Bielliptic Surfaces and their Geometry Master’s thesis in Engineering mathematics and computational science XUDONG LIU DEPARTMENT OF MATHEMATICAL SCIENCES CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2024 www.chalmers.se www.chalmers.se Master’s thesis 2024 Bielliptic Surfaces and their Geometry XUDONG LIU Department of Mathematical Sciences Division of algebraic geometry and number theory Chalmers University of Technology Gothenburg, Sweden 2024 Bielliptic Surfaces and their Geometry XUDONG LIU © XUDONG LIU, 2024. Supervisor: Dennis Eriksson, Department of Mathematical Sciences Examiner: Dennis Eriksson, Department of Mathematical Sciences Master’s Thesis 2024 Department of Mathematical Sciences Division of algebraic geometry and number theory Chalmers University of Technology SE-412 96 Gothenburg Telephone +46 31 772 1000 Typeset in LATEX, template by Kyriaki Antoniadou-Plytaria Printed by Chalmers Reproservice Gothenburg, Sweden 2024 iv Bielliptic Surfaces and their Geometry XUDONG LIU Department of Mathematical Sciences Chalmers University of Technology Abstract Geometry is a high concern in modern mathematics. One way to begin the study is by handling a nice example. The bielliptic surfaces can play such a role. It is constructed using elliptic curves, some nice curves in some way equivalent to a torus. The prerequisites of bielliptic surfaces involve algebraic geometry and elliptic curves. The final result is about the intersection of bielliptic surfaces, so the intersection theories of surfaces will also be introduced. Works of classification and works of Néron-Severi lattices are crucial for the study of bielliptic surfaces in the last section. Algebraic geometry focuses on the method of solving geometry problems in alge- braic ways. The fundamental of the study is abstract algebra. It studies curves, surfaces, and some other higher-dimension objects like hyperspaces. The key point is describing geometry structures by zeros of polynomials. Many results are derived over the complex field, where many nice properties can be found. The elliptic curve is a kind of algebraic curve of genus one. Weierstrass equations are the algebraic forms of elliptic curves. The composition law defines an operation on the elliptic curves. Another important property is that the lattices over the complex field determine the elliptic curves, which can be derived from the construction of the Weierstrass ℘-function. Isogenies are introduced as the maps between elliptic curves. The topic of intersection theory on surfaces concerns the intersection number of two curves on the given surface, which is the number of intersection points counted with algebraic multiplicity. The definition of intersection number can be generalized to n varieties in high dimensions. In the article, the situation of two curves on a surface is enough. One important result is Bézout’s theorem, a theorem of the intersection number of plane curves. The definition of bielliptic surfaces is based on the elliptic curves. With all the knowledge before, the final result about the intersection number of bielliptic sur- faces can be given. Keywords: elliptic curve, bielliptic surfaces, intersection theory, intersection number. v Acknowledgments In the period I was in Goteborg, my family gave me lots of financial support for living and studying. My teacher Dennis Eriksson, who was introduced by Minchen Xia, helped me many times when I had difficulties in understanding mathematics and writing the article. As both the administrator of the master’s program and my functional analysis teacher, Håkan Andreasson also benefited me. Xudong Liu, Gothenburg, March 2024 vii Nomenclature Below is the nomenclatures that have been used throughout this thesis. K Algebraic closed field C Algebraic curve S Algebraic surface E,F Elliptic curve Λ Lattice in C ix x Contents Nomenclature ix List of Tables xiii 1 Introduction 1 1.1 Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Intersection Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Bielliptic Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Algebraic Geometry Foundations 5 2.1 Algebraic Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Maps between curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Elliptic Curves 11 3.1 General Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1.1 Weierstrass Equations . . . . . . . . . . . . . . . . . . . . . . 11 3.1.2 Group Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Elliptic Curves over C . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.1 Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.2 Weierstrass ℘-Function . . . . . . . . . . . . . . . . . . . . . . 16 3.3 Maps of Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3.1 Isogeny . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3.2 The Dual Isogeny . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3.3 Maps on C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4 Intersection Theory of Surfaces 21 4.1 Intersection Number . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Self-Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.3 Bézout’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5 Bielliptic Surfaces 27 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.2 Néron–Severi Group . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.3 Intersection Theory of Bielliptic Surfaces . . . . . . . . . . . . . . . . 30 xi Contents Bibliography 33 xii List of Tables 5.1 Classification of bielliptic curves . . . . . . . . . . . . . . . . . . . . . 28 5.2 Classification of Néron–Severi group . . . . . . . . . . . . . . . . . . . 29 5.3 Classification of Néron-Severi lattice . . . . . . . . . . . . . . . . . . . 30 xiii List of Tables xiv 1 Introduction This section is about the outline of the article. There are some interpretations, but only a few details are given. 