DIPARTIMENTO DI MATEMATICA E FISICA Corso di Laurea Magistrale in Matematica An Introduction to Elliptic Curves and Modular Forms Summary Relatore: Candidato: Prof. Francesco Pappalardi Federico Campanini no matricola 428445 Anno Accademico 2014-2015 Ottobre 2015 AMS Classification: 11F06, 11F11, 11F25, 11G05, 11E05. Keywords: elliptic curves, modular curves, modular groups, modular forms, Hecke operators. Summary The theory of elliptic curves and modular forms is one subject where the most diverse branches of Mathematics like complex analysis, algebraic geometry, representation theory and number theory come together. Our point of view will be number theoretic. A well-known feature of number theory is the abundance of conjectures and theorems whose statements are accessible to high school students but whose proofs either are unknown or, in some cases, are the culmination of decades of research and use some of the most powerful tools of twentieth century mathematics. In addressing problems such as these, which have no easy solution, it is essential to create strong tools that penetrate in different and apparently completely distinct branches of mathematics. The aim of this thesis is to explore the theory of elliptic curves and modular forms and to develop tools that are essential in the resolution of some remarkable problems, like the Ramanujan conjectures and the Fermat's Last Theorem. Chapter 1 is devoted to the study of elliptic curves. We have chosen an analytic approach, due to Weierstrass, which involves the theory of elliptic functions. We talk about this topics before to give the definition of elliptic curves. A lattice Λ is a subgroup of C which is free of dimension 2 over Z and which generates C over R. If f!1;!2g is a basis of Λ over Z, then we also write Λ = [!1;!2] or Λ = !1Z ⊕ !2Z. Moreover, we assume the normalizing convention that Im(!1=!2) > 0. A complex torus is a quotient of the complex plane by a lattice, C=Λ = fz + Λ j z 2 Cg: 1 An elliptic function f with respect to Λ is a meromorphic function on C which is Λ-periodic, i.e. f(z + !) = f(z) 8z 2 C; 8! 2 Λ: The most important example of a non-constant elliptic function with respect to a lattice Λ is the Weierstrass }-function, which is defined as 1 X 1 1 }(z) = + − ; z2 (z − !)2 !2 !2Λ∗ where Λ∗ = Λ n f0g. The power series development at the origin of this function is given by 1 1 X }(z) = + (2n + 1)G z2n; z2 2n+2 n=1 where X 1 G (Λ) = G = m m !m !2Λ∗ is the Eisenstein series of weight m and it is an example of modular forms, as we will see later. } satisfies the differential relation 02 3 } = 4} − g2} − g3 0 where g2 = 60G4 and g3 = 140G6, and so the points (}(z);} (z)) lie on the curve defined by the equation 2 3 Y = 4X − g2X − g3: The cubic polynomial on the right-hand side has three distinct roots and 3 2 so its discriminant ∆ = g2 − 27g3 does not vanish. Note that we have 0 a map z + Λ 7! (}(z);} (z)) from the torus C=Λ to the points of the 2 3 0 curve Y = 4X − g2X − g3, because of the periodicity of } and } . If we want to embed the points in the projective space 2 , we can write PC 0 z + Λ 7! (}(z);} (z); 1). In this way we obtain a map from the torus C=Λ to the set E(C) of the complex projective points on the homogenized curve 2 3 2 3 Y Z = 4X − g2XZ − g3Z , where the lattice points are the points going 2 to infinity on the curve. Namely, we have a map Φ: C=Λ −! E(C) 0 + Λ 7! (0; 1; 0) z + Λ 7! (}(z);}0(z); 1) and it is easy to see that it is a bijection. Now we introduce the notion of elliptic curve over C. An elliptic curve over C is a cubic projective curve, defined over C, given by the equation 2 3 2 3 E : Y Z = 4X + a2XZ + a3Z ; a2; a3 2 C: (1) Sometimes we write the equation in its non-homogeneous form, namely 2 3 E : Y = 4X +a2X +a3. The equation (1) or its non-homogeneous form will be referred to as the Weierstrass equation. We say that E is non-singular if the cubic polynomial on the right-hand side has three distinct roots, i.e. 3 2 its