they coincide.Since MP is a K[C]P -module, so is the quotient MP /N , and notice that

    • MP · (MP /N) = (N +M2P )/N

    • N +M2P = (K · t+M2P )K[C]P

    ThereforeMP · (MP /N) =MP /N

    and by Nakayama’s lemma this can happen only if MP = N = (t).

    Now let f ∈∞⋂h=1

    MhP =∞⋂h=1

    (th) and write f = th · fh. Since fh = t · fh+1 we

    obtain a chain of inclusions

    (f1) ⊆ (f2) ⊆ ..

    which has to stabilize due to the Noetherianity of K[C]P . Therefore there existsm > 0 such that (fm) = (fm+1), hence

    fm+1 = afm = atfm+1 for some a ∈ K[C]P

    Either at = 1 or fm+1 = 0. Since t ∈ MP , t is not invertible, and we canconclude that fm+1 = 0 =⇒ f = 0.

    Definition. Any such t ∈MP rM2P is called uniformizer of C at P .Define ordP (·) : K[C]P −→ Z ∪ {∞} as

    ordP (f) =

    {+∞ if f = 0min{l ∈ Z>0 | f ∈M lP rM

    l+1P } if f ̸= 0

    (1.2.1)

    Remark 1.11. (i) If f ̸= 0 and l = ordP (f), then f = a · tl for some a ∈K[C]×P .

  • 1.2. ALGEBRAIC CURVES 7

    (ii) We can extend ordP to the whole K(C), namely if F = f/g with f, g ∈K[C]P and g ̸= 0 then define

    ordP (F ) = ordP (f)− ordP (g)

    This is well defined since if l = ordP (F ) then l is the unique power forwhich F/tl is a unit in K[C]P .

    Proposition 1.12. ordP : K(C) −→ Z∪{∞} satisfies the following properties:

    (a) ordP (F ·G) = ordP (F ) + ordP (G)

    (b) ordP (F +G) ≥ min{ordP (F ), ordP (G)}

    (c) K[C]P = {F ∈ K(C) | ordP (F ) ≥ 0}

    (d) MP = {F ∈ K(C) | ordP (F ) > 0}

    (e) ordP (F ) =∞⇐⇒ F = 0

    In other words, K[C]P is a discrete valuation ring with valuation given byordP .

    Theorem 1.13. Let C be a projective curve, P ∈ C be a nonsingular pointand let W be a projective variety. If F : C −→ W is a rational map, then F isregular at P .In particular it follows that if C is nonsingular, then F is automatically a mor-phism.

    Proof. Let F = [f0, .., fn] with fj ∈ K(C) and let t be a uniformizer at P .Define m = min

    0≤j≤n{ordP (fj)}, then ordP (t−m · fj) ≥ 0 ∀j = 0, .., n and equality

    holding for some j = j0. This means that t−m · fj is regular at P and (t−m ·

    fj0)(P ) ̸= 0.Hence taking g = t−m exhibits that F is regular at P .

    Definition. A proper subring R ⊂ K(C) containing K is called discrete valua-tion ring of K(C) over K if there exists a function v : K(C) −→ Z∪{∞} calledvaluation, such that:

    (a) v(FG) = v(F ) + v(G);

    (b) v(F +G) ≥ min{v(F ), v(G)};

    (c) v(F ) =∞ ⇐⇒ F = 0;

    (d) R = {F ∈ K(C) | v(F ) ≥ 0}.

    In particular if R is a discrete valuation ring of K(C) over K, then R is a

    local ring with maximal ideal MRdef= {F ∈ R | v(F ) > 0}.

    In particular, for a nonsingular projective curve C we have the following:

    Proposition 1.14. Let C be a nonsingular projective curve and let R be adiscrete valuation ring of K(C) over K with valuation v. Then there exists P ∈ Csuch that v = c · ordP for some c ∈ Z>0, moreover for such P , R = K[C]P andMR =MP .

  • 1.2. ALGEBRAIC CURVES 8

    Proof. Let [x0, .., xn] be the projective coordinates of Pn, then we can regard xjas a function from C to K, mapping a point P to its j-th coordinate.For what follows, do not consider the xj ’s such that xj ∈ I(C) = {f ∈K[X0, .., Xn] | f(P ) = 0 ∀P ∈ C}. For the remaining indices, regard xj/x0as an element of K(C)

    • If v(xj/x0) ≥ 0 ∀j pass to the affine curve C ∩ U0 = C0 (which is notempty since x0 ̸= 0 somewhere in C).Elements of the affine coordinate ring K[C0] are polynomials in the xj/x0’s,therefore v ≥ 0 on the whole ring. It follows that K[C0] ⊂ R.

    • If v(xj/x0) < 0 for some j, then let j0 be the index for which v(xj0/x0) isthe smallest possible. Then v(xj/xj0) = v(xj/x0) − v(xj0/x0) > 0 for allj.Changing j0 with 0 we can assume C0 is not empty and K[C0] ⊆ R.

    Therefore Idef= MR ∩K[C0] is a proper ideal of K[C0] ∼= K[X1, .., Xn]/I(C).

    Let J be the inverse image of I in K[X1, .., Xn] (so in particular I(C) ⊂ J). Thenthere are some points P ∈ An that annihilate all the elements of J , thereforethose P belongs to C and for each f ∈ I we have ordP (f) ≥ 0.We claim that ∀F ∈ K(C)

    v(F ) > 0 and ordP (F ) ≥ 0 =⇒ ordP (F ) > 0 (1.2.2)

    In fact since C ∩ U0 ̸= ∅, by Lemma 1.9 we can identify K(C) with K(C0) andK[C]P with K[C0]P . Let F ∈ MR ∩ K[C0]P , F = f/g for f, g ∈ K[C0] andordP (g) = 0; g ∈ K[C0] ⊂ R, implies that f = F · g belongs to I and thereforeordP (f) > 0. It follows that ordP (F ) = ordP (f)− ordP (g) > 0.

    The contrapositive of (1.2.2) is ordP (F ) = 0 =⇒ v(F ) ≤ 0; which appliedto 1/F yields

    ordP (1/F ) = −ordP (F ) = 0 =⇒ 0 ≥ v(1/F ) = −v(F ) =⇒ v(F ) ≥ 0

    and consequentelyordP (F ) = 0 =⇒ v(F ) = 0 (1.2.3)

    Finally choose an uniformizer t ∈ K[C0] at P : since K[C0] ⊆ R we have thatv(t) ≥ 0 and if F ∈ K(C), F = tℓ · F0 with ℓ = ordP (F ) and ordP (F0) = 0 (sothat by (1.2.3) v(F0) = 0), then

    v(F ) = v(tℓ) + v(F0) = ℓ · v(t) = v(t) · ordP (F )

    Since R is a proper subring of K(C), v cannot be identically zero and v(t) ≥ 0:therefore v(t) > 0. We obtained that v = v(t) · ordP and c := v(t) > 0 as wewanted.

    Theorem 1.15. Let L be a field of trancendental degree one and finitely gen-erated over K and let CL be the set of discrete valuation rings R of L over K.Then there exists a nonsingular projective curve C such that K(C) ∼= L and CLcan be canonically identified with the set of points of C.

    Sketch of the proof.

  • 1.2. ALGEBRAIC CURVES 9

    Step 1 If B is an integral domain finitely generated as K-algebra then B is iso-morphic to a coordinate ring of an affine variety. Namely, let x1, .., xngenerate B over K, then the map Xj ↦→ xj yields a homomorphismK[X1, .., Xn] � B which kernel I is prime since B is a domain. There-fore VI is the required affine variety.

    Step 2 Let x ∈ L r K and consider the polynomial ring K[x]: then L is a finiteextension of the quotient field K(x). Now if B is the integral closure of K[x]in L, the construction in the previous step generate a nonsingular affinecurve (since in dimension 1 a ring is regular if and only if it is integrallyclosed)

    Step 3 Let R ∈ CL, choose x ∈ R r K, then K[x] ⊆ R and since R is a discretevaluation ring, it is integrally closed in L. Hence B ⊆ R. Now let N =MR∩B, then N is maximal in B and therefore it corresponds to a uniquepoint of the affine curve constructed from B.

    Step 4 Finally maps CL diagonally into the product of finitely many projectiveclosures (of affine curves). The image of CL will be the desired curve C.

    Theorem 1.16. Let C1 be a projective nonsingular curve over K and let C2be any curve over K. If F : C1 −→ C2 is a morphism, then either F (C1) is apoint (i.e. F is constant) or F (C1) = C2 (i.e. F is surjective). In the latter caseF ∗ is injective, K(C1) is a finite field extension of F

    ∗(K(C2)) and F is a finitemorphism.

    Proof. See [Har77] II.6.8

    Theorem 1.17. Let C1 and C2 be projective nonsingular curves over K and leti : K(C2) −→ K(C1) be an injective map fixing K. Then there exists a uniquenon constant map F : C1 −→ C2 such that F ∗ = i

    Proof. Let C1 ⊂ Pm, and without loss of generality assume that C2 * V (x0).Then for each j = 0, ..,m, let fj := xj/x0 ∈ K(C2), and define

    F = [1, i(f1), .., i(fm)] : C1 −→ C2

    Such F satisfies F ∗ = i. If G = [g0, .., gm] is another map with the propertyG∗ = i, then gj/g0 = G

    ∗(fj) = F∗(fj) = i(fj) for all j = 0, ..,m, therefore

    F = G.

    Definition. Let F : C1 −→ C2 be a map between nonsingular projective curvesover K. Let us define the degree of F as

    deg(F ) =

    {0 if F is constant

    [K(C1) : F∗(K(C2))] otherwise

    Moreover if F is not constant, then F is separable, inseparable, or purely insep-arable if the field extension K(C1)/F

    ∗(K(C2)) is respectively separable, insepa-rable or purely inseparable.

    Remark 1.18. (i) If C1 and C2 are defined over K0, then

    [K(C1) : F∗(K(C2))] = [K0(C1) : F

    ∗(K0(C2))]

  • 1.2. ALGEBRAIC CURVES 10

    (ii) Since any algebraic extension can be obtained as a separable extensionfollowed by a purely inseparable one, we denote degsF the degree of theseparable part and degiF the degree of the purely inseparable part of theextension K(C1)/F

    ∗(K(C2)). Then deg(F ) = degs(F ) · degi(F )

    Corollary 1.19. Let F : C1 −→ C2 be a non constant map between nonsingularprojective curves over K. If deg(F ) = 1 then F is an isomorphism.