1.1 Algebraic Geometry As a part of modern mathematics, algebraic geometry is the method of studying geometry by abstract algebra. It is the foundation of studying bielliptic surfaces. Although this branch of mathematics is wide and complex, the knowledge needed in the thesis is just some simple facts. The simplest situation is probably a line in space C2 given by the linear equation aX + bY + c = 0. A slightly different example is the line in space C3 a1X + b1Y + c1Z + d1 = 0 a2X + b2Y + c2Z + d2 = 0. The equations are not unique, but the ideal generated by the two equations is. Similarly, a geometry structure V denoted by some equations f1 = 0, f2 = 0, ..., fn = 0 can be also denoted by the zeros of the unique ideal I = (f1, f2, ..., fn). Conversely, for any ideal I in C[X1, X2, ..., Xn], the relation f(P ) = 0, P ∈ Cn,∀f ∈ I also denotes a unique geometry structure V . Thus some geometry structures could be described by algebraic expressions, which makes commutative algebra tools use- ful in geometry study. In this way, the zero set V of such an ideal I is called variety. An important space used in the thesis is projective space Pn. It is based on the equivalence relation of Kn \ {(0, 0, ..., 0)} (x1, x2, ..., xn) ∼ (kx1, kx2, ..., kxn), k ∈ K∗. In the projective space, the geometry structure can be described by homogeneous polynomials. Homogeneous polynomials are f with all nonzero terms having the 1 1. Introduction same degree, like X3 + X2Y + Y 3. Some problems could be simplified once it is embedded in the projective space. With these foundations, the geometry concepts can be better defined by algebraic expressions, like curves C,D, surface S, intersection multiplicity mP (C,D) of C,D at point P , Néron–Severi group NS(S) of some surface S. The details of definitions are omitted in the introduction, but the concepts are used to give some basic expla- nations. Following, the elliptic curves and intersection theory are introduced as the necessary knowledge of bielliptic surfaces. 1.2 Elliptic Curves Any elliptic curve over C could be described by a Weierstrass equation in C[X, Y, Z], or a variety E in the projective space P3. Let x = X/Z, y = Y/Z, the equation is of form E : y2 = 4x3 − 27c4x− 54c6. Composition law, or the additional group law of E, is defined by points P,Q,R of a line L intersecting E. That is, P,Q,R ∈ L ∩ E. Take O = [0 : 1 : 0] as the zero element given by the Weierstrass equation, by the composition law there is P ⊕Q⊕R = O. A lattice Λ with bases ω1, ω2 ∈ C is defined as Λ := {k1ω1 + k2ω2 : k1, k2 ∈ Z}. Over C/Λ, an interesting result is constructing E with Weierstrass ℘-Function. It is a periodic function of the form ℘(z; Λ) := 1 z2 + ∑ 0̸=w∈Λ ( 1 (z − w)2 − 1 w2 ). With the help of ℘(z; Λ), there exists an isomorphism ϕ keeping the group structure to the elliptic curve E(C) ϕ : C/Λ → E(C) ⊂ P2(C), z 7→ [℘(z), ℘′(z), 1]. One important result, which gives more details about the relationships between C/Λ and elliptic curve E, is that these categories are equivalent: 1. Objects: Elliptic curves E over C. Maps: Isogenies. 2 1. Introduction 2. Objects: Elliptic curves E over C. Maps: Complex analytic maps taking O to O. 3. Objects: Lattices Λ ⊂ C, up to homothety. Maps: Map(Λ1,Λ2) = {α ∈ C : αΛ1 ⊂ Λ2}. 1.3 Intersection Theory To illustrate the intersection number, the concept of intersection multiplicitymP (C,D) of curve C,D at point P is necessary. There is an example. If two curves C,D are given by Y = X2, X = 0, then m(0,0)(Y −X2, X) = dimOA2,(0,0)/(Y −X2, X) = 2 with the fact of degX2 = 2. This case shows how to calculate the multiplicity algebraically. For two curves C,D, the number of intersection points #(C ∩ D) (counted with multiplicity) is an intuitive problem, denoted as C · D. However, the intersection theory can be generalized to the intersection of some high-dimension geometry ob- jects V1 · V2 · ... · Vn. The situation of bielliptic surfaces is simple: two curves C,D on a surface S. The intersection number can be defined as C ·D = ∑ P∈C∩D mP (C,D). One important and interesting result is Bézout’s theorem, a theorem about how the degrees define the global intersection number. For distinct plane curve C,D ⊂ P2, it is of form C ·D = degC · degD. An example is the fundamental theorem of algebra: any nontrivial polynomial f(x) has d = deg f roots counting with multiplicity. The roots of f(x) correspond to the intersection point of f(X, Y ) = Y df(x) = 0, x = X/Y and Y = 0. The degrees of two polynomials are deg(f(X, Y )) = d, deg Y = 1. The result is also true for homogeneous polynomials. In this way, the fundamental theorem of algebra is a special case of Bézout’s theorem. There is also a general version of Bézout’s theorem. For n distinct hypersurfaces H1, H2, ..., Hn defined as zeros of homogeneous polynomials f1, f2, ..., fn, if the in- tersection number is finite, there is H1 ·H2 · ... ·Hn = deg f1 · deg f2 · ... · deg fn. 3 1. Introduction 1.4 Bielliptic Surfaces Let E,F be two elliptic curves, G acts on E,F as automorphism, the bielliptic surface S has form S ∼= E × F/G. G, a fixed point free action, acts on E,F in the following way: for some injective homomorphism a : G → Aut(F ), g ∈ G acting on (x, y) ∈ (E,F ) is g(x, y) = (x+ g, a(g)y). Another concept here is Néron–Severi group NS(S) for the bielliptic surfaces S, an equivalence relation of intersection theory. Denote [C] for the class of curve C, then for any C ′ ∈ [C] and any other curve D, C ·D = C ′ ·D. There is an important fact: any curve C in S can be factored into a[E]+b[F ], where a, b are coefficients, and [E], [F ] are the equivalent classes of E,F . The result of the intersection theory in the last section is [E]2 = [F ]2 = 0, [E] · [F ] = #G. Factor the curves C1, C2, then there is [C1] · [C2] = (a1[E] + b1[F ]) · (a2[E] + b2[F ]) = (a1b2 + a2b1)#G. The coefficients a, b are similar to the degrees, so the result works similarly with Bézout’s theorem. 4 2 Algebraic Geometry Foundations In this section, some backgrounds of algebraic geometry will be established. It concentrates on describing geometry structure by algebraic terms, especially poly- nomials. An example is y = x2 related to a parabola in the R2 coordinate. These foundations are the bases of elliptic curves and bielliptic surfaces. An algebraic variety (Definition 2.1.1,2.1.4) describes geometry objects in the form of polynomial rings and ideals. Curve and surface (Definition 2.1.8) are defined in this way as a 1-dimensional and 2-dimensional projective variety. Another important topic is the degree of map (Definition 2.3.1). Some properties of degrees are introduced, as the necessary tools used in the last section. For a curve C, a divisor (Definition 2.2.1) D = ∑ P∈C nP (P ), where all but finite nP are zeros, is the formal additional group of points P with coefficients nP , in the thesis are mainly zeros and poles with multiplicities for a rational function. The genus of a curve, a parameter that can be used to classify, can be calculated by the Riemann-Roch theorem (Theorem 2.4.4) in the form of divisors. 