discriminant ∆ = a2 − 27a3 6= 0. The set of the complex projective points on the curve is denoted by E(C). Let Pz = Φ(z +Λ) denotes a generic point in E(C). We can use the one-to-one correspondence Φ to define a commutative group law on E(C), namely for any z1 + Λ; z2 + Λ in C=Λ, we can define Pz1 + Pz2 = Pz1+z2 : In this way we obtain an addition law that has two remarkable properties: first, there is a geometric interpretation for adding points on an elliptic curve and second, we can express the coordinates of Pz1+z2 as rational functions of the coordinates of Pz1 and Pz2 . In this way, Φ defines an analytic group isomorphism between the complex 2 3 torus C=Λ and the elliptic curve E(C) defined by the equation Y = 4X − g2(Λ)X − g3(Λ). It follows that for any complex torus C=Λ there exists ∼ an elliptic curve E(C) such that C=Λ = E(C). A result known as the Uniformization Theorem say us that the converse holds as well: in fact 2 3 for any elliptic curve given by an equation E : Y = 4X + a2X + a3, there 3 exists a lattice Λ such that a2 = −g2(Λ), a3 = −g3(Λ) and C=Λ is isomorphic to E(C) via Φ. The Uniformization Theorem allows us to treat complex analytic objects, like complex tori, and algebraic objects, like elliptic curves, in the same manner. So, the term \(complex) elliptic curve" can be used as synonym of \complex torus". We are interested in class-isomorphism of elliptic curves. If Λ and Λ0 are two lattices, then there exists an analytic group homomorphism from C=Λ to 0 0 C=Λ if and only if there exists a non-zero element m 2 C such that mΛ ⊆ Λ and such homomorphism is an isomorphism if and only if mΛ = Λ0. It follows that if Λ = [!1;!2] is a lattice and if Λτ denotes the lattice of the form [τ; 1], ∼ then C=Λ = C=Λτ , with τ = !1=!2. Therefore for every lattice Λ there exists another one of the form Λτ = [τ; 1] for some τ 2 C with Im(τ) > 0 such that ∼ 0 0 0 C=Λ = C=Λτ . Moreover if Λ = [!1;!2] and Λ = [!1;!2] are two lattices, 0 a b then Λ = Λ if and only if there exists M = c d 2 SL2(Z) such that ! ! !0 ! 1 = M 1 0 !2 !2 where SL2(Z) denotes the modular group ( ! ) a b SL2(Z) = 2 M2(Z) j ad − bc = 1 : c d 0 Combining these two results we can show that for any τ; τ 2 C with positive ∼ imaginary part, C=Λτ = C=Λτ 0 if and only if there exists a matrix M 2 SL2(Z) τ 0 τ such that 1 = M ( 1 ). The last condition can be rewritten by saying that 0 aτ+b there exist integers a; b; c; d such that ad − bc = 1 and τ = cτ+d . This fact motivates the definition of an action of SL2(Z) on the upper half plane H = fz 2 C j Im(z) > 0g. The modular group acts on H through fractional linear transformations as follows: ! az + b a b Mz = ;M = 2 SL2(Z); z 2 C: cz + d c d In Chapter 2 we introduce the main results about this group action and about the action of some subgroups of SL2(Z) on H. Let N be a positive 4 integer. The principal congruence subgroup of level N is ( ! ! ! ) a b a b 1 0 Γ(N) = 2 SL2(Z) j ≡ (mod N) : c d c d 0 1 (The matrix congruence is interpreted entrywise.) A subgroup Γ of SL2(Z) is + a congruence subgroup of level N if Γ(N) ⊆ Γ for some N 2 Z . The most important congruence subgroups are ( ! ! ! ) a b a b ∗ ∗ Γ0(N) = 2 SL2(Z) j ≡ (mod N) c d c d 0 ∗ and ( ! ! ! ) a b a b 1 ∗ Γ1(N) = 2 SL2(Z) j ≡ (mod N) c d c d 0 1 satisfying Γ(N) ⊆ Γ1(N) ⊆ Γ0(N) ⊆ SL2(Z): The results in Chapter 1 show that every τ 2 H determinates an elliptic ∼ ∼ curve E(C) = C=Λτ . However the choice of τ is not unique, but E(C) = ∼ C=Λτ = C=Λτ 0 if and only if there exists a matrix M 2 SL2(Z) such that τ 0 = Mτ. This fact motivates the definition of an equivalence relation between elements on H under the action of SL2(Z). We say that two points 0 τ; τ 2 H are equivalent relative to the modular group SL2(Z) or 0 SL2(Z)-equivalent if there exists a matrix M 2 SL2(Z) such that τ = Mτ. We write τ ∼ τ 0. \∼" defines an equivalence relation and the set of all equivalence classes is denoted by Y (1) = SL2(Z) n H; to indicate that SL2(Z) acts on H on the left.