    Proof. See [Sil09] II.4.1

    Definition. Let F : C1 −→ C2 be a non constant map between nonsingularprojective curves. The ramification degree of F at a point P ∈ C1 is the positiveinteger

    eF (P ) = ordP (F∗(t)) = ordP (t ◦ F )

    where t = tF (P ) is a uniformizer at F (P ).F is unramified at P if eF (P ) = 1, and it is unramified if it is unramified ateach point of C1.

    Notice that tF (P ) ◦F = teP (F )P g for tP uniformizer at P and some g in K(C1)

    such that ordP (g) = 0.

    Proposition 1.20. Let F : C1 −→ C2 be a non constant map between nonsin-gular projective curves, then

    (a) For all Q ∈ C2 we have∑

    P∈F−1(Q)eF (P ) = deg(F ).

    (b) For all but finitely many Q ∈ C2 we have |F−1(Q)| = degs(F )

    Proof. (a) See [Sha77] III.2.1

    (b) See [Har77] II.6.8

    Corollary 1.21. A map F : C1 −→ C2 between nonsingular projective curvesis unramified if and only if |F−1(Q)| = deg(F ) ∀Q ∈ C2.

    Proof. By Proposition 1.20, we have that

    |F−1(Q)| = deg(F ) ⇐⇒∑

    P∈F−1(Q)

    eF (P ) = |F−1(Q)|

    Since eF (P ) ≥ 1, this happens if and only if eF (P ) = 1

    1.2.1 Fields of positive characteristic

    We now want to take a look on what happens in the case of charK ̸= 0: theproblem in such situation is that field extensions of fields with prime character-istic are not necessarely separable.Let p be a fixed rational prime and Fp be the field of p elements with algebraicclosure Fp. Recall that for any q = pr, r ∈ Z>0, there exists a unique field Fqof order q in Fp, namely the splitting field of Xq − X over Fp. In particularFp =

    ⋃r≥1

    Fpr .

  • 1.2. ALGEBRAIC CURVES 11

    Definition. The q-th Frobenius map φq on Fp is φq : Fq −→ Fq , x ↦→ xq

    Proposition 1.22. Let q = pr, r ∈ Z>0, then:

    (a) Fq = Fp

    (b) Let α ∈ Fq, then α ∈ Fqn if and only if φnq (α) = α

    (c) φq is an automorphism of Fq; in particular ∀x, y ∈ Fq

    φq(x+ y) = φq(x) + φq(y) and φq(xy) = φq(x)φq(y)

    Proof. (a) It is a more general fact that ifM ⊂ L with L algebraic overM thenM = L.In fact if α ∈ L and L is algebraic over M , then α is algebraic over M , thusα ∈M . The other inclusion is trivial and equality follows.

    (b) Let us first prove that Fq = {α ∈ Fq | αq = α}. Since F×q ⊂ Fq is a group oforder q − 1 we have that αq−1 = 1 for all α ∈ F×q and therefore ∀α ∈ Fq wehave αq = α. This proves the containment Fq ⊂ {α ∈ Fq | αq = α}.Now recall that a polynomial f(X) has multiple roots if and only if f(X) andits formal derivative f ′(X) have common roots. Since we are in characteristicp,

    d

    dX(Xq −X) = qXq−1 − 1 = −1

    which means that Xq − X has no multiple roots and therefore we haveq = def(Xq −X) dinstict α ∈ Fq such that αq = α. Hence |Fq| = |{α ∈ Fq |αq = α}| and Fq ⊆ {α ∈ Fq | αq = α} therefore they must coincide. Finally,(b) follows by substituting q with qn.

    (c) Remark that(pk

    )= p!k!(p−k)! and for all k = 1, .., p− 1 we have a factor p at

    its numerator which does not cancel by the denominator, thus

    ∀k = 1, .., p− 1(p

    k

    )= 0 mod p

    It follows that φp(x+ y) = (x+ y)p = xp + yp = φp(x) + φp(y). Now since

    if q = pr then φq = φrp, we have

    φq(x+ y) = φq(x) + φq(y) ∀x, y ∈ Fq

    On the other hand the multiplicativity of φq follows by commutativity ofFq.Therefore φq is a homomorphism of fields, and this also gives us injectivity.Now if α ∈ Fq, then α ∈ Fqn for some n ∈ Z>0 and therefore α = φnq (α) =φq(φ

    n−1q (α)). Hence α ∈ im(φq), therefore φq is surjective.

    Fact 1.23. For q = pr, Fq/Fp is a Galois extension with cyclic Galois group oforder r generated by φp.

    Definition. The Frobenius map on Fnp is φp : Fn

    p −→ Fn

    p , (x1, .., xn) ↦→(xp1, .., x

    pn)

  • 1.2. ALGEBRAIC CURVES 12

    Remark 1.24. φp : Fn

    p −→ Fn

    p is a bijection with fixed points Fnp , which inducesa bijection at level of projective classes

    φp : Pn(Fp) −→ Pn(Fp) [x0, .., xn] ↦→ [xp0, .., xpn]

    For brevity write X = (X0, .., Xn), e = (e0, .., en) and consider a homoge-neous polynomial f(X) =

    ∑eaeX

    e ∈ Fp[X0, .., Xn], then define

    f (q)(X) =∑e

    φq(ae)Xe ∈ Fp[X0, .., Xn]

    and since we are in characteristic p if follows that

    f (q)(Xq) = (f(X))q (1.2.4)

    Remark 1.25. Let q = pr, and let C be a projective curve over Fp with idealof polynomials I(C) = ⟨f1, .., fr⟩. Then we can define a new curve C(q) over Fpwhich ideal of polynomial I(C(q)) is the ideal generated by {f (q) | f ∈ I(C)}.We have that the Frobenius map on Pn induces a q-th Frobenius morphism

    φq : C −→ C(q) given by [x0, .., xn] ↦→ [xq0, .., xqn]

    In fact ∀[x0, .., xn] ∈ C

    f (q)(φq(x0, .., xn)) = f(q)(xq0, .., x

    qn)

    (1.2.4)=

    (f(x0, .., xn)

    =0

    )qIn particular notice that if C/Fp then C(p) = C

    Theorem 1.26. Notation as above, we have:

    (a) φ∗q(K(C(q))) = K(C)q

    def= {fq | f ∈ K(C)}

    (b) φq is purely inseparable;

    (c) deg(φq) = q

    Proof. (a) Remark that every algebraically closed field is perfect, therefore eachelement of K is a q-th power. Hence (K[X0, .., Xn])

    q = K[Xq0 , .., Xqn].

    Now let f/g ∈ K(C) with f, g homogeneous polynomial of the same degree.Then for any φ∗q(f/g) ∈ φ∗q(K(C(q))) we have

    φ∗q(f/g) = f(Xq0 , .., X

    qn)/g(X

    q0 , .., X

    qn) = f

    q0 (X

    q0 , .., X

    qn)/g

    q0(X

    q0 , .., X

    qn) ∈ K(C)q

    where f0 and g0 are polynomials which coefficients to the q-th power arethe coefficients of f and g respectively.

    On the other hand K(C)q is the subfield of K(C) of elements

    f(X0, .., Xn)q/g(X0, .., Xn)

    q

    Therefore they coincide.

    (b) Follows from (a).

  • 1.2. ALGEBRAIC CURVES 13

    (c) deg(φq) = ordP (tφq(P ) ·φq) = q where the first equality follows from Propo-sition 1.20 together with the fact that φq is a injective.

    Corollary 1.27. Let C1, C2 be nonsingular projective curves, defined over afield of prime characteristic p. Then any map F : C1 −→ C2 factor as F = Fs◦φqwhere q = degi(F ), φq : C1 −→ C

    (q)1 is the q-th Frobenius morphism and

    Fs : C(q) −→ C2 is a separable morphism.

    Proof. Let L = F ∗(K(C2))sep be the maximal separable extension of F∗(K(C2))

    in K(C1). Then K(C1)/L is purely inseparable of degree q = degi(F ), and inparticular we have that K(C1)

    q ⊂ L.From Theorem 1.26 K(C1)

    q = φ∗q(K(C(q)1 )) and [K(C1) : φ

    ∗q(K(C

    (q)1 ))] = q.

    By comparing the degrees, it follows that L = φ∗q(K(C(q)1 )). Thus we have the

    inclusionsF ∗(K(C2)) ⊂ φ∗q(K(C

    (q)1 )) ⊂ K(C1)

    which correspond to

    C1φq−→ C(q)1

    Fs−→ C2

    1.2.2 Riemann-Roch theorem

    Definition. The abelian group of divisors of a nonsingular curve C, Div(C), isthe free abelian group generated by the points of C. In other words D ∈ Div(C)is a formal sum

    D =∑P∈C

    nP (P ) with nP ∈ Z, nP = 0 for almost all P

    The degree of a divisor D is the (actual) sum deg(D) =∑

    P∈CnP ∈ Z.

    The subgroup of degree-0 divisors is

    Div0(C) = {D ∈ Div(C) | deg(D) = 0}

    Remark that if C is defined over K0, then Aut(K/K0) acts on Div(C) by

    Dσ =∑P∈C

    nP (Pσ)

    We say that D is defined over K0 if Dσ = D for all σ ∈ Aut(K/K0) The divisor

    of f ∈ K(C)× is

    div(f) =∑P∈C

    ordP (f) · (P ) =∑

    P∈f−1(0)

    ordP (f) · (P )−∑

    P∈f−1(∞)

    ordP (f) · (P )

    Divisors of the form div(f) are called principal divisors and its set is denoted byDivl(C). Notice that principal divisors have degree 0. Define the Picard groupof C as the quotient

    Pic(C) = Div(C)/Divl(C)

  • 1.2. ALGEBRAIC CURVES 14

    Remark 1.28. If F : C1 −→ C2 is a dominant morphism between nonsingularprojective curves, then F induces maps on divisors in both directions:

    F ∗ :Div(C2) −→ Div(C1) F∗ : Div(C1) −→ Div(C2)

    (Q) ↦→∑

    P∈F−1(Q)

    eF (P ) · (P ) (P ) ↦→ (F (P ))

    Definition. Let C be a nonsingular projective curve, the space of meromorphicdifferential 1-forms on C, denoted ΩC , is the K-vector space generated by thesymbols dx for x ∈ K(C) such that ∀x, y ∈ K(C), ∀u ∈ K:

    1. d(x+ y) = dx+ dy

    2. d(xy) = xdy + ydx

    3. du = 0

    If F : C1 −→ C2 is a nonconstant map between nonsingular projectivecurves, then the map between function fields F ∗ : K(C2) −→ K(C1) induces amap on differentials F ∗ : ΩC2 −→ ΩC1 given by∑

    fidxi ↦→∑

    F ∗(fi)d(F∗xi)

    Proposition 1.29. Let C be a curve, then

    (a) ΩC is a 1-dimensional K(C)-vector space.

