Lecture 17: Abel-Jacobi Map, Elliptic Curves

Lecture 17 18.725 Algebraic Geometry I Lecture 17: Abel-Jacobi Map, Elliptic Curves Few more remarks on the analytics theory. Last time we let X be a smooth compact C-manifold of dimension 1, obtained from a normal, complete curve over C. (In fact, any smooth compact C-manifold of dimension 1 is obtained from an algebraic curve; note that this fails for dimension ≥ 2). In this case, Pic◦ (X) = Div(X)/PDiv(X). We remarked that we have a map from it to Γ(Ω1 (X))∗ /H1 (X, Z) = Cg /Z2g (the AbelJacobi map). Theorem 1.1. X can be reconstructed from the lattice H1 (X, Z) ⊆ Γ(Ω1 )∗ . This can be generalized to smooth complete varieties in any dimension. Instead of degree, we consider a map Div → Hn−2 (X), and principal divisors are the preimages of 0. There is another Pic(X) → Hn−2 (X) with kernel Pic◦ (X), and the theorem reads Pic◦ (X) = Γ(Ωn−1 (X))∗ /H1 (X, Z). Proposition 1. Pic◦ (X) is itself a complex variety as well as a compact abelian Lie group. In fact, one can define an algebraic group Jac(X) on it, such that for a curve X, the A-J map is algebraic. To formally define the Jacobian, one defines a functor it represents. More explicitly, for a variety S, one define a family of invertible sheaves of X parametrized by S, which is essentially an invertible sheaf on S × X, modulo the line bundles pulled back from S. Theorem 1.2. Let g be the (geometric) genus of X and assume it equals