2.1 Algebraic Variety In affine space, point sets are associated with ideals. It is similar in projective space, the difference being that polynomials mentioned in projective space are ho- mogeneous. Definition 2.1.1. The affine n-space over K is the set An = An(K) := {P = (x1, x2, ..., xn) : xk ∈ K}. For the ideal I in the polynomial ring K[X] = K[X1, X2, ..., Xn], the associated set is VI := {P ∈ An : f(P ) = 0,∀f ∈ I}. V is an algebraic set if it has the form VI . The ideal of V is I(V ) := {f ∈ K[X] : f(P ) = 0,∀P ∈ V }. 5 2. Algebraic Geometry Foundations Definition 2.1.2. The projective n-space over K, denoted by Pn or Pn(K), is (An+1 \ {(0, 0, ..., 0)})/ ∼, where the equivalence relation ∼: (x0, x1, ..., xn) ∼ (kx0, kx1, ..., kxn), k ∈ K. Each equivalent class x is denoted by [x0 : x1 : ... : xn]. Definition 2.1.3. f ∈ K[X] is a homogeneous polynomial if f(kx0, kx1, ..., kxn) = kn+1f(x0, x1, ..., xn),∀k ∈ K. An ideal in K[X] is homogeneous if it can be generated by homogeneous polynomials. Definition 2.1.4. For the ideal homogeneous I in the polynomial ring K[X] = K[X1, X2, ..., Xn], the associated set is VI := {P ∈ Pn : f(P ) = 0,∀f(X) ∈ I, f(X) is homogeneous} V is a projective algebraic set if it has form VI . The homogeneous ideal of V is I(V ) := {f(X) ∈ K[X] : f(P ) = 0,∀P ∈ V, f(X) is homogeneous}. The ideal I ⊂ K[X] gives a natural equivalence relation f ∼ g : f − g ∈ I for the polynomial ring, where comes the coordinate ring. Definition 2.1.5. A (projective) algebraic set V is called a (projective) variety if its (homogeneous) ideal I(V ) is a prime ideal. The coordinate ring K[V ] is defined by K[V ] := K[X]/I(V ). The rational ring K(V ) is generated by polynomials f and g−1, where f ∈ K[X], g ∈ K[X] \K[V ], h1 ∼ h2 ∈ K(V ) if and only if h1 − h2 ∈ I(V ). A morphism between varieties is a regular map, as the definition below. Definition 2.1.6. Between two projective varieties V1 → V2, a map ϕ = [f1 : f2 : ..., fn], fi ∈ K(V1) is regular at P ∈ V1 if there exists g ∈ K(V1), s.t. 1. gfi is well defined for all i. 2. (gfi)(P ) ̸= 0 for some i. ϕ is a morphism if it is regular everywhere, i.e. regular at any P ∈ V1. The algebraic curve and surface are defined after the dimension, defined in the view of field extension, of variety. And the nonsingularity is also defined. Definition 2.1.7. The dimension of an affine variety V , denoted by dim(V ), is the transcendence degree of K(V )/K. For a projective variety V ⊂ Pn, there exists An ⊂ Pn, s.t. V ∩ An ̸= ϕ. The dimension of V is defined in An as dim(V ) := dim(V ∩ An). Definition 2.1.8. A curve C (or surface S) is a projective variety of dimension 1 (or 2). 6 2. Algebraic Geometry Foundations Definition 2.1.9. For a variety V ∋ P , where the ideal I(V ) generated by f1, f2, ..., fm ∈ K[X], V is nonsingular (or smooth) at P if the rank is n − dim V for the m × n matrix ( fi Xj (P )). V is nonsingular (or smooth) if it is smooth at every point P . 2.2 Divisors In this section, C is always a smooth curve over K. Divisor is the formal sum of points on the curve. The points here are often the zeros and poles with coefficients as the multiplicities. Definition 2.2.1. The divisor group of a curve C is defined as Div(C) := {D = ∑ P∈C nP (P )}. It is an abelian group generated by points P ∈ C, where nP ∈ Z and nP = 0 for all but finitely many P . The degree of D is degD := ∑ P∈C nP . Div0(C) is the degree-0 subgroup of Div(C) Div0(C) := {D ∈ Div(C) : degD = 0}. Let ordP (f) be the order of zeros of f at P . For 0 ̸= f ∈ K(C)∗, where C is a smooth curve, define div(f) := ∑ P∈C ordP (f)(P ). Definition 2.2.2. A divisor D is principal if it has form D = div(f) for some f ∈ K(C). The divisors D1, D2 are linearly equivalent when D1 − D2 is principal with the notation D1 ∼ D2. The Picard group Pic(C) of a curve C is the quotient of Div(C) by its subgroup of principal divisors. Proposition 2.2.3. Let f ∈ K(C)∗. 1. div(f) = 0 if and only if f ∈ K. 2. deg div(f) = 0. Proof. See [1, p.28, Proposition 3.1]. The canonical divisor (Definition 2.2.4), used in the Riemann-Roch theorem, is defined by differential form. The proposition following the definition ensures the definition is proper. Definition 2.2.4. The space of (meromorphic) differential forms of a curve C, denoted by ΩC, is the K(C)-vector space spanned by form dx, x ∈ K(C) satisfying 1. d(x+ y) = dx+ dy,∀x, y ∈ K(C). 7 2. Algebraic Geometry Foundations 2. d(xy) = ydx+ xdy,∀x, y ∈ K(C). 3. dα = 0,∀α ∈ K. The canonical divisor class KC := div(ω) is the class of ω ∈ ΩC in Pic(C), where div(ω) := ∑ P∈C ordP (ω)(P ) ∈ Div(C) which is well defined by Proposition 2.2.5. Proposition 2.2.5. For all ω1, ω2 ∈ ΩC, there exists an f ∈ K(C), s.t. ω1 = fω2, div(ω1) = div(f) + div(ω2). Proof. See [1, p.32, Remark 4.4]. 2.3 Maps between curves ϕ : C1 → C2 is always a map of curves C1, C2 over K in this part. For a nonconstant rational map ϕ, its composition induces an injection ϕ∗ : K(C2) → K(C1), ϕ∗f = f ◦ ϕ. The following contents are about the degree of ϕ, as important facts used in the last section. Definition 2.3.1. For map ϕ : C1 → C2, the degree deg ϕ = 0 when ϕ is constant, otherwise deg(ϕ) := dim[K(C1) : ϕ∗K(C2)]. ϕ is separable, inseparable, or purely inseparable if the extension K(C1) : ϕ∗K(C2) has the corresponding property. The notation is degs(ϕ) for the separable degree. Particularly, degs(ϕ) = deg(ϕ) when K = C. Proposition 2.3.2. Let ϕ : C1 → C2 be a nonconstant map of smooth curves, for all but finite Q ∈ C2, #ϕ−1(Q) = degs(ϕ). Particularly, #ϕ−1(Q) = deg(ϕ) when K = C, since degs(ϕ) = deg(ϕ). Proof. See [1, p.23, Proposition 2.6]. Proposition 2.3.3. For nonconstant map ϕ : C1 → C2 of smooth curves, (deg ϕ∗D) = (deg ϕ)(degD), where D ∈ Div(C2). Proof. See [1, p.29, Proposition 3.6]. 8 2. Algebraic Geometry Foundations 2.4 Genus The Riemann-Roch theorem gives the genus in the form of l(D) = dimL(D), where L(D) is a linear function space, where D is a divisor. Definition 2.4.1. A divisor D = nP (P ) is positive if nP ≥ 0, P ∈ C, denoted by D ≥ 0. The notation D1 ≥ D2 means that D1 −D2 is positive. Definition 2.4.2. For D ∈ Div(C), L(D) := {f ∈ K(C) : div(f) +D ≥ 0} ∪ {0}. L(D) is a finite-dimensional K-vector space as [2, p.122, Theorem 5.19], its dimen- sion l(D) := dimK L(D). Proposition 2.4.3. If degD < 0, D ∈ Div(C), then L(D) = {0}, l(D) = 0. Proof. Let 0 ̸= f ∈ L(D), then there is a contradiction by Proposition 2.2.3 degD = deg(D + div(f)) − deg div(f) ≥ − deg div(f) = 0. Theorem 2.4.4. (Riemann–Roch theorem) For the canonical divisor KC on a smooth curve C, the genus g ∈ N of C satisfies l(D) − l(KC −D) = degD − g + 1,∀D ∈ Div(C). Proof. See [2, p.295, Theorem 1.3]. Corollary 2.4.5. By the Riemann–Roch theorem, 1. l(KC) = g. 2. degKC = 2g − 2. 