    (b) Let x ∈ K(C), then dx is a basis for ΩC over K(C) if and only if K(C)/K(x)is a finite separable extension.

    (c) Let F : C1 −→ C2 be a non constant map between curves, then F isseparable if and only if F ∗ : ΩC2 −→ ΩC1 is injective (equivalently, nonzero).

    (d) Let P ∈ C and let t = tP be a uniformizer at P . Then for every ω ∈ ΩCthere exists a unique function f ∈ K(C) (depending on ω and t), such that

    ω = fdt

    Proof. See [Sil09] II.4.2, II.4.3

    Definition. Let C be a curve, let ω ∈ ΩC , let P ∈ C and t be a uniformizer atP . The order of ω at P is ordP (ω) = ordP (f) if ω = fdt. Then we can definethe divisor associated to ω as

    div(ω) =∑P∈C

    ordP (ω) · (P ) ∈ Div(C)

    We say that the differential ω is regular (or holomorphic) if ordP (ω) ≥ 0 for allP ∈ C.

    Notice that if ω1, ω2 ∈ ΩC are nonzero, then there exists f ∈ K(C)× suchthat ω2 = fω1 and consequently div(ω2) = div(f) + div(ω1).

  • 1.2. ALGEBRAIC CURVES 15

    Definition. Therefore define the canonical class of C as the image in Pic(C)of div(ω) for any non zero differential ω ∈ ΩC .

    Remark that for a curve C, there is a partial order in Div(C) as follows:we say that a divisor D =

    ∑P∈C

    nP (P ) is positive (and write D ≥ 0) if nP ≥

    0 ∀P ∈ C. Then for D1, D2 ∈ Div(C) the partial order is given by D1 ≥ D2 ifD1 −D2 ≥ 0

    Definition. Let D be a divisor on a curve C, the linear system associated toD is the (finite-dimensional) K-vector space

    L(D) = {f ∈ K | div(f) +D ≥ 0}

    Denote its dimension over K by ℓ(D)

    Definition. Let C be a curve defined over K, define the genus of C, denotedby g = g(C) as the dimension of L(κ) for any κ canonical divisor on C

    Let κ be a canonical divisor, say κ = div(ω), then L(κ) = {f ∈ K(C) |div(f) + κ ≥ 0}, or in other words f ∈ L(κ), hence 0 ≤ div(f) + κ =div(f) + div(ω) = div(fω). This means that fω is holomorphic. Viceversa,if fω is holomorphic, then f ∈ L(κ). It follows that

    L(κ) ∼= {ω ∈ ΩC | ω is holomorphic} =: Ω1(C)

    Proposition 1.30. Let D be a divisor on a curve C, then

    (a) If deg(D) < 0, then L(D) = {0} and ℓ(D) = 0.

    (b) If D′ ∈ Div(C) is linearly equivalent to D i.e. there exists f ∈ K(C)× suchthat D′ = D + div(f), then L(D) ∼= L(D′) and consequently ℓ(D′) = ℓ(D).

    Proof. (a) Let f ∈ L(D) r {0} then div(f) + D ≥ 0 then 0 ≤ deg(div(f)) +deg(D) = deg(D) ≤ 0 so we get a contradiction. Thus L(D) = {0}.

    (b) D′ = D + div(f) then the map L(D′) −→ L(D) such that g ↦→ g · f is anisomorphism of K-vector spaces.

    Theorem 1.31. (Riemann-Roch) Let C be a nonsingular projective curveand let κ be a canonical divisor on C. Then for all divisors D ∈ Div(D)

    ℓ(D)− ℓ(κ−D) = deg(D)− g + 1

    Proof. See [Har77] IV.1

    Corollary 1.32. Let κ be a canonical divisor on C and let D be a divisor onC, then:

    (a) deg(κ) = 2g − 2

    (b) If deg(D) > 2g − 2 then ℓ(D) = deg(D)− g + 1

    Proof. (a) Apply Riemann-Roch theorem with D = κ, so that

    g − 1 = ℓ(κ)− ℓ(0) = deg(κ)− g + 1

    thus deg(κ) = 2g − 2.

  • 1.3. RIEMANN SURFACES 16

    (b) By (a), deg(κ−D) < 0, hence by Proposition 1.30 ℓ(κ−D) = 0, and thenthe result follows by Riemann-Roch.

    Proposition 1.33. Let C/K0 be a nonsingular projective curve and let D be adivisor defined over K0. Then the K-vector space L(D) has a basis defined overK0 i.e. consisting of elements of K0(C).

    Proof. See [Sil09] II.5.8

    Theorem 1.34. (Hurwitz) Let F : C1 −→ C2 be a nonconstant separablemap between nonsingular projective curves. Let g1 = g(C1) and g2 = g(C2).Then

    2g1 − 2 ≥ (deg(F ))(2g2 − 2) +∑P∈C1

    (eF (P )− 1)

    Moreover, we have equality if and only if either char(K) = 0 or char(K) = p > 0and p does not divide eF (P ) for all P ∈ C1.

    Proof. See [Sil09] II.5.9

    1.3 Riemann Surfaces

    From now untill the end of the chapter, suppose that K = C. Then affineand projective spaces come with the complex topology, in addition to the Zariskitopology. Then one can naturally give the points of a variety over C a topol-ogy inherited from the subspace topology. A little extra work (with the inversefunction theorem and other analytic arguments) shows you that, if the variety isnonsingular, you have a nonsingular complex manifold. Nonetheless, in generalthe converse is false: there are many complex manifolds that do not arise fromnonsingular algebraic varieties in this manner.However, in dimension 1, a miracle happens, and the converse is true: all com-pact one dimensional complex manifolds (i.e. compact Riemann Surfaces) areanalytically isomorphic to the complex points of a nonsingular projective onedimensional variety (i.e. Curves), endowed with the complex topology insteadof the Zariski topology. (See [Gun66] Chapter 10)

    It follows that all the terminology and results we developed about curves overan arbitrary algebraically closed field hold for compact Riemann surfaces. Inparticular the genus g of a compact Riemann surface has the costumary topo-logical meaning and one can show that the two definition coincide.

    Let X be a Riemann surface of genus g ≥ 1, since X is a closed orientablesurface of genus g, the ordinary homology group H1(X,Z) is free abelian on2g generators. Let α1, .., αg, β1, .., βg be the standard basis over Z i.e. with theproperty that

    αiβj = δij and αiαj = 0 = βiβj ∀i ̸= j

    Now consider the map∫: H1(X,Z) −→ Ω1(X)∨

    def= HomC(Ω

    1(X),C) given by

    [γ] ↦→[ω ↦→

    ∫γ

    ω

    ](1.3.1)

  • 1.3. RIEMANN SURFACES 17

    Since∫

    : H1(X,Z) −→ Ω1(X)∨ is injective, we can view the elements of thefirst homology groups as mapping holomorphic differentials on X to C, moreexplicitely:

    Proposition 1.35. Let X be a compact Riemann surface of genus g ≥ 1 andlet ω1, .., ωg be a basis over C for Ω1(X) (which has dimension g since it isisomorphic to L(κ) for a canonical divisor κ). Then the 2g vectors

    λ1 =

    ⎛⎜⎝∫α1ω1...∫

    α1ωg

    ⎞⎟⎠ , . . . , λ2g =⎛⎜⎝∫βgω1...∫

    βgωg

    ⎞⎟⎠ ∈ Cgare linearly independent over R and therefore λ1, .., λ2g are a Z basis for a latticeΛ(X) in Cg.

    Proof. See [Swi74] I.1.7

    In particular since dimC(Ω1(X)) = g, that is also the complex dimension

    of its dual space. It follows that H1(X,Z) free abelian subgroup of rank 2g,generated by 2g indipendent elements over R, hence it is a lattice in Ω1(X)∨.

    Definition. The Jacobian variety of a compact Riemann surface X is

    J(X) = Ω1(X)∨/H1(X,Z)

    which is realized as the g-dimensional complex torus Cg/Λ(X)

    Now fix a point x0 ∈ X and define the Abel-Jacobi map Φ : X −→ J(X) by

    x ↦→(∫ x

    x0ω1, . . . ,

    ∫ xx0ωg)

    Remark 1.36. Since J(X) is a group, we can extend, by Z-linearity, Φ to amap Φ : Div(X) −→ J(X) , D =

    ∑x∈X

    ordx(D) · (x) ↦→∑x∈X

    ordx(D) · (Φ(x))

    For a generic divisor D, this definition depends on the base point x0, howeverfor a 0-degree divisor we have:

    Lemma 1.37. The Abel-Jacobi map Φ : Div(X) −→ J(X) restricted toDiv0(X) is independent of the base point x0.

    Proof. Let y0 be another base point and let γ be a path from x0 to y0. Then inthe formula of Φ(x), if we change x0 to y0, we see that the image changes by

    λ =(∫

    γω1, . . . ,

    ∫γωg)

    mod Λ(X)

    Such element λ ∈ J(X) is indipendent of x, hence if∑nx = 0, then Φ

    (∑nx ·

    (x))changes by

    ∑nx · (λ) = (λ) ·

    ∑nx = 0.

