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Elliptic Curves and Their Moduli

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Elliptic Curves and Their Moduli ELLIPTIC CURVES AND THEIR MODULI Maxim Jeffs December 26, 2019 INTRODUCTION In the study of Riemann surfaces, elliptic curves arise in three forms: 1. As plane cubic curves X ⊆ P2; ∼ 2. As quotients C=Λ of C by a lattice Λ = Z2; 3. As double-covers of P1 branched at four points. In this short note, we will explain how to relate each of these three perspectives, which will lead us to very classical topics in mathematics such as elliptic functions and integrals. We shall then use these three different perspectives to study the classification problem for1 genus- curves. CUBIC CURVES By the degree-genus formula, a degree d = 3 homogeneous polynomial P (X; Y; Z) defines a curve X ⊆ P2 of genus: (d − 1)(d − 2) (3 − 1)(3 − 2) g = = = 1 2 2 whenever rP is nowhere-vanishing (except possibly at 0). After an appropriate change of variables, the equation P = 0 may be written in the inhomogeneous form y2 = f(x) for f a cubic polynomial (see [Sil09]). After a further change of variables, this can be written in one of the two following forms: the Weierstrass form 2 3 y = 4x − g2x − g3 or the Legendre form: 2 y = x(x − λ1)(x − λ2) 3 − 2 The condition that X is non-singular translates into the condition that the discriminant ∆ = g2 27g3 be non-zero. First, let us describe the topology of this curve via a branched covering of P1. In the homogeneous form, the cubic curve X has a map X ! P1 given by projecting to the variables [X; Z]. On the affine open subset Z = 1, since y2 = f(x) has two solutions for f(x) =6 0, this covering is generically two- sheeted, with branch points at roots of f. There is a further branch point on the line at 1, lying over 1 2 3 2 3 1 = [1 : 0] 2 P , given by setting Z = 0 in Y Z = X − g2XZ − g3Z . In the Legendre form, these roots are exactly at 0; λ1; λ2, with again a branch point over 1. Conversely, by the Riemann-Hurwitz 1 formula, a degree 2 covering of P1 with 4 branch points will have Euler characteristic χ(X) = 2χ(P1) − R = 2(2) − 4 = 0 and hence will be a genus-1 curve. We are yet to see that these may all be described as cubic curves. Secondly, we may also describe X as a quotient C=Λ where Λ is called the period lattice: by the uniformization theorem, we know that the universal covering space of X is C, and that X is the quotient of C by some subgroup Γ of holomorphic automorphisms of C with no fixed points. If X ∼ is a compact surface, then Γ must be the group of translations by a rank-2 lattice Λ = Z2. Notice that the identification of X with C=Λ gives a distinguished point on X, corresponding to the origin in 0 2 C. Once this distinguished point has been chosen, X has the structure of an abelian group under addition, with this point at the identity element. This is why in number theory an elliptic curve is often defined to be a genus-1 curve along with a choice of a distinguished point. Let’s now describe how this isomorphism works explicitly. On a cubic curve, we have a nowhere-vanishing holomorphic 1-form given (on the affine open subset z =6 0) by dx 2 dy α = = y f 0(x) More explicitly, this has the form dx α = p f(x) where f is a cubic polynomial: indefinite integrals of this form α over X are called elliptic integrals (or at least, one kind of elliptic integrals). This indefinite integral must be interpreted as the integral along a path γ on X: since α is closed, this only depends on the choice of homotopy class of γ. Fixing a basepoint ∗ 2 X, any two paths γ; γ0 from ∗ to a point p 2 X can be concatenated to give an element δ 2 π1(X; ∗): then Z Z Z α − α = α 0 γ R γ δ and so the two integrals differ precisely by δ α. If we take γ1; γ2 to be integral generators of π1(X), then we can define the periods of X by Z τ1 = α Zγ1 τ2 = α γ2 Since the indefinite integral Z p p 7! α ∗ is well-defined up to addition of τ1; τ2, this yields a well-defined a map X ! C=Λ, where Λ is the lattice generated by the periods. In fact, this map will be a group homomorphism. By pulling α back to the universal covering space X~ of X, one may easily show that the indefinite integral of α map induces an ∼ isomorphism