3. If degD > 2g − 2, then l(D) = degD − g + 1. Proof. 1. Let D = 0, then L(0) = K, l(0) = 1. 2. Let D = KC . 3. deg(KC −D) = degKC −degD < 0, thus by Proposition 2.4.3, l(KC −D) = 0. 9 2. Algebraic Geometry Foundations 10 3 Elliptic Curves This section will focus on elliptic curves, which are denoted E. A fact is any E over C is related to a torus, by the isomorphism between E and a lattice C/Λ. The knowledge of elliptic curves is an important basis for the last section since two elliptic curves E,F can construct a bielliptic surface S. Elliptic curves (Definition 3.1.1) are non-singular curves of genus 1 with a point O. Another description of E is the Weierstrass equation. There is also composition law (Definition 3.1.3), or "geometry group law", on E defined by intersections of E and some lines. This section will focus on E over a complex field C. One important result, trans- forming geometry structures into algebraic structures, is that each E is related to a lattice C/Λ, where the Weierstrass ℘-function (Definition 3.2.6) connects the two forms of expressions. Therefore an elliptic curve E over C is isomorphic to a torus since the lattice C/Λ has the same topology structure. Isogeny (Definition 3.3.1), or isomorphism up to a finite kernel between two el- liptic curves E1, E2, is another important subject. Each isogeny ϕ has its dual ϕ̂ (Definition 3.3.4). For E over C, the isogeny is also a homomorphism of lattices C/Λ1,C/Λ2. 3.1 General Knowledge 3.1.1 Weierstrass Equations The general definition of elliptic curves is based on the genus, where O in the defi- nition is "0" in the group law shown after. For calculation, each E can be associated to a Weierstrass equation. Definition 3.1.1. An elliptic curve is a pair (E,O), where E is a non-singular curve of genus 1, O ∈ E. Write E/K if E is defined over K as a curve and O ∈ E(K). For any elliptic curve E, there exists a1, ..., a6 ∈ K and base point O = [0, 1, 0], that E has an equation of the form Y 2Z + a1XY Z + a3Y Z 2 = X3 + a2X 2Z + a4XZ 2 + a6Z 3. 11 3. Elliptic Curves Let x = X/Z, y = Y/Z E : y2 + a1xy + a3y = x3 + a2x 2 + a4x+ a6. There are simple forms E : y2 = 4x3 + b2x 2 + 2b4x+ b6, char(K) ̸= 2 E : y2 = 4x3 − 27c4x− 54c6, char(K) ̸= 2, 3 where b2 = a2 1 + 4a2, b4 = 2a4 + a1a3, b6 = a2 3 + 4a6 c4 = b2 2 − 24b4, c6 = −b3 2 + 36b2b4 − 216b6. Proposition 3.1.2. Let E be an elliptic curve defined over K. 1. For a given (E,O), there is an isomorphism ϕ : E → P2, ϕ = [x, y, 1] map E/K onto a curve given by the Weierstrass equation C : y2 + a1xy + a3y = x3 + a2x 2 + a4x+ a6 where x, y ∈ K(E), a1, ..., a6 ∈ K,ϕ(O) = [0 : 1 : 0]. 2. For any two Weierstrass equations of E, there exists a transformation x = u2x′ + r, y = u3y′ + su2x′ + t where r, s, t ∈ K, u ∈ K. 3. Each Weierstrass equation gives E/K with the base point O = [0 : 1 : 0]. Proof. See [1, p.59, Proposition 3.1]. 3.1.2 Group Law E under the composition law, or "geometry group law", forms an Abelian group. The multiplying-by-m map [m] of any point P ∈ E is defined based on this group structure. Definition 3.1.3. Composition Law. The following process gives the group law of P,Q ∈ E: For the line L1 ∋ P,Q (P = Q then L is tangent at P ), there exists a third inter- section point R ∈ L1. The line L2 ∋ R,O intersects E at a third point S ∈ L2 as the sum of P,Q, with the notation P ⊕Q := S Proposition 3.1.4. For the composition law of E: 1. P ⊕O = P . 2. P ⊕Q = Q⊕ P . 3. An inverse ⊖P exists for any P , satisfying P ⊕ (⊖P ) = O. 12 3. Elliptic Curves 4. (P ⊕Q) ⊕R = P ⊕ (Q⊕R). Proof. See [1, p.51, Proposition 2.2]. Notation. ⊕ and ⊖ will be written + and −. For m ∈ Z and P ∈ E, let [0]P = O, [m]P − [m− 1]P = P That is, [m] is the map of multiplying by m. For given E, the composition law relates to divisors. The relation connects the geometry and algebraic structure of E. Lemma 3.1.5. For P,Q ∈ C a curve of genus 1, the divisors (P ) ∼ (Q) if and only if P = Q. Proof. ⇐: Obvious. ⇒: Since C has genus 1, l(Q) = deg(Q) = 1 by the Riemann–Roch theorem. If (P ) ∼ (Q), then for any f satisfies div(f) = (P ) − (Q) that is f ∈ L(Q). f ∈ K since H0(0) ⊂ H0(Q) and both of dimension 1, thus P = Q. Proposition 3.1.6. For a given (E,O) with genus 1, 1. There exists σ mapping D ∈ Div0(E) to σ(D) = P ∈ E σ : Div0(E) 7→ E, s.t.D ∼ (P ) − (O). 2. There exists a bijection κ : E ∼→ Pic0(E), P 7→ divisor class of (P ) − (O) . 3. For the Weierstrass equation of E, σ induces a homomorphism from the com- position law to Pic0(E). Proof. 1. l(D + (O)) = 1 by The Riemann–Roch theorem. Let 0 ̸= f ∈ L(D + (O)), since div(f) ≥ −D − (O) and deg(div(f)) = 0, there exists a P ∈ E, div(f) = −D − (O) + (P ). Thus there is a point P satisfies D ∼ (P ) − (O). 2. The map is injective by Lemma 3.1.5 since κ(P ) = κ(Q) ⇔ (P ) − (O) ∼ (Q) − (O) ⇔ (P ) ∼ (Q) ⇔ P = Q. The map is surjective since σ gives κ−1 by letting D = (P ) − (O) ∈ Div0(C). 13 3. Elliptic Curves 3. It suffices to show that κ(P +Q) = κ(P ) + κ(Q). For (E,O) in P2, let f(X, Y, Z) = αX + βY + γZ = 0. give the line L intersects E at P,Q,R, and let f ′(X, Y, Z) = α′X + β′Y + γ′Z = 0 be the line L′ intersects E at P +Q,R,O. By the fact that the line Z = 0 intersects E at O with multiplicity 3, we have div(f/Z) = (P ) + (Q) + (R) − 3(O) div(f ′/Z) = (R) + (P +Q) − 2(O). Then (P+Q)−(P )−(Q)+(O) = div(f ′/f) ∼ 0, thus κ(P+Q)−κ(P )−κ(Q) = 0. 3.2 Elliptic Curves over C 3.2.1 Elliptic Functions From now on, Λ will always be a lattice, a discrete subgroup of C, in this section. The fundamental parallelogram is a single "unit" of a given lattice Λ. Elliptic functions are periodic functions on a lattice Λ. Definition 3.2.1. A lattice Λ with bases ω1, ω2 ∈ C is defined as Λ := {k1ω1 + k2ω2 : k1, k2 ∈ Z}. A fundamental parallelogram for Λ is a set D := {a+ t1ω1 + t2ω2 : 0 ≤ t1, t2 < 1} where a ∈ C and {ω1, ω2} is a basis for Λ. Note that the natural map D → C/Λ is bijective. Definition 3.2.2. An elliptic function f(z) relative to a lattice Λ is a meromorphic function on C that satisfies f(z + ω) = f(z),∀ω ∈ Λ, z ∈ C. The set of all such f(z) is denoted by C(Λ), which is a field. The following proposition is similar to Liouville’s theorem [4, p.122] in complex analysis. 14 3. Elliptic Curves Proposition 3.2.3. A holomorphic elliptic function, or an elliptic function with no poles, is constant. Similarly, an elliptic function with no zeros is constant. Proof. For any holomorphic elliptic function f(z) on the fundamental parallelogram D̄, there is sup C f(z) = sup D̄ f(z). f(z) is continuous on the compact set D̄, so it is bounded on D̄, even on C. Thus f(z) is a constant by Liouville’s theorem. Notation. The notation ∑ w∈C/Λ denotes a sum over a fundamental parallelogram D for Λ. By implication, w is independent of the choice of D, and only finitely many terms of the sum are nonzero. Divisors for Λ over C are more concrete than the general situation. The order of a function in C(Λ) can be defined after the theorem below showing the number of poles and number of zeros are equal. Theorem 3.2.4. For f ∈ C(Λ), 1. ∑ w∈C/Λ resw(f) = 0. 2. ∑ w∈C/Λ ordw(f) = 0. 3. ∑ w∈C/Λ ordw(f)w ∈ Λ. Proof. See [1, p.162, Theorem 2.2]. Definition 3.2.5. For an elliptic function f(z), the order ∑ w∈D ordw(f) is the order of zeros, which equals the order of poles, of f at w in a fundamental parallelogram D. The divisor group on C/Λ, denoted as Div(C/Λ), consists of the formal sums∑ w∈C/Λ nw(w), nw ∈ Z where nw = 0 for all but finitely many w. For D = nw(w) ∈ Div(C/Λ), define degree of D degD = ∑ nw and 0-degree divisor group Div0(C/Λ) = {D ∈ Div(C/Λ) : degD = 0}. The principal divisor of f ∈ C(Λ) is div(f) = ∑ w∈C/Λ ordw(f)(w) ∈ Div0(C/Λ). Which gives a homomorphism div : C(Λ) → Div0(C/Λ). 15 3. Elliptic Curves 3.2.2 Weierstrass ℘-Function One important fact of the elliptic curves over C is that each of them is associated to an elliptic function. Weierstrass ℘-function connects the two forms. In this part about the Weierstrass ℘-function, the lattice Λ is fixed. Definition 3.2.6. Let Λ ⊂ C be a lattice. The Weierstrass ℘-function for Λ is ℘(z; Λ) := 1 z2 + ∑ 0̸=w∈Λ ( 1 (z − w)2 − 1 w2 ). The following theorem seems obvious. However, the convergence needs to be checked as proven in the reference. Theorem 3.2.7. The Weierstrass ℘-function is an even elliptic function. Proof. See [1, p.165, Theorem 3.1]. The theorem below shows the relation between Weierstrass ℘-function and elliptic functions. That is, elliptic functions can be represented by ℘. The main idea is to construct a function g(z) in the form of ℘ having the same zeros and poles as the elliptic function f(z), then by Proposition 3.2.3 there exists a constant c s.t. g = cf . Theorem 3.2.8. Let Λ ⊂ C be a lattice, every elliptic function is a rational combi- nation of ℘, ℘′, i.e. C(Λ) = C(℘(z), ℘′(z)). Proof. For an elliptic function f(z) ∈ C(Λ) f(z) = 1 2(f(z) + f(−z)) + 1 2(f(z) − f(−z)). It suffices to prove the theorem for functions that are either odd or even. Further, if f(z) is odd, then f(z)℘′(z) is even by Theorem 3.2.7, so we are reduced to the case that f is an even elliptic function. Let D be the fundamental parallelogram {t1ω1 + t2ω2 : t1, t2 ∈ [−1 2 , 1 2 ]} for Λ, and let H be the positive “half” {t1ω1 + t2ω2 : t1 ∈ [0, 1 2 ], t2 ∈ [−1 2 , 1 2 ]} of D. Suppose the divisor of f has the form ∑ w∈H nw((w) + (−w)) (3.1) for certain integers nw ∈ Z. Consider the function g(z) = ∏ w∈H\{0} (℘(z) − ℘(w))nw . The divisor of ℘(z) − ℘(w) is (w) + (−w) − 2(0), so we see that f and g have the same zeros and poles except possibly at w = 0, and by Theorem 3.2.4 they have the same order at 0. Then f(z)/g(z) is a holomorphic elliptic function, thus a constant by Proposition 3.2.3. Therefore there exists a constant c f(z) = cg(z) ∈ C(℘(z), ℘′(z)). 16 3. Elliptic Curves Thus it suffices to show (3.1). The assumption that f is even implies for every w ∈ C ordwf = ord−wf. So 3.1 holds for 2w ∈ D \ Λ. The rest is to show that ordwf is even for 2w ∈ Λ, or w ≡ −w (mod Λ). To see this, we differentiate f(z) = f(−z) repeatedly to obtain f (i)(z) = (−1)if (i)(−z). If 2w ∈ Λ, then f (i)(z) has the same value at w and −w, so f (i)(w) = f (i)(w − 2w) = f (i)(−w) = (−1)if (i)(w). f (i)(w) = 0 for odd values of i, so ordwf is even. Thus 3.1 holds. The rest of this part is the relation between ℘ and E/C. The theorem below shows ℘, ℘′ satisfies the Weierstrass equation. If the defined curve of ℘, ℘′ is non-singular, by Theorem 3.1.2 the relation gives an elliptic curve. This is done at the end of this part. Therefore we have ℘, a bridge connecting C(Λ) and E/C . Theorem 3.2.9. For all z ∈ C/Λ, ℘(z) and its derivative ℘′(z) satisfy the relation ℘′(z)2 = 4℘(z)3 − 60G4℘(z) − 140G6 where for k ∈ Z+, G2k = ∑ w∈Λ/{0} w−2k. Proof. See [1, p.169, Theorem 3.5]. Notation. Standard notations are g2 = g2(Λ) = 60G4(Λ), g3 = g3(Λ) = 140G6(Λ). Then the relation of ℘(z), ℘′(z) is ℘′(z)2 = 4℘(z)3 − g2℘(z) − g3. Proposition 3.2.10. 1. Roots for the polynomial f(x) = 4x3 − g2x− g3 are distinct. Thus the discriminant ∆(Λ) = g3 2 − 27g2 3 is nonzero. 2. Let E/C be the elliptic curve E : y2 = 4x3 − g2x− g3. Then there exists a complex analytic isomorphism ϕ : C/Λ → E(C) ⊂ P2(C), z 7→ [℘(z), ℘′(z), 1]. Proof. See [1, p.170, Proposition 3.6]. 17 3. Elliptic Curves 3.3 Maps of Elliptic Curves 3.3.1 Isogeny A special class of morphisms between E1, E2 is an isogeny ϕ, as the homomorphism of group structures on E1, E2. Definition 3.3.1. An isogeny is a morphism of elliptic curves ϕ : E1 → E2, ϕ(O) = O, and E1, E2 are isogenous if ϕ(E1) ̸= {O}. Theorem 3.3.2. For an isogeny ϕ : E1 → E2, ϕ(P + Q) = ϕ(P ) + ϕ(Q) for all P,Q ∈ E1. Proof. Obvious when ϕ(E1) = {O}. Otherwise, ϕ induces a homomorphism of the divisor group ϕ∗ : Pic0(E1) → Pic0(E2), class of ∑ ni(Pi) 7→ class of ∑ ni(ϕPi). On the other hand, we have group isomorphisms by Theorem 3.1.6 κi : Ei → Pic0(Ei), P 7→ class of (P ) − (O). Then there is the commutative diagram: E1 Pic0(E1) E1 Pic0(E2) ∼= κ1 ϕ ϕ∗ ∼= κ2 κ1, κ2, and ϕ∗ are all group homomorphisms. Thus ϕ is also a homomorphism. By convention, set deg[0] = 0. Thus for all chains of isogenies E1 ϕ→ E2 ψ→ E3, deg(ψ ◦ ϕ) = deg(ψ) deg(ϕ). 3.3.2 The Dual Isogeny The pullback of ϕ, a map of degree 0 Picard groups, ϕ∗ : Pic0(E2) → Pic0(E1) induces a special isogeny ϕ̂, the dual isogeny, as follows. Theorem 3.3.3. Let ϕ : E1 → E2 be a nonconstant isogeny of degree m. 1. There exists a unique isogeny ϕ̂ : E2 → E1, ϕ̂ ◦ ϕ = [m]. 2. As a group homomorphism, ϕ̂ equals E2 → Div0(E2) ϕ∗ −→ Div0(E1) sum→ E1 Q 7→ (Q) − (O) ∑ nP (P ) 7→ ∑ [nP ]P . 18 3. Elliptic Curves Proof. See [1, p.81, Theorem 6.1]. Definition 3.3.4. For an isogeny [0] ̸= ϕ : E1 → E2 be, the dual isogeny is ϕ̂ : E2 → E1 given by Theorem 3.3.3. The theorem below is about the operational laws of dual. Theorem 3.3.5. Let ϕ : E1 → E2 be an isogeny of degree m, λ : E2 → E3, ψ : E1 → E2 are other isogenies 1. ϕ̂ ◦ ϕ = [m] on E1, ϕ ◦ ϕ̂ = [m] on E2. 2. λ̂ ◦ ϕ = ϕ̂ ◦ λ̂. 3. ϕ̂+ ψ = ϕ̂+ ψ̂. 4. [̂d] = [d], deg[d] = d2,∀d ∈ Z. 5. deg ϕ̂ = deg ϕ. 6. ˆ̂ ϕ = ϕ. Proof. See [1, p.83, Theorem 6.2]. 3.3.3 Maps on C Let Λ1 and Λ2 be lattices in C, and suppose that αΛ1 ⊂ Λ2 for some α ∈ C. Then scalar multiplication by α induces a holomorphic homomorphism ϕα : C/Λ1 → C/Λ2, ϕα(z) = αz (mod Λ2). ϕα gives all the possible isogenies as the theorem below. Theorem 3.3.6. For holomorphic maps ϕ, 1. There is a bijection {α ∈ C : αΛ1 ⊂ Λ2} → {C/Λ1 → C/Λ2 with ϕ(0) = 0} α → ϕα . 2. For E1 and E2 corresponding to lattices Λ1 and Λ2, the natural inclusion in- duces a bijection {isogenies ϕ : E1 → E2} → {holomorphic maps ϕ : C/Λ1 → C/Λ2, ϕ(0) = 0}. Proof. 1. To show it is bijective, sufficient to show there is an inverse. That is, any ϕ0 : C/Λ1 → C/Λ2, ϕ0(0) = 0 associates to a unique α, or ϕα. There exists a unique α ∈ C such that ϕ0(1) = α = ϕα(1). Let ϕ := ϕ0 − ϕα, since C is simply connected, we can lift ϕ to a holomorphic map f : C → C, f(0) = 0 so that the following diagram commutes: C C C/Λ1 C/Λ2 f ϕ . 19 3. Elliptic Curves Thus f(z + ω) ≡ f(z) (mod Λ2) for all ω ∈ Λ1 and all z ∈ C. The difference f(z + ω) − f(z) must be independent of z. Differentiating, we find that f ′(z + ω) = f ′(z) for all ω ∈ Λ1 and all z ∈ C as a holomorphic elliptic function, f ′(z) is constant by Proposition 3.2.3, so f(z) = βz + γ for some β, γ ∈ C. The assumption that f(0) = 0 implies that γ = 0, since ϕ(z) = ϕ0(z) − ϕα(z), f(1) = ϕ(1) = ϕ0(1) − ϕα(1) = α− α = 0 tells us that β = 0, so ϕ = f = 0. Thus ϕ0 = ϕα. 2. See [1, p.171 , Theorem 4.1]. The corollary below results from the association of E/C and C(Λ) given by ℘(z). Corollary 3.3.7. Let E1/C and E2/C be elliptic curves corresponding to lattices Λ1 and Λ2. Then E1 and E2 are isomorphic over C if and only if Λ1 and Λ2 are homothetic, i.e., there exists some α ∈ C∗ such that Λ1 = αΛ2. Proof. Omitted. The relation between E/C and E(Λ) is summarized as follows theorems. Theorem 3.3.8. (Uniformization Theorem) For A,B ∈ C, A3 − 27B2 = 0, there exists a unique lattice Λ ⊂ C satisfying g2(Λ) = A and g3(Λ) = B. Proof. See [1, p.173, Theorem 5.1]. Theorem 3.3.9. The following categories are equivalent: 1. Objects: Elliptic curves over C. Maps: Isogenies. 2. Objects: Elliptic curves over C. Maps: Complex analytic maps taking O to O. 3. Objects: Lattices Λ ⊂ C, up to homothety. Maps: Map(Λ1,Λ2) = {α ∈ C : αΛ1 ⊂ Λ2}. Proof. See [1, p.175, Theorem 5.3]. 20 4 Intersection Theory of Surfaces A topic for algebraic surfaces S is the properties of intersection points P of two curves C,D ⊂ S. The situation of the intersection theory of a surface is studied in this section, for the last result of bielliptic surfaces. There are two results of the intersection theory in this section. The first is self- intersection. It is derived from a generalization form of intersection in the Picard group. For surfaces in P2, the second result is Bézout’s theorem giving the intersec- tion number depends on the degree of curves. A curve is irreducible if its divisor has only one component, and it is reduced if the coefficients of all the irreducible components are 1. Two distinct reduced curves are curves that have no common component and no nontrivial factorization of com- ponents. In this section, C,D are distinct reduced irreducible curves on a smooth projective surface S over K. C,D are defined by equations f, g ∈ K(S) locally, that is, ∀P ∈ C(or D),∃U ⊂ S, s.t.P ∈ U where C|U(or D|U) is defined by f |U(or g|U), U is open. Define the function in the coordinate ring ḡ := g mod I(C). 4.1 Intersection Number The multiplicity needs to be defined to introduce the intersection number. The properties of the intersection number will be shown afterward to illustrate the pairing structure. Definition 4.1.1. For distinct irreducible curves C,D of equations f, g, the multi- plicity of a single point P is mP (C,D) := dimK OX,P/(f, g) where OX,P is the localization of regular functions in k[X] at P . Remark 4.1.2. Notice that mP is well-defined. At P , C,D might be locally defined by different f1 ̸= f2, g1 ̸= g2, where f1, f2, g1, g2 ∈ K(X). But the ideals (f1) = OC,P = (f2), (g1) = OD,P = (g2). 21 4. Intersection Theory of Surfaces Therefore (f1, g1) = ((f1), (g1)) = ((f2), (g2)) = (f2, g2). Thus mP (C,D) = dimK OX,P/(f1, g1) = dimK OX,P/(f2, g2). Thus mP (C,D) is well-defined. Example 4.1.3. One simple situation of the multiplicity is {y = x2} ∩ {y = 0} at (0, 0) is 2, as the part of the intersection theory in the first section (introduction). There is a more complex but similar example. Consider the multiplicity at a point P for two curves C,D defined by functions f, g. Let X = K[x, y], f = x, P = (0, 0), g = ∑ j,k aj,kx jyk, ajk ∈ K, then OX,P/(f, g) = OA2,(0,0)/(x, ∑ j,k aj,kx jyk) = OA,(0)/( ∑ k bky k) where bk = a0,k. Thus mP (C,D) = dimK OX,P/(f, g) = min{k : bk ̸= 0}. Definition 4.1.4. For distinct irreducible curves C,D, the intersection number is C ·D := ∑ P∈C∩D mP (C,D). The intersection number defines a symmetric and bilinear pairing of the Picard group (Pic(S), P ic(S)) → Z. There is also an equivalence relation based on such a pairing, as the theorem below. Theorem 4.1.5. For any g ∈ K(X) and curve C in X, if div(g) is distinct from C, then C · div(g) = 0. Proof. Since C is locally defined by f , and ḡ = g mod I(C) is well-defined for C being distinct from div(g), mP (C, div(g)) = dimOX,P/(f, g) = dim(OX,P/(f))/(g) = dimOC,P/(ḡ) = ordP (ḡ) . Therefore C · div(g) = ∑ C∩div(g) mP (C, div(g)) = ∑ P ordP ḡ = deg ḡ. Thus by Proposition 2.2.3 C · div(g) = deg ḡ = 0. 22 4. Intersection Theory of Surfaces Remark 4.1.6. The intersection number (C,D) → C · D is defined for distinct C,D, but it is not enough for more general situations. Factoring C = ∑ j ajRj, D =∑ k bkRk into irreducible components, then C ·D = ( ∑ j ajRj) · ( ∑ k bkRk) = ∑ j,k (ajbk)Rj ·Rk. Now Rj · Rk is well defined for Rj ̸= Rk, but the self-intersection is needed for the general C ·D. That is, there should be a definition for the form R2 j = Rj ·Rj. 4.2 Self-Intersection To give a well-defined self-intersection structure, the following is necessary. Proposition 4.2.1. For any curve C, the intersection number C · (C − div(g)), g ∈ K(X) is well-defined. That is, 1. Existence: there exists some functions g, that C,C − div(g) are distinct. 2. Consistency: for any gj, gk, C · (C − div(gj)) = C · (C − div(gk)). Proof. 1. Existence: There is a uniformizer g by [1, p.18, Remark 1.1.1], s.t. ordC(g) = 1, g ∈ K(C). So C ⊂ div(g), and (div(g) − C), C not distinct ⇒ ordC(g) > 1. Thus C,C − div(g) are distinct. 2. Consistency: By Corollary 4.1.5 C · (C − div(gj)) − C · (C − div(gk)) = −C · (div(gj) − div(gk)) = −C · (div(gj − gk)) = 0 . Definition 4.2.2. The self-intersection of C is C2 := C · (C − div(g)), g ∈ K(X). 23 4. Intersection Theory of Surfaces Remark 4.2.3. The structure of pairing can be illustrated, the intersection number (C,D) → C ·D is a pairing Pic(S) × Pic(S) → Z. The equivalence relation, which is well-defined by Theorem 4.1.5, is 0 ∼ div(f), f = g/h ∈ K(S). It is easy to check that such an equivalence relation is well-defined for projective space. 4.3 Bézout’s theorem A lemma is introduced to give a simplified version of Bézout’s theorem. Lemma 4.3.1. Any plane curve C ⊂ P2 is defined by a homogeneous polynomial F . Proof. Firstly, there are some polynomials F , s.t. C ⊂ div(Fj). To illustrate the statement, let C ′ := C|Z=1 = C ∩ A2. By [2, p.7, Proposition 1.13], C ′ is defined by an irreducible polynomial f ∈ K[x, y], x = X/Z, y = Y/Z. There exist some homogeneous polynomials Fj ∈ K[X, Y, Z], j ∈ Z, where f = Fj|Z=1. There is C ⊂ div(Fj), since C ′|Z=1 = C|Z=1, (X, Y, Z) ∈ C ⇒ (x, y, 1) = (X/Z, Y/Z, 1) ∈ C ′, for Z ̸= 0 where C ′ is defined by f . Notice that div(Fj) is closed, by taking its boundary, all the points of {Z = 0} ∩ C are also taken. The rest is to show, there exists a Fj, s.t. C = div(Fj). For j, k ∈ Z, Fj ̸= Fk, (Fj/Fk)|Z=1 = (Fj|Z=1)/(Fk|Z=1) = fj/fk = 1. Since only the constant function in K[x, y] can always be 1, Fj/Fk has only factors without X, Y but Z, so Fj = ZaFk. Therefore C is defined by the irreducible F without any factor Z, which means degF = deg f . Otherwise, there is another component other than the curve. Theorem 4.3.2. (Bézout’s theorem) For distinct plane curves C,D ∈ P2, the in- tersection number is C ·D = ab, a = degC, b = degD. 24 4. Intersection Theory of Surfaces Proof. C,D are given by nonzero functions F,G by the lemma, where degF = a, degG = b. Since C ∈ P2, for line L : X = 0, s.t. F/La is a homogeneous rational function, then by Remark 4.2.3 [C] = [F/La · La] = [F/La] + [La] = a[L]. Similarly, for D there is a distinct line M : Y = 0, s.t. [D] = b[M ]. Thus C ·D = aL · bM = (ab)L ·M. If L : X = 0,M : Y = 0, then L ·M = 1 by 4.1.3. Thus C ·D = ab. 25 4. Intersection Theory of Surfaces 26 5 Bielliptic Surfaces Bielliptic surfaces are constructed by elliptic curves. The classification of bielliptic surfaces, which won’t be proved here, is based on such structure. These results can conclude the intersection theory of bielliptic surfaces. Only some simple introductions of bielliptic surfaces are illustrated to avoid too many other topics. The first thing is the structure: bielliptic surfaces have the form E × F/G, the product of two elliptic curve quotients in a group (Definition 5.1.1). The next part focuses on the Néron–Severi group (Definition 5.2.1), an important concept in intersection theory. The structure of the Néron–Severi group is also introduced (Theorem 5.2.3). Based on these facts, there is the result of the intersection theory of S (Proposition 5.3.2). In this section, S is always a bielliptic surface, E,F are elliptic curves, G is the group acting on S, and a : G → Aut(F ) is always an injective homomorphism. 5.1 Introduction The structure of bielliptic surfaces is S = (E×F )/G, where E,F are elliptic curves and G is the group acting on S. Under the equivalence relation of G, bielliptic surfaces can be classified into finite classes. Definition 5.1.1. For any bielliptic surface S, there exist elliptic curves E,F and the group G acting on S, s.t. S = (E × F )/G where for some injective homomorphism a : G → Aut(F ), g ∈ G acting on (x, y) ∈ (E,F ) is g(x, y) = (x+ g, a(g)y). There are two maps of S f : S → E/G, g : S → F/a(G) ∼= P1. Proposition 5.1.2. There exists a finite subgroup G0 of translations of F , and an isomorphism a(G) ∼= G0 × Z/nZ, n ∈ {2, 3, 4, 6} for some injective homomorphism a : G → Aut(F ). For char(K) = 0, the classifi- cation of bielliptic curves is listed in the table 5.1. 27 5. Bielliptic Surfaces Type G G0 n (a1) Z/2Z {0} 2 (a2) (Z/2Z)2 Z/2Z 2 (b1) Z/3Z {0} 3 (b2) (Z/3Z)2 Z/3Z 3 (c1) Z/4Z {0} 4 (c2) (Z/2Z) × (Z/4Z) Z/2Z 4 (d) Z/6Z {0} 6 Table 5.1: Classification of bielliptic curves Proof. See [3, p.20, List 1.17]. Remark 5.1.3. The actions of group G on S are listed below, where the additions are defined as the composition law (Definition 3.1.3). (a1) There exists a nontrivial 2-torsion point a ∈ E, s.t. G ∼= (Z/2Z) ∼= ⟨a⟩, a.(x, y) = (x+ a,−y). (a2) There exists nontrivial 2-torsion points a, b, c ∈ E, s.t. G ∼= (Z/2Z)2 ∼= ⟨b⟩ × ⟨a⟩, a.(x, y) = (x+ a,−y), b.(x, y) = (x+ b, y + c). (b1) There exists a nontrivial 3-torsion point a ∈ E, where α(a) = ω is an automor- phism of order 3, s.t. G ∼= (Z/3Z) ∼= ⟨a⟩, a.(x, y) = (x+ a, ω(y)). (b2) There exists nontrivial 2-torsion points a, b ∈ E, where α(a) = ω is an auto- morphism of order 3, s.t. G ∼= (Z/3Z)2 ∼= ⟨b⟩ × ⟨a⟩, a.(x, y) = (x+ a, ω(y)), b.(x, y) = (x+ b, y + c). (c1) There exists a nontrivial 4-torsion point a ∈ E, s.t. G ∼= (Z/4Z) ∼= ⟨a⟩, a.(x, y) = (x+ a, i(y)). (c2) There exists nontrivial 4-torsion points a, b ∈ E, s.t. G ∼= (Z/2Z) × (Z/4Z) ∼= ⟨b⟩ × ⟨a⟩, a.(x, y) = (x+ a, i(y)), b.(x, y) = (x+ b, y + c). (d) There exists a nontrivial 6-torsion point a ∈ E, where α(a) = ω is an automor- phism of order 3, s.t. G ∼= (Z/6Z) ∼= ⟨a⟩, a.(x, y) = (x+ a,−ω(y)). The composition law of E is isomorphic to the addition of C/Λ by Proposition 3.2.10, so one way to understand how G acts on S is to consider the addition and multiplication of C/Λ. There are some examples for case (d). Consider 28 5. Bielliptic Surfaces E = C/(Z + Zτ1), F = C/(Z + Zτ2) then a.(x, y) = (x+ 1 6 , e 1 3πiy), x ∈ E, y ∈ F. F can also be the elliptic curve given by the Weierstrass equation w2 = z3 + 1 G acts on F as a.(z, w) = (e 2 3πiz,−w), (z, w) ∈ F where there is the natural result a6.(z, w) = (z, w). 5.2 Néron–Severi Group Recall that a connected set is a topology subspace that is not the union of multiple disjoint non-empty open sets. Then there is the definition of the Néron–Severi group as below. Definition 5.2.1. Consider a relationship for D,D′ if there is a curve C, ∃ϕ : S → C, s.t.P, P ′ ∈ C, where D = ϕ−1(P ), D′ = ϕ−1(P ′). This generates the equivalence relationship D ∼ D′, and NS(S) is defined as the quotient of Div(S) by this relationship. A particular fact is that if D ∼ D′ in the same NS(S) class, then they are in the same intersection class, i.e. for any other curve R, [D] = [D′], D ·R = D′ ·R. For the intersection theory of surfaces, an element of NS(S) is just an equivalence class [C], which could be better illustrated in [2, p.140, Remark 6.10.3]. The classi- fication of NS(S) by [3, p.27, Table 2.2] is listed in the table 5.2. The proposition following gives the relation between NS(S) and maps f, g of bielliptic surfaces S. Type NS(X) (a1) Z2 ⊕ (Z/2Z)2 (a2) Z2 ⊕ Z/2Z (b1) Z2 ⊕ Z/3Z (b2) Z2 (c1) Z2 ⊕ Z/2Z (c2) Z2 (d) Z2 Table 5.2: Classification of Néron–Severi group 29 5. Bielliptic Surfaces Definition 5.2.2. The Néron-Severi lattice is the quotient of NS(S) by its torsion subgroup: Num(S) := NS(S)/NS(S)tors. The structure of Num(S) for elliptic surfaces S is described in the next theorem. Theorem 5.2.3. For S ∼= (E×F )/G, if the characteristic of the field K is different from 2 and 3, then the classification of the Néron-Severi lattice is listed in the table 5.3 Type Basis of Num(S) (a1) {1 2 [E], [F ]} (a2) {1 2 [E], 1 2 [F ]} (b1) {1 3 [E], [F ]} (b2) {1 3 [E], 1 3 [F ]} (c1) {1 4 [E], [F ]} (c2) {1 4 [E], 1 2 [F ]} (d) {1 6 [E], [F ]} Table 5.3: Classification of Néron-Severi lattice In this list, the classes of E and F refer to the general fibers of g and f , which are in fact isomorphic to E and F. Proof. See [3, p.39, Theorem 3.3]. 5.3 Intersection Theory of Bielliptic Surfaces In this part, K = C. Intersection theory could be applied to bielliptic surfaces, based on the structure of the Néron-Severi group (Theorem 5.2.3). The essential point is to factor [C] · [D] into linear combinations of [E], [F ] (Remark 4.1.6), where [C], [D], [E], [F ] ∈ NS(S). That is [C] = a[E] + b[F ], [D] = c[E] + d[F ]. The intersection number is the bilinear form [C] · [D] = ac[E]2 + (ad+ bc)[E] · [F ] + bd[F ]2. The next proposition is about the intersection numbers [E]2, [F ]2, [E] · [F ], by which there is [C] · [D] = (ad+ bc)#G. For any divisor of curve R = ∑ P nP (P ), the degree degR = ∑ P nP . The above relation can be considered the bielliptic surfaces version of Bézout’s theorem since it tells how the coefficients a, b, c, d define the intersection number. 30 5. Bielliptic Surfaces Lemma 5.3.1. In the Néron–Severi group NS(S), all the fibers of f construct the class [F ], and all smooth fibers of g consist of the class [E]. That is, for any P ∈ E/G,Q ∈ F/α(G), f−1(P ) = [F ], g−1(Q) = [E] where f, g are the projections f : S → E/G, g : S → F/a(G) ∼= P1. Proof. For f−1(P ), all the preimages form a class by Remark 5.2.1. Let 0E be the class of 0 in E/G, then F ∼= {(0E, y) : y ∈ F} ∈ f−1(0E). Therefore, f−1(P ) = [F ]. The proof of g−1(Q) = [E] is similar. Proposition 5.3.2. For bielliptic surface S ∼= E × F/G and [E], [F ] ∈ NS(S), 1. [E]2 = [F ]2 = 0. 2. [E] · [F ] = #G. Proof. 1. By Proposition 5.3.1, [E], [F ] ∈ NS(S) consist of elements as f−1(P ), g−1(Q), where the classes are independent of choices of P,Q. Since for P1 ̸= P2, Q1 ̸= Q2, by 5.2.1 f−1(P1) ∩ f−1(P2) = g−1(Q1) ∩ g−1(Q2) = ϕ. Thus [E]2 = [F ]2 = 0. 2. Denote by R = E×F and S = (E×F )/G and consider the natural projection σ : R → S, E/G R S F/a(G) σ f g . Taking inverse images of points (P,Q) ∈ S under σ has the effect of multiplying intersection multiplicities by #G. This means that #{(fσ)−1(P ) × (gσ)−1(Q)} = #{σ−1 ( f−1(P ) × g−1(Q) ) } = #G#{f−1(P ) × g−1(Q)}. (5.1) 31 5. Bielliptic Surfaces Then it is enough to understand #{(fσ)−1(P ) × (gσ)−1(Q)}, for any given points P,Q in E/G and F/a(G). To this effect, we describe the inverse images of E × F → E/G and E × F → F/a(G). First of all, the points in the inverse image of P ∈ E/G under the map E ×F → E/G are the points of the form ⋃ g∈G g(P0) ×F , where P0 is a point in E mapping to P . Similarly, the points in the inverse image of Q ∈ F/a(G) under the map E × F → F/a(G) are of the form ⋃ g∈GE × g(Q0) where Q0 is a point in F mapping to Q. All the intersections, in total (#G)2 of them, to be computed to find the left-hand side of (5.1) are of the form P0 × F and E ×Q0. Now, the intersection of two such objects is independent of the choice of P0 and Q0 since they determine the same classes in the Néron-Severi group of E × F. Note that both P0 × F and E × Q0 are locally given by equations, namely ℓ = 0 and ℓ′ = 0, where ℓ = 0 and ℓ′ = 0 denote local equations for P0 and Q0 in E and F , considered over E × F. Now the multiplicity is defined as m(P0,Q0)((P0 × F ), (E ×Q0)) = dim OE×F,P0,Q0/(ℓ, ℓ′) = 1 since the two curves intersect transversally. This means that #{(fσ)−1(P ) · (gσ)−1(Q)} = (#G)2. By (5.1), we find that #{f−1(P ) · g−1(Q)} = #G which was to be proven. 32 Bibliography [1] Silverman, J. H. (2009). The arithmetic of elliptic curves (Vol. 106). Springer. [2] Hartshorne, R. (2013). Algebraic geometry (Vol. 52). Springer Science & Busi- ness Media. [3] De Narvaez, D. B. (2021). Moduli of bielliptic surfaces [Phdthesis]. Technische Universität München. [4] Martin, D., & Ahlfors, L. (1966). Complex analysis. New York: McGraw-Hill. 33 DEPARTMENT OF SOME SUBJECT OR TECHNOLOGY CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden www.chalmers.se www.chalmers.se Nomenclature List of Tables Introduction Algebraic Geometry Elliptic Curves Intersection Theory Bielliptic Surfaces Algebraic Geometry Foundations Algebraic Variety Divisors Maps between curves Genus Elliptic Curves General Knowledge Weierstrass Equations Group Law Elliptic Curves over C Elliptic Functions Weierstrass -Function Maps of Elliptic Curves Isogeny The Dual Isogeny Maps on C Intersection Theory of Surfaces Intersection Number Self-Intersection Bézout's theorem Bielliptic Surfaces Introduction Néron–Severi Group Intersection Theory of Bielliptic Surfaces Bibliography