    Theorem 1.38. (Abel) Let X be a compact Riemann surface of genus g ≥ 1and let D be a divisor on X. Then

    D is principal ⇐⇒ deg(D) = 0 and Φ(D) = 0 ∈ J(X)

    Or in other words the map from Pic0(X) to J(X) given by[∑x

    nx · x

    ]↦→∑x

    nx

    ∫ xx0

    is an isomorphism.

  • 1.3. RIEMANN SURFACES 18

    Proof. See [Mir95] VIII.2.2

    Corollary 1.39. Let X be a compact Riemann surfact of genus g ≥ 1. Thenthe Abel-Jacobi map is injective

    Proof. Assume is not, then let Φ(x) = Φ(y) for some x ̸= y ∈ X. Since Φis additive, Φ(x − y) = 0. Therefore by Abel’s theorem, x − y is a principaldivisor, which means that there exists some meromorphic function f on X witha simple pole at y and no other poles. This implies that f is an isomorphismbetween X, of genus greater than 1, and P1(C), which has genus 0, hence acontradiction.

    Proposition 1.40. Let X be a compact Riemann surface of genus 1, then J(X)is isomorphic to X.

    Proof. Since X has genus one, J(X) is a one-dimensional complex torus, henceit is also a compact Riemann surface of genus one. Therefore for X of genusone, Φ is an injective holomorphic map between compact Riemann surfaces,therefore an isomorphism.

  • Chapter 2

    Elliptic Curves

    2.1 Elliptic curves over an arbitrary field

    2.1.1 Weierstrass form & abstract elliptic curves

    Definition. A Weierstrass equation over a field K is a cubic equation (in ho-mogeneous coordinates) of the form

    E : Y 2Z + a1XY Z + a3Y Z2 = X3 + a2X

    2Z + a4XZ2 + a6Z

    3, a1, .., a6 ∈ K0(2.1.1)

    If a1, .., a6 ∈ K0 then we say that E is defined over K0 .

    We identify the Weierstrass equation E with the projective curve in P2 givenby the points that annihilate F (X,Y, Z)

    def= Y 2Z + a1XY Z + a3Y Z

    2 − X3 −a2X

    2Z − a4XZ2 − a6Z3 i.e.

    E = {[X,Y, Z] ∈ P2 | Y 2Z+a1XY Z+a3Y Z2−X3−a2X2Z−a4XZ2−a6Z3 = 0}

    Remark that the only K-rational point on the line at infinity {Z = 0} is O =[0, 1, 0], which is nonsingular since ∂F (X,Y,Z)∂Z (O) = 1 ̸= 0.Thus we can study the curve by working with the non-homogeneous coordinatesx=X/Z, y=Y/Z:

    E : y2 + a1xy + a3y = x3 + a2x

    2 + a4x+ a6 (2.1.2)

    with associated curve

    E = {(x, y) ∈ A2 | y2 + a1xy+ a3y−x3− a2x2− a4x− a6 = 0}∪ {∞ = [0, 1, 0]}

    Let us now simplify the equation in (2.1.2) under some assumption regardingthe characteristic of K.First assume that Char(K) ̸= 2, then the variable change

    y ↦→ (y − a1x− a3)2

    completes the square and yields

    y2 = 4x3 + b2x2 + 2b4x+ b6 (2.1.3)

    19

  • 2.1. ELLIPTIC CURVES OVER AN ARBITRARY FIELD 20

    where b2 = a21 + 4a2, b4 = 2a4 + a1a3 and b6 = a

    23 + 4a6.

    Moreover, if char(K) ̸= 3, then

    (x, y) ↦→(x− 3b2

    36,y

    108

    )produces the equation

    y2 = x3 − 27c4x− 54c6 (2.1.4)

    where c4 = b22 − 24b4 and c6 = −b32 + 36b2b4 − 216b6

    Definition. Let b8 = a21a6 + 4a2a6 − a1a3a4 + a2a23 − a24, then define the dis-

    criminant ∆ of the curve given by the equation in (2.1.2) by

    ∆ = ∆(E) = −b22b8 − 8b34 − 27b26 + 9b2b4b6 (2.1.5)

    If ∆ ̸= 0 it makes sense to define the so-called j-invariant of E

    j = jE = c34/∆ (2.1.6)

    Lastly, define the invariant differential associated to E.

    ω =dx

    2y + a1x+ a3=

    dy

    3x2 + 2a2x+ a4 − a1y(2.1.7)

    Remark that for char(K) ̸= 2, 3 we have

    ∆ =c34 − c261728

    (2.1.8)

    Definition. An admissible change of variable in a Weierstrass equation is oneof the form

    X = u2X ′ + r ; Y = u3Y ′ + su2X ′ + t ; Z = Z ′ for u, r, s, t ∈ K, u ̸= 0(2.1.9)

    This change of variable fixes the point at infinity [0, 1, 0] and carries the line{Z = 0} to itself. Moreover one can compute all the new coefficients and seethat

    u12∆′ = ∆ ; j′ = j and ω′ = uω (2.1.10)

    Proposition 2.1. The curve E given by a Weierstrass equation is nonsingularif and only if ∆ ̸= 0

    Proof. Let E = {(x, y) ∈ A2 | f(x, y) = y2+a1xy+a3y−x3−a2x2−a4x−a6 =0} ∪ {∞}.We already saw that the point at infinity is never singular.Now suppose that E is singular at a some point P = (x0, y0) and consider theadmissable change of variable given by x′ = x−x0 and y′ = y− y0 which leaves∆ invariant. Therefore we can assume that E is singular at the point (0, 0) ∈ A2.Singularity means that

    0 = f(0, 0) = −a6 ; 0 =∂f(x, y)

    ∂x(0, 0) = −a4 ; 0 =

    ∂f(x, y)

    ∂y(0, 0) = a3

  • 2.1. ELLIPTIC CURVES OVER AN ARBITRARY FIELD 21

    It follows that f(x, y) = y2 + a1xy − a2x2 − x3 and consequently

    b2 = a21 + 4a2 and b4 = b6 = b8 = 0 =⇒ ∆ = −(a2 + 4a2)2 · 0 + 0 = 0

    where ∆ is calculated using the formula in (2.1.5).For the other implication assume that char(K) ̸= 2 to simplify the computation.As in (2.1.3),

    f(x, y) = y2 − 4x3 − b2x2 + 2b4x+ b6Then E is singular if and only if there is a point (x0, y0) such that y

    20 = 4x

    30 −

    b2x20 + 2b4x0 + b6 and

    0 =∂f

    ∂x(x0, y0) = 12x

    20 + 2b2x0 + 2b4 ; 0 =

    ∂f

    ∂y(x0, y0) = 2y0

    The last equality forces y0 = 0 and thus (x0, 0) is singular if and only if it is adouble root of the polynomial q(x) = 4x3 − b2x2 + 2b4x + b6 if and only if thediscriminant of q(x) is 0. Recall that for such polynomial its discriminant is

    ∆q = b22 · (2b4)2 − 16(2b4)3 − 4b32b6 − 27 · 16b26 + 18 · 4b2(2b4)b6 =

    = 4(b22b24 − 32b34 − b32b6 − 108b26 + 36b2b4b6) =

    (∗)= 4(−4b22b8 − 32b34 − 108b26 + 36b2b4b6 = 16∆(E)

    where the equality in (∗) follows from the fact that b24 − b2b6 = −4b8. Thiscompletes the proof for char(K) ̸= 2. The case of characteristic 2 is treated in[Sil09] A.1.2

    Proposition 2.2. Let E be a singular curve given by a Weierstrass equation.Then E is birational to P1.

    Proof. As in the proof of Proposition 2.1, after an admissible linear change ofvariables we assume that E is singular at the point (0, 0) and thus E = {(x, y) ∈A2 | y2 + a1xy = x3 + a2x2}.Then the rational maps

    E →→ P1 P1 →→ E

    (x, y) ✤ →→ [x, y] [x, y] ✤ →→(

    y2+a1xy−a2xx2 ,

    y3+a1xy2−a2x2y

    x3

    )are mutually inverse.

    Definition. An elliptic curve is a pair (E,O), where E us a nonsingular curveof genus 1 and O belongs to E. The elliptic curve is defined over K0 if E isdefined over K0 and O ∈ E(K0).

    Proposition 2.3. Let (E,O) be an elliptic curve defined over K0. Then thereexist f, g ∈ K0(E) such that the morphism

    F : E −→ P2, F = [f, g, 1]

    induces an isomorphism of E to a nonsingular curve given by a Weierstrassequation

    Y 2Z + a1XY Z + a3Y Z2 = X3 + a2X

    2Z + a4XZ2 + a6Z

    3

    for a1, .., a6 ∈ K0 such that F (0) = [0, 1, 0].

  • 2.1. ELLIPTIC CURVES OVER AN ARBITRARY FIELD 22

    Proof. • Claim 1: There exists f, g ∈ K0(E) that satisfy a Weierstrass equa-tion.

    By Riemann-Roch theorem we have that, since g = g(E) = 1, the dimensionof the linear system L(n(O)) is n for any n ≥ 1. By Proposition 1.33 we canchoose a basis of L(n(O)) defined over K0. Therefore let f ∈ K0(E) be such that{1, f} is a basis over K0 for L(2(O)) and then choose any g so that {1, f, g} isa basis over K0 for L(3(O)). Notice that f must have a pole of order exactly 2at O and g a pole of order exactly 3 at O. Therefore we have that

    Space L(2(0)) L(3(0)) L(4(0)) L(5(0)) L(6(0))Dimension 2 3 4 5 6Generators {1, f} {1, f, g} {1, f, g, f2} {1, f, g, f2, fg} {1, f, g, f2, fg, f3, g2}

    Since seven functions are in L(6(O)), they must be linearly dependent. In par-ticular the coefficients of f3 and g2 cannot be 0, otherwise we would have a poleof order 6 which doesn’t cancel. After scaling coefficients we will have a linearrelation

    f3 + a2f2 + a4f + a6 = g

    2 + a1gf + a3g

    Therefore we have a map F : E −→ P2 given by F = [f, g, 1] whose image is inthe curve {X3 + a2X2Z + a4XZ2 + a6Z3 = Y 2Z + a1XY Z + a3Y Z2} =: C.By Theorem 1.13 and Theorem 1.16 F is a surjective morphism. Moreover re-mark that since f (respectively g) has a pole of order 2 (resp. 3) at O, locallywe have f = 1z2 (1 + ...) (resp. g =

    1z3 (1 + ...)), therefore F (O) = [f, g, 1]|O =

    [z3f, z3g, z3]|O = [0, 1, 0].