X~ ! C, descending to the give an isomorphism X = C=Λ. 2 ELLIPTIC FUNCTIONS ∼ Now suppose X = C=Λ is a genus-1 curve described as the quotient of C by a lattice: we want to present this as a cubic curve. We will need to begin by constructing meromorphic functions on X, that is, Λ-periodic meromorphic functions on C, called elliptic functions. Firstly, note that by Liouville’s theorem, there are no non-constant Λ-periodic holomorphic functions. Moreover, if we look at the integral of a Λ-periodic function around the boundary of the fundamental parallelogram, then we get 0. Hence by Cauchy’s theorem, the sum of the residues of poles contained in the fundamental parallelogram must be zero. In particular, the meromorphic functions we are looking for should have at least two poles (counted with multiplicity). We define the Weierstrass elliptic function } by ( ) 1 X 1 1 }(z) = + − z2 (z − λ)2 λ2 λ2Λnf0g This is clearly doubly-periodic with a double pole at z = 0. We may rewrite ! ( ) ( ) 1 ( ) 1 1 1 λ2 1 1 1 X z n − = − 1 = − 1 = (n + 1) (z − λ)2 λ2 λ2 (z − λ)2 λ2 (1 − z/λ)2 λ2 λ n=1 and hence } has a Laurent series expansion given by 1 1 X }(z) = + (2n + 1)G z2n z2 2n+2 n=1 where Gk is the Eisenstein series: X 1 G = k λk λ2Λnf0g which for k odd are easily seen to be zero because of the reflection symmetry Λ = −Λ. Importantly, } satisfies a cubic differential equation: 0 2 3 (} ) = 4} − g2} − g3 where g2 = 60G4 and g3 = 140G6. An easy way to see this is to use the expansion of } in a Laurent series above to show that: 6 }00(z) = + 2c + ··· z4 1 1 }(z)2 = + c + ··· z4 1 and hence that }00 − 6}2 is an elliptic function with no poles: it is hence constant by Liouville’s theorem. This exactly implies that } satisfies such a cubic differential equation, and we may easilyfind the coefficients using the Laurent series expression. As }0 is also a Λ-periodic meromorphic function, we have a well-defined map X ! P2 given by [} : }0 : 1]. Because of the above differential equation, the image lands in the elliptic curve definedby 2 3 y = 4x − g2x − g3. This map is in fact an isomorphism: the inverse is exactly the map X ! C=Λ constructed above. Define a function via the indefinite integral considered in the previous section Z z dx F (z) = p ∗ f(x) 3 3 for f(x) = 4x − g2x − g3. Then the inverse of F would satisfy the cubic differential equation: ( ) d 2 1 F −1(z) = = f(F −1(z)) dz F 0(z)2 which is exactly the cubic differential equation satisfied by }. One can use this fact to show that } is indeed the inverse. Elliptic functions first appeared in precisely this way, as inverses of elliptic integrals. Next, we wish to express X = C=Λ as a branched cover of P1. One way we can do this is by composing the previous map X ! P2 with the projection P2 ! P1 given by the variable x. But this is just the meromorphic function } : X ! P1. We can clearly see that there is one branch point at 0 since } has a double pole there. More generally, the fact that }(−z) = }(z) implies that branch points occur at half-periods: points z 2 C where z ≡ −z mod Λ. A more geometric way to see this is as follows: once a basepoint has been chosen, every elliptic curve X has an involution σ : z 7! −z coming from the abelian group structure. The fixed points of this involution have 2z ≡ 0 and are called half-periods: when we identify X = C=Λ, for Λ generated by τ1; τ2 over R, then there are four half-periods given by 0; τ1=2; τ2=2; τ1=2 + τ2=2. Hence the quotient map X ! X/σ is a double cover with four branch points. To identify the quotient X/σ with P1 we observe that since }(−z) = }(z), } descends to give a meromorphic function on the quotient X/σ. Since z = 0 is a 2-fold branch point, and } has a double pole at 0, on the quotient X/σ, } has a single simple pole at [0], identifying X/σ with P1. In the usual topological picture of the torus, this involution looks like flipping the torus 180◦ around an axis that intersects it in 4 points. Figure 1: Flipping a torus. The values of these branch points are not easy to compute, so one instead works with theta functions, which will instead give us a presentation of X in Legendre form, where the branch values are easily identified.
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