    • Claim 2: F has degree 1, so that it is an isomorphism by Corollary 1.19.

    In fact by Proposition 1.20 the morphisms [f, 1] : E −→ P1 and [g, 1] : E −→ P1have degree 2 and 3 respectively. Hence [K0(E) : K0(f)] = 2 and [K0(E) :K0(g)] = 3, and this implies that [K0(E) : K0(f, g)] divides 2 and 3 therefore[K0(E) : K0(f, g)] = 1.

    • Claim 3: C is a nonsingular curve.

    Suppose not, then by Proposition 2.2 there is a degree 1 rational map h : C −→P1 and therefore h◦F is a degree 1 morphism between nonsingular curves, hencean isomorphism by Corollary 1.19. This is a contradiction since E has genus 1and P1 has genus 0 but the genus of a curve is a topological invariant. We canconclude that C is nonsingular and F is an isomorphism.

    Also the converse holds true:

    Proposition 2.4. If a curve E is given by a nonsingular Weierstrass equation(2.1.2) over K and O is taken as the usual point at infinity, then (E,O) is anelliptic curve.

    Proof. To see this we only need to compute the genus of E. One first provesthat for the invariant differential ω, div(ω) is a canonical divisor of degree 0(See [Sil09] III.1.5). Then Riemann-Roch theorem with D = div(ω) yields

    g = dimL(div(ω)) = deg(div(ω)) + dimL(0)− g + 1 = 0 + 1− g + 1

  • 2.1. ELLIPTIC CURVES OVER AN ARBITRARY FIELD 23

    Hence g = −g + 2 ⇒ g = 1.

    Proposition 2.5. Let (E,O) and (E′, O′) be two elliptic curves defined overK0 given by Weierstrass equations over K0 with O and O

    ′ corresponding to∞. If there exists an isomorphism F : E −→ E′ defines over K0 such thatF (∞) = ∞, then E and E′ are related by an admissible change of variable(2.1.9) with coefficients in K0.

    Proof. Suppose that E, respectively E′, corresponds to the set of coordinates(f, g), resp. (x, y), as in Proposition 2.3. Then {1, f} and {1, x} are bases ofL(2(O)) and L(2(O′)) respectively, while {1, f, g} and {1, x, y} are bases ofL(3(O)) and L(3(O′)). Since F is an isomorphism of curves, it induces an iso-morphism between function fields. Hence f ′

    def= F ∗(x) = x ◦ F ∈ L(2(O)) and

    g′ = F ∗(y) ∈ L(3(O)). In other words, there exist u1, u2, r, s1, t ∈ K0, u1, u2 ̸= 0such that

    f ′ = u1f + r and g′ = u2g + s1f + t

    Since (f, g), (f ′, g′) satisfy the Weierstrass equation in E, we obtain that u31 =u22. Letting u = u2/u1 and s = s1/u

    2 gives a change of variable as in (2.1.9).

    Let us now introduce a group operation on the elliptic curve.

    Definition. Let (E,O) be an elliptic curve, regarded as a nonsingular curve inP2. Let P,Q ∈ E and consider a line ℓ through these two points (we understandthat the line is tangent to the curve at P if P = Q). By Bezout’s theorem thereis a third point R in ℓ ∩E, thus let ℓ′ be the line through R and O. Define thethird point in ℓ′ ∩ E as P +Q.

    Proposition 2.6. This operation makes E into an abelian group with O asidentity element. Moreover the inclusion E(K0) ⊂ E(K) is a group homomor-phism.

    Proof. The core of the proof is to show associativity (See[Kna93] Theorem3.8). The fact that the operation is commutative and that O is the identityelement follows immediately from the above definition. Since by definition andassociativity, three points P,Q,R on a line sum to O means that Q+R = −Pi.e. we have an additive inverse.

    Fact 2.7. One can prove that for an elliptic curve over K0 given by a Weierstrassequation over K0, addition and negative are morphisms defined over K0.

    Proof. See [Sil09] III.3.6

    We can now reformulate Proposition 1.40 and obtain a description of theaddition on E by mean of divisors.

    Theorem 2.8. Let (E,O) be an elliptic curve over K, then the map φ :Div(E) −→ E given by

    ∑nP (P ) ↦→

    ∑[nP ]P induces an isomorphism

    Pic0(E) = Div0(E)/Divl(E)∼−→ E

    Therefore the principal divisors of E are characterized by∑nP (P ) ∈ Divl(E) ⇐⇒

    ∑nP = 0 and

    ∑[nP ]P = O (2.1.11)

  • 2.1. ELLIPTIC CURVES OVER AN ARBITRARY FIELD 24

    2.1.2 Isogenies

    From now on we write E for an elliptic curve, and drop the reference to thegroup identity, which will be denoted O and for Weierstrass equations will bethe point at ∞.

    Definition. Let E and E′ be elliptic curves over K. An isogeny between themis a non-constant morphism F : E −→ E′ such that F (O) = O.

    Notice that by Theorem 1.16 all isogenies are surjective morphisms.

    Proposition 2.9. Let F : E −→ E′ be a morphism such that F (0) = 0, thenF is a group homomorphism.

    Proof. If F = 0 there is nothing to prove. Assume F ̸= 0, then F is a finitemorphism and by Remark 1.28 we have a homomorphism F∗ : Div(E) −→Div(E′) which induces a homomorphism, denoted again by F∗, Pic

    0(E) −→Pic0(E′). Therefore by Theorem 2.10, we can write F as the composition ofthree homomorphisms

    F : E ∼= Pic0(E) F∗−→ Pic0(E′) ∼= E′

    Example 2.10. Let E be an elliptic curve then

    1. The multiplication by an integer, denoted by [m], is an isogeny from E toE.

    2. If E is defined over Z/pZ then the Frobenius map is an isogeny.

    Let E and E′ be elliptic curves over K, denoted by Hom(E,E′) the set ofall isogenies from E to E′.

    Remark 2.11. Hom(E,E′) is an abelian group with pointwise sum.

    Proof. Let F,G ∈ Hom(E,E′), then F +G is the composition of

    E −→ E × E such that P ↦→ (P, P )

    F ×G : E × E −→ E′ × E′ such that (P,Q) ↦→ (F (P ), G(Q))+ : E′ × E′ −→ E′ such that (P,Q) ↦→ P +Q

    Moreover, if E = E′, Hom(E,E) = End(E) is a ring, with multiplicationgiven by composition of morphisms.For P ∈ E define a translation map TP : E −→ E by Q ↦→ Q+ P . Then T ∗P isan automorphism of K(E).

    Proposition 2.12. Let F : E −→ E′ be a nonzero isogeny between ellipticcurves over K. Then ker(F ) is finite and the map P ↦→ T ∗P induces an isomor-phism

    ker(F ) ∼= Aut(K(E)/F ∗(K(E′))

    )Furthermore, if F is separable then F is unramified, deg(F ) = #ker(F ) andK(E)/F ∗(K(E′)) is a Galois extension.

  • 2.1. ELLIPTIC CURVES OVER AN ARBITRARY FIELD 25

    Proof. Since F is non constant, it surjects E to E′ and it is a finite morphism,thus is particular has finite fibers, hence finite kernel.Now if P ∈ ker(F ) and f ∈ F ∗(K(E′)) then f = g ◦ F for some g ∈ K(E′).Hence T ∗P (g ◦ F ) = g ◦ F ◦ TP = g ◦ F = f since F ◦ TP = F ∀P ∈ ker(F ).Therefore T ∗P fixes F

    ∗(K(E′)) for any P ∈ ker(F ). Moreover

    T ∗P ◦ T ∗Q(f) = f ◦ TQ ◦ TP = f ◦ TP+Q = T ∗P+Q(f) = T ∗Q+P (f)

    Hence P ↦→ T ∗P is a homomorphism.By Corollary 1.20 (b), #F−1(Q) = degs(F ) for all but finitely many Q ∈ E′.Now if F is an isogeny, the equality holds for all Q ∈ E′, in fact ∀Q1, Q2 ∈ E′there exists (by surjectivity of F ) R ∈ E such that F (R) = Q1−Q2 and since Fis a homomorphism, the map TR : F

    −1(Q1) −→ F−1(Q2) sending P ∈ F−1(Q1)to P + R ∈ F−1(Q2) is a bijection. It follows that #F−1(Q) = degs(F ) for allQ ∈ E′. In particular #ker(F ) = #F−1(O) = degs(F ) and by Galois theory

    #Aut(K(E)/F ∗(K(E′))

    )≤ degs(F ) = #ker(F )

    It follows that it is enough to prove that the map P ↦→ T ∗P is injective. But thisis obvious since if T ∗P = T

    ∗Q then TP = TQ and therefore

    ∀ R ∈ E, R+ P = TP (R) = TQ(R) = Q+R

    Thus in particular P = O + P = O +Q = Q.Finally assume that F is separable, then

    #F−1(Q) = degs(F ) = deg(F ) ∀Q ∈ E′

    hence F is unramified by Corollary 1.21 and setting Q = O produces

    #ker(F ) = #F−1(O) = deg(F )

    By the isomorphism P ↦→ T ∗P it follows

    [K(E) : F ∗(K(E′))] = deg(F ) = #ker(F ) = #Aut(K(E)/F ∗(K(E′))

    )

    Lemma 2.13. 1− φq is a separable isogeny for any q = pr, r ≥ 1.

    Proof. Assume not, then 1− φq factors as

    1− φrp = 1− φq = f ◦ φsp

    for some f : E −→ E separable morphism and s ≥ 1. Hence

    1 = f ◦ φsp + φrq =

    {(f ◦ φs−rp + 1) ◦ φr−1p ◦ φp if s ≥ r(f + φr−sp ) ◦ φs−1p ◦ φp if r > s

    which contradicts the fact that 1 is an isomorphism, while the right hand sidecannot be since φp is an inseparable map of degree p.

  • 2.1. ELLIPTIC CURVES OVER AN ARBITRARY FIELD 26

    Remark 2.14. Let E be an elliptic curve defined over Fp, then Proposition2.12 gives us a way of computing #E(Fp). We saw that the Frobenius map φfixes exactly those points of E that are in E(Fp), or in other words

    Q ∈ E(Fp) ⇐⇒ φp(Q) = Q ⇐⇒ ([1]−φp)(Q) = O ⇐⇒ Q ∈ ker([1]−φp)

    hence #E(Fp) = #ker([1]− φp).Since 1− φp is a separable isogeny, by Proposition 2.12

    #E(Fp) = #ker([1]− φp) = deg([1]− φp) (2.1.12)

    Lemma 2.15. Let F : E1 −→ E2 and G : E1 −→ E3 be nonzero isogeniesbetween elliptic curves over K. Let F be separable and such that ker(F ) ⊆ker(G). Then there exists a unique isogeny H : E2 −→ E3 such that G = H ◦F .Moreover, if F and G are both defined over K0, so is H.

    Proof. By Proposition 2.12 we have

    Aut(K(E1)/F

    ∗(K(E2)))∼= ker(F ) ⊂ ker(G) ∼= Aut

    (K(E1)/G

    ∗(K(E3)))

    Hence is σ fixes F ∗(K(E2)), then it also fixes G∗(K(E3)). Moreover since F

    is separable K(E1)/F∗(K(E2)) is a Galois extension, therefore G

    ∗(K(E3)) ⊂F ∗(K(E2)) is an inclusion of fields. Theorem 1.17 yields a nonconstant morphismH : E2 −→ E3 such that F ∗ ◦H∗ = G∗. By uniqueness follows that G = H ◦F ,and this implies that H is an isogeny since O = G(O) = H(F (O)) = H(O) asrequired.If F and G are defined over K0 we can apply Theorem 1.17 to G

    ∗(K0(E3)) ⊂F ∗(K0(E2)) to produce H defined over K0 aswell.

    Theorem 2.16. Let F : E −→ E′ be a nonzero isogeny of degree m, betweenelliptic curves over K. Then there exists a unique isogeny F̂ : E′ −→ E suchthat F̂ ◦ F = [m]. Such F̂ is called dual isogeny of F .

    Proof. Factor F as F = Fs ◦ φq for Fs a separable morphism and the q-thFrobenius map φq. Then it is enough to prove existence of dual isogeny for each

    factor. In fact, assume there exist F̂s and φ̂q, then

    (φ̂q ◦ F̂s) ◦ (Fs ◦ φq) = φ̂q ◦ [degs(F )] ◦ φq = [degs(F )] ◦ φ̂q ◦ φq = [m]

    Hence φ̂q ◦ F̂s = (F̂s ◦ φ1).

    • Let Fs be a separable morphism of degree m, then #ker(Fs) = m andtherefore [m] : E −→ E annhilates ker(F ). It follows that ker(F ) ⊂ker([m]) and we found ourselves in the assumption of Lemma 2.15 withFs : E −→ E′ and [m] : E −→ E. Therefore there exists a uniqueH : E′ −→ E such that [m] = H ◦ Fs. Thus H = F̂s.

    • Let φq be the q-th Frobenius map for q = pr, then φq = φrp. Thus itit enough to prove existence of φ̂p. Recall that deg(φp) = p, and sincechar(K) = p, multiplication by p is not separable1. Since [p] is not sepa-

    rable, it has a factorization [p] = Gs ◦ φhp , thus take φ̂p = Gs ◦ φh−1p .1 It follows from Proposition 1.29 (c), since [p]∗ : ω ↦→ 0, where ω is the invariant differential

  • 2.2. ELLIPTIC CURVES OVER C 27

    Finally let us prove uniqueness: let F̂ , F̂ ′ be two dual isogeny of F , then

    (F̂ − F̂ ′) ◦ F = [m]− [m] = 0

    But by assumption F is nonzero, therefore F̂ = F̂ ′.

    Definition. We say that E is isogenous to E′ if there exists a non zero isogenyfrom E to E′. The Theorem implies that ”being isogenous” is an equivalencerelation.

    The next result is a sort of converse of Proposition 2.12

    Theorem 2.17. Let E be an elliptic curve over K and let S be a finite subgroupof E. Then there exist an elliptic curve E′ unique up to isomorphism and aseparable isogeny F : E −→ E′ such that ker(F ) = S. Moreover if E is definedover K0 and S is stable under Aut(K/K0), then E

    ′ is also defined over K0. Suchelliptic curve is often denoted E/S.

    Proof. We give a proof in the case char(K) = 0, for a proof in positive charac-teristic see [DS06 Section 7.8].To any P ∈ S associate the automorphism T ∗P : f ↦→ f ◦ TP , ∀f ∈ K(E). SinceT ∗P+P ′ = T

    ∗P ◦ T ∗P ′ , S can be identified with a finite subgroup of Aut(K(E)),

    namelyS ∼= {T ∗P | P ∈ S}

    Let K(E)S be the subfield of K(E) fixed by the elements in S, then K(E)/K(E)S

    is a Galois extension with Galois group Gal(K(E)/K(E)S) ∼= S. Moreover,K(E)S is a function field of dimension one over K, thus there exist a uniquenonsingular projective curve E′ over K and a K-isomorphism F ∗ : K(E′)

    ∼−→K(E)S ↪→ K(E). The fields-curves correspondence in Theorem 1.17 yields asurjective morphism F : E −→ E′.We are left to show that such F is unramified and the genus of E′ is 1. For anyX ∈ E, let Y = F (X) ∈ E′, then ∀P ∈ S, ∀f ∈ K(E)S we have

    f(X + P ) = f ◦ TP (X) = T ∗P f(X) = f(X) =⇒ F (X + P ) = F (X) = Y

    =⇒ X + S = {X + P | P ∈ S} ⊆ F−1(Y )

    On the other hand we have

    #F−1(Y ) ≤∑

    X∈F−1(Y )

    eX(F ) = deg(F ) = [K(E) : F∗(K(E′)] = [K(E) : K(E)S ] = #S

    where the last equality follows by Galois theory. Hence we can conclude thatF−1(Y ) = X + S and therefore F is unramified and deg(F ) = #S.Finally Theorem 1.34 in characteristic 0 yields that the genus of E′ is 1.

    2.2 Elliptic curves over CThe theory of elliptic curves over C becomes much easier, since it is equiva-

    lent (in a categorial sense) to the theory of lattices in C or, equivalently again,to the one of complex tori.

  • 2.2. ELLIPTIC CURVES OVER C 28

    Definition. (i) A lattice Λ in C is a subgroup of (C,+) of the form Λ =Zw1 ⊕ Zw2 such that Λ⊗Z R = Rw1 +Rw2 = C i.e. {w1, w2} is a R-basisfor C.

    (ii) An ordered basis

    (w1w2

    )of a lattice Λ is normalized if w1/w2 ∈ H

    where H := {z ∈ C | Im(z) > 0}.

    (iii) Two lattices Λ and Λ′ are homothetic if there exists α ∈ C× such thatΛ′ = αΛ.

    Remark 2.18. (a) (i) is equivalent to saying that w1, w2 ∈ C× and w1/w2 ∈Cr R.

    (b) Λ is a lattice if and only if Λ is a free rank 2 discrete subgroup of (C,+).

    Let us denote by L the set of all lattices and by B the set of normalizedbases.

    Then we have a surjective map π : B → L given by(w1w2

    )↦→ Zw1 + Zw2.

    Lemma 2.19. The fibers of π are all of the form SL2(Z) · b for b ∈ B. Hence πinduces a bijection π : SL2(Z)\B −→ L

    Proof. Notice that b =

    (w1w2

    )and b′ =

    (w′1w′2

    )are the same basis for a lattice Λ

    if and only if there exists a matrix γ =

    (a bc d

    )∈ GL2(Z) such that(

    w′1w′2

    )=

    (a bc d

    )(w1w2

    )=

    (aw1 + bw2cw1 + dw2

    )Setting τ = w1/w2 and τ

    ′ = w′1/w′2 gives that τ, τ

    ′ belong to C r R and τ ′ =aτ+bcτ+d . Noticing that Im(τ

    ′) = det(γ)|cτ+d|2 Im(τ) we deduce that b, b′ are normalized

    basis if and only if det(γ) > 0, therefore γ ∈ SL2(Z).

    Notice that there is an action of C× both on L and B. Namely, let α ∈ C×,

    Λ ∈ L and(w1w2

    )∈ B, then αΛ ∈ L and

    (αw1αw2

    )∈ B.

    In particular we have an identification B/C× −→ H given by[w1w2

    ]↦→ w1/w2.

    The actions of C× and SL2(Z) on B commute: therefore the action of SL2(Z)on B induces an action of SL2(Z) on H by fractional linear transformation

    SL2(Z)× H −→ H((a bc d

    ), τ)↦→ aτ + b

    cτ + d

    (2.2.1)

    Corollary 2.20. The bijection π induces a natural bijection

    π̃ : SL2(Z)\H −→ L/C×

    In particular any lattice Λ is homothetic to a lattice of the form Λτ = Zτ+Zfor some τ ∈ H.

  • 2.2. ELLIPTIC CURVES OVER C 29

    Remark 2.21. We will see that the set L/C× = {[Λ] = homothety class of the lattice Λ}is in bijection with the set of isomorphic classes of complex tori {[C/Λ] =isomorphism class of C/Λ} where isomorphism means holomorphic isomorphism.Moreover, the set of isomorphism classes of 1-dimensional complex tori is in bi-jection with the set of isomorphism classes of elliptic curves over C (with respectto algebraic isomorphism).

    2.2.1 The Weierstrass ℘-function

    Definition. Let Λ = Zw1 + Zw2 be a lattice in C. A meromorphic functionf : C −→ C is an elliptic function of period Λ if f(z +w) = f(w) for all w ∈ Λ.This amount to saying that f(z + wi) = f(z) for i = 1, 2.

    Definition. Let Λ = Zw1 + Zw2 be a lattice in C, the function ℘ : C −→ P1given by

    ℘Λ(z) = ℘(z) =1

    z2+∑w∈Λw ̸=0

    ( 1(z − w)2

    − 1w2

    )is called Weierstrass ℘-function for Λ.

    Proposition 2.22. Let Λ = Zw1 +Zw2 be a lattice in C, and ℘ be the Weier-strass ℘-function for Λ, then:

    (a) ℘(z) is an even function whose singularities are double poles at lattice points;

    (b) ℘ is an elliptic function of period Λ;

    (c) deg(℘) = 2;

    (d) ℘ is surjective.

    Proof. (a) One first proves that∑

    w∈Λw ̸=0

    (1

    (z−w)2 −1w2

    )converges absolutely and

    uniformely on compact subsets of Cr Λ.Then to see that it has a double pole at lattice points, notice that thefunction hv(z) = ℘(z)− 1(z−v)2 for some v ∈ Λ converges uniformely on (CrΛ)∪{v}. Hence in a neighbourhood of v, ℘(z) = 1(z−v)2 +holomorphic part.Finally absolute and uniform convergence on compact subsets allow us toreorder the summands and therefore since Λ = −Λ

    ℘(−z) = 1(−z)2

    +∑w∈Λw ̸=0

    ( 1(−z − w)2

    − 1w2

    )=

    1

    z2+∑w∈Λw ̸=0

    ( 1(z − (−w)2

    − 1(−w)2

    )= ℘(z)

    (b) Since ℘ is meromorphic on C, its derivative ℘′ is also meromorphic. Moreover℘′(z) = − 2z3 +

    ∑w∈Λw ̸=0

    (−2

    (z−w)3

    )= −2

    ∑w∈Λw ̸=0

    (1

    (z−w)3

    ), hence ℘′(z + w) =

    ℘′(z) for all w ∈ Λ since w + Λ = Λ. Therefore ℘′ is an elliptic function,hence ℘(z +wi) = ℘(z) + ci for some ci ∈ C. Taking z = −wi/2 yields that℘(wi2 ) = ℘(−

    wi2 ) + ci. But by (a), ℘ is even, thus ci = 0 for both i = 1, 2.

    We conclude that ℘ is an elliptic function.

    (c) Follows from Proposition 1.20 together with the fact that ℘ has poles oforder 2 at any lattice point.

  • 2.2. ELLIPTIC CURVES OVER C 30

    (d) Since ℘ is an elliptic function of period Λ, it factors through the complecttorus C/Λ, which is compact. Now ℘ is non constant therefore it must surjectto P1(C).

    Lemma 2.23. ℘′ has degree 3 and it has three distinct zeroes in the funda-

    mental domain∏ def

    = {t1w1 + t2w2 ∈ C | 0 ≤ t1, t2 < 1}, at w12 ,w22 and

    w32 ,

    where w3 = w1 + w2, each of multiplicity one.

    Proof. ℘′ has only poles of order 3 in w ∈ Λ, thus deg(℘′) = 3. Since ℘ is even,℘′ is odd, and we saw that ℘′ is elliptic with period Λ. Therefore

    ℘′(wi2

    )= −℘′

    (− wi

    2

    )= −℘′

    (− wi

    2+ wi

    )= −℘′

    (wi2

    )Hence ℘′

    (wi2

    )= 0 for i = 1, 2, 3 and as deg(℘′) = 3 these must be all the zeroes

    of ℘′ in∏

    and therefore have multiplicity one.

    Definition. For n ≥ 3, we call Eisenstein series of weight n the quantity

    En(Λ) =∑w∈Λw ̸=0

    1

    wn

    Theorem 2.24. Define g2 = g2(Λ) = 60E4(Λ) and g3 = g3(Λ) = 140E6(Λ),then for all z ∈ C:

    (℘′(z))2 = 4℘(z)3 − g2℘(z)− g3 (2.2.2)

    Proof. Recall the equality 1 + t+ t2 + .. = 11−t for |t| < 1. Deriving we obtain

    1

    (1− t)2= 1 + 2t+ 3t2 + ... =

    ∞∑k=0

    (k + 1)tk

    Hence for z near 0 we have

    1

    (z − w)2− 1w2

    =1

    w2

    ( 1(z/w − 1)2

    −1)=

    1

    w2(1+2(z/w)+...−1

    )=

    ∞∑k=1

    k + 1

    wk+2zk

    Therefore we can write ℘(z) as

    ℘(z) =1

    z2+

    ∞∑k=1

    (k+1)(∑

    w∈Λw ̸=0

    1

    wk+2

    )zk =

    1

    z2+

    ∞∑k=1

    Ek+2(Λ)zk =1

    z2+3E4(Λ)z2+5E6(Λ)z4+..

    and

    ℘′(z) = − 2z3

    + 6E4(Λ)z + 20E6(Λ)z3 + .. =⇒ (℘′(z))2 =4

    z6− 24E4(Λ)

    1

    z2+ ...

    Therefore setting h(z) = (℘′(z))2 − 4℘(z)3 + g2℘(z) + g3 we see that h has nopoles in 0 ∈

    ∏since all terms with zn for n < 0 cancel out, and h(0) = 0. As

    ℘ and ℘′ are holomorphic in∏

    r{0}, h is holomorphic in∏

    and Λ-periodic.Therefore h is constant and h(0) = 0 implies that h ≡ 0 as we wanted toshow.

  • 2.2. ELLIPTIC CURVES OVER C 31

    Definition. For a lattice Λ ⊂ C, define its discriminant as

    ∆(Λ) = g2(Λ)3 − 27g3(Λ)2

    Remark 2.25. Recall that for a cubic polynomial of the form P (x) = 4x3 −g2x− g3 its discriminant is ∆P = g32 − 27g23 = 16(e1 − e2)2(e1 − e3)2(e2 − e3)2where e1, e2, e3 are the roots of P (x). It follows that for any lattice Λ in C,

    ∆P = ∆(Λ) ̸= 0

    Proof. This is equivalent to prove that the roots of the polynomial are distinct.Since ℘ is even, ℘′ is odd. Now let v ∈ {w12 ,

    w22 ,

    w32 } where Λ = Zw1 +Zw2 and

    w3 = w1+w2; then by Lemma 2.23 ℘′(v) = 0 and they are simple zeroes. Hence

    by the functional equation

    0 = ℘′(v)2 = 4℘(v)3 − g2℘(v)− g3

    Therefore ℘(wi2 ) is a root of P (x). We are reduced to show that ℘(wi) ̸= ℘(wj)for i ̸= j so that we have 3 distinct roots and nonzero discriminant.Consider the function ℘(z + wi2 ) − ℘(

    wi2 ) which is even since ℘ is and it is 0

    at z = 0. Therefore it has a zero of order at least 2 at z = 0, but since it hasdegree 2, it has order exactly 2 at z = 0 and no other zeroes. Thus in particularfor z =

    wi+wj2 with i ̸= j, we obtain 0 ̸= ℘(

    wj2 )− ℘(

    wi2 ) and this concludes the

    proof.

    Therefore E = {[x, y, z] ∈ P2 | y2z = 4x3− g2(Λ)xz2− g3(Λ)z3} is a nonsin-gular projective curve for any lattice Λ.Moreover it makes sense to define the quantity

    j(Λ) = 1728g2(Λ)

    3

    ∆(Λ)

    for any lattice Λ in C, called j-invariant.

    Theorem 2.26. Let π : C −→ T = C/Λ be the canonical surjection on thequotient. Consider Φ : T −→ P2 given by

    t = π(z) ↦→

    {[℘(z), ℘′(z), 1] if z /∈ Λ[0, 1, 0] if z ∈ Λ

    Then Φ is holomorphic (as a map of complex manifolds) with image

    Φ(T ) = Edef= {[x, y, z] ∈ P2 | y2z = 4x3 − g2xz2 − g3z3}

    Moreover, Φ is injective and has maximal rank (one) on T . Thus since T iscompact, Φ : T −→ E is a biholomorphism, hence T ∼= E.

    Proof. Φ is holomorphic on T r {0}: in fact let t ∈ T , t ̸= 0, then Φ(t) ∈ U2 ={[x0, x1, x2] ∈ P2 | x2 ̸= 0} ⊂ P2. We must check that for V ⊆ T r {0}, the mapF := ϕ2 ◦ Φ ◦ (π−1V )−1 is holomorphic on Ṽ = π

    −1V ⊆ C, where ϕ2[x0, x1, x2] ↦→(

    x0x2, x1x2

    )∈ C2. We have

    F (z) = ϕ2(Φ(πV (z))) = ϕ2([℘(z), ℘′(z), 1]) = (℘(z), ℘′(z))

  • 2.2. ELLIPTIC CURVES OVER C 32

    which are holomorphic in T r {0}.Moreover we can immidiatly see that Φ has maximal rank in T r {0}: in factthis is the case if and only if for all z ∈ Cr Λ,

    JC(F ) =

    (℘′(z)℘′′(z)

    )̸=(00

    )Suppose not, then ℘′(z) = 0, thus z = wi2 for i = 1, 2, 3, but we saw that theseare simple zeroes, therefore ℘′′

    (wi2

    )̸= 0.

    Let us now show that Φ is holomorphic at t = 0: we have Φ(0) = [0, 1, 0] ∈ U1.Let Ṽ be a neighbourhood of 0 ∈ C and let z ̸= 0 in Ṽ , then

    F0(z) = (ϕ1 ◦ Φ ◦ π)(z) =( ℘(z)℘′(z)

    ,1

    ℘′(z)

    )(We can divide by ℘′(z) since we are near 0). Recall that

    ℘(z) =1

    z2(1 + a1z + ...) ℘

    ′(z) = − 2z3

    (1 + b1z + ..)

    =⇒ F0(z) =(− z

    2(1 + c1z + ...),−

    z3

    2(1 + d1z + ..)

    )Therefore F0(z) is holomorphic at 0.Using the same expasion near 0 we see that

    JC(F0) =

    ((℘/℘′

    )′(z)(

    1/℘′)′(z)

    )|z=0

    =

    (− 12 + z(...)− 32z

    2 + z(...)

    )|z=0

    =

    (− 120

    )̸=(00

    )

    So we can conclude that Φ has maximal rank everywhere on T .To see that Φ(T ) = E , let P = [x, y, z] ∈ E: if z = 0 then x = 0, thusP = [0, 1, 0] = Φ(0) ∈ Φ(T ). Thus assume z ̸= 0 and write P = [x, y, 1], since ℘is surjective, let x̃ ∈ C be such that x = ℘(x̃). As P ∈ E we get

    y2 = 4x3 − g2x− g3 = 4℘(x̃)3 − g2℘(x̃)− g3 = (℘′(x̃))2

    Hence y = ℘′(x̃) or y = −℘′(x̃). For the former case we have P = Φ(π(x̃)), forthe latter notice that y = −℘′(x̃) = ℘′(−x̃) and x = ℘(x̃) = ℘(−x̃), thereforeP = Φ(π(−x̃)).The other inclusion Φ(T ) ⊂ E is trivial by Theorem 2.27.We are left to prove that Φ is injective: let z1, z2 ∈

    ∏be such that Φ(π(z1)) =

    Φ(π(z2)). If z1 = 0 then Φ(π(0)) = [0, 1, 0] = Φ(π(z2)), which yields that z2 = 0.If z1, z2 ̸= 0, then {

    ℘(z1) = ℘(z2) (i)

    ℘′(z1) = ℘′(z2) (ii)

    (i) implies that either z1 = z2 ( and we are done) or z1 = −z2 + w for somew ∈ Λ. Then

    ℘′(z2) = ℘′(z1) = ℘

    ′(−z2 + w) = −℘′(z2)

    hence ℘′(z2) = 0 so that z2 =wi2 and z1 =

    wj2 . Since ℘(z1) = ℘(z2), i = j and

    z1 = z2 as we wanted.

  • 2.2. ELLIPTIC CURVES OVER C 33

    Notice that the cubic equation y2 = 4x3 − g2x− g3 is in the form of (2.1.3)therefore E defined by such cubic is an elliptic curve over C.The Theorem shows how to associate to a lattice Λ in C an elliptic curve andimplements this association by the biholomorphism Φ. However also the converseis true, for any elliptic curve E there exists a lattice Λ such that E is realizedas C/Λ.

    2.2.2 Isogenies over CLet E and E′ be elliptic curves isomorphic to C/Λ = T and C/Λ′ = T ′

    respectively, for Λ,Λ′ lattices in C. Then every isogeny of E into E′ correspondsto a holomorphic homomorphism of T onto T ′, and viceversa.

    Proposition 2.27. With the same notation as above, we have that

    Hom(E,E′) ∼= Hom(T, T ′) = {α ∈ C | αΛ ⊂ Λ′}

    Proof. Assume that f is non-constant and consider the following diagram:

    C

    π1

    ↓↓

    C

    π2↓↓

    Tf →→ T ′

    Now by the Hurwitz theorem, f has no ramification points, therefore it is acovering of T ′ (as well as π2) and π1 is a covering of T . Since we have twouniversal covering of T ′ given by π2 and f ◦ π1, there exists F : C −→ Cisomorphism of covering such that π2 ◦ F = f ◦ π1, and in particular F iscontinuous. Fixing a point O = π1(0) ∈ T , we have that f(O) = O ∈ T ′and ∀b ∈ π−12 (O) there exists a unique F isomorphism of coverings such thatF (0) = b. In particular since we require that F is a group homomorphism, thereis a unique F so that F (0) = 0.Let us now show that F (z) = αz for some α ∈ C such that αΛ ⊂ Λ′: the factthat F passes to quotients means that for all w ∈ Λ there exists w′ ∈ Λ′ suchthat F (z+w) = F (z)+w′. Now notice that w′ = F (z+w)−F (z) is indipendentfrom z since the map F (∗ + w) − F (∗) is continuous, C is connected and Λ′ isdiscrete. Hence taking derivatives yields the equality F ′(z + w) = F ′(z) , or inother words F ′ is an elliptic function of period Λ. Therefore it defines a mapg : T −→ C holomorphic such that F ′ = g◦π1. On the other hand T is compact,hence g is constant, and so F ′ is : say F ′(z) = α ̸= 0 ∀z ∈ C and consequentlyF (z) = αz since F (0) = 0.In particular for any w ∈ Λ, F (z +w) = F (z) +w′ implies aw = w′ ∈ Λ′ henceαΛ ⊂ Λ′.If f is constant, then F is constant, and in particular it is 0 (since F (0) = 0).Viceversa, let T = C/Λ, T ′ = C/Λ′ be complex tori and F (z) = αz with αΛ ⊂Λ′. Then there exists f : T −→ T ′ holomorphic homomorphism, in fact such Fpasses to the quotient since π2 ◦ F (Λ) = π2(αΛ) = 0 = π1(Λ). Therefore thereexists f continuous such that f ◦ π1 = π2 ◦ F . Finally f is holomorphic sincelocally f|U = π2|

    V ′◦ F ◦ π−11|V is composition of holomorphic maps.

    In particular we have that

    End(E) ∼= {α ∈ C | αΛ ⊂ Λ}

  • 2.2. ELLIPTIC CURVES OVER C 34

    and this always contains Z, since multiplication by integers maps Λ into Λ.

    Definition. We say that an elliptic curve E has complex multiplication ifEnd(E) % Z.

    Proposition 2.28. Let Λ = Zw1 + Zw2 and Λ′ = Zw′1 + Zw′2 be normalizedlattices in C. Then E ∼= T = C/Λ and E′ ∼= T ′ = C/Λ′ are isogenous (resp.isomorphic) if and only if there exists an element γ ∈ GL+2 (Q) ∩M2(Z) withGL+2 (Q) = {γ ∈ GL2(Q) | det(γ) > 0} (resp. SL2(Z)) such that γ · τ ′ = τ ,where τ = w1/w2, τ

    ′ = w′1/w′2 and the action of γ on H is by fractional linear

    transformation i.e.

    (a bc d

    )· τ ′ = aτ

    ′+bcτ ′+d

    Proof. If 0 ̸= α ∈ Hom(E,E′) then αΛ ⊂ Λ′, therefore there exist a, b, c, d ∈ Zsuch that

    αw1 = aw′1 + bw

    ′2

    αw2 = cw′1 + dw

    ′2

    Thus we obtain γ =

    (a bc d

    )∈M2(Z) ∩GL2(Q) such that τ = γ · τ ′. Moreover

    det(γ) > 0 since both bases are normalized.

    Conversely, if γ · τ ′ = τ for γ =(a bc d

    )∈ M2 ∩ GL+2 (Q) then for λ = cτ ′ + d

    we have

    λ

    (τ1

    )=

    (a bc d

    )(τ ′

    1

    )=⇒ (λw′2/w2)

    (w1w2

    )=

    (a bc d

    )(w′1w′2

    )Hence αΛ ⊂ Λ′ for α = λw

    ′2

    w2. In particular αΛ = Λ′ if and only if γ ∈ SL2(Z).

    Proposition 2.29. Let Λ = Zw1 + Zw2 with τ = w1/w2 ∈ H. Then C/Λhas complex multiplication if and only if there exists a nonscalar element ξ ∈GL+2 (Q) such that ξ · τ = τ .

    Proof. Let α ∈ End(E), then since αΛ ⊂ Λ, there exist a, b, c, d ∈ Z such that{ατ = aτ + b

    α = cτ + d=⇒ α

    (τ1

    )=

    (a bc d

    )(τ1

    )=: ξ

    (τ1

    )for ξ ∈M2(Z)∩GL+2 (Q)

    (2.2.3)Now notice that α is an eigenvalue for ξ for how ξ is defined, on the other handα is an eigenvalue for ξ since

    α

    (τ1

    )= ξ

    (τ1

    )thus, in a compact way, we have

    (τ τ1 1

    )(α 00 α

    )= ξ

    (τ τ1 1

    )Therefore α ∈ Z ⇐⇒ b = 0 = c which yields a = d ⇐⇒ ξ is scalar.

    Proposition 2.30. Let Λ = Zw1 + Zw2 with τ = w1/w2 ∈ H. Then C/Λ hascomplex multiplication if and only if Q(τ) is a quadratic imaginary field andEnd(E) is an order in Q(τ).

  • 2.2. ELLIPTIC CURVES OVER C 35

    Proof. C/Λ has complex multiplication if and only if α as in Proposition 2.32is not in Z hence it is not real (since τ ∈ H) and ξ is not scalar. Thereforeeliminating τ from (2.2.2) we obtain the quadratic equation

    α2 − (a+ d)α+ ad− bc = 0 (∗)

    Since α = cτ + d, it follows that Q(τ) = Q(α) is a quadratic imaginary field.Moreover, (∗) tells us that α is integral over Z, thus contained in the ring ofintegers OQ(τ). Therefore End(E) ⊂ OQ(τ) and it is free of rank 2 because OQ(τ)is torsion free and End(E) contains 1 and an imaginary element: we concludethat End(E) is an order in Q(τ).

    From now on suppose that the elliptic curve E ∼= T = C/Λ has complexmultiplication by the maximal order OK , for a quadratic imaginary field Kand that Λ is of the form Zτ + Z. Under these assumptions, the conditionEnd(E) = OK is equivalent to saying that OKΛ ⊂ Λ. Therefore Λ is a fractionalOK-ideal.

    Fact 2.31. If I is a fractional OK-ideal then there exists λ ∈ K× such thatλI ⊂ OK and therefore it is an OK-ideal.

    In particular, we have m ∈ Z>0 such that mΛ ⊂ OK and consequently mΛis a nonzero ideal of OK .Conversely, if a is a fractional OK-ideal, then the torus C/a satisfies End(C/a) =OK . In fact OKa ⊂ a, thus End(C/a) ⊃ OK and equality follows by maximalityof OK in K.Recall that the class group of K is

    Pic(OK) =(multiplicative group of fractional ideals)

    (subgroup of principal fractional ideals)=

    =(multiplicative semigroup of nonzero ideals of OK)

    ∼where I1 ∼ I2 if and only if there exist α1, α2 ∈ OK nonzero, such that α1I1 =α2I2.Therefore it follows that:

    Proposition 2.32. For a complex torus T , denote by [T ] its isomorphism classin the category of complex tori. After fixing one of the two possible embeddingsι : OK −→ C, the map

    Pic(OK) −→ {[T ] | End(T ) = OK} defined by [ι(a)] ↦→ [C/ι(a)]

    is a bijection.

    Corollary 2.33. {[T ] | End(T ) = OK} is finite since Pic(OK) is.

    2.2.3 Automorphisms of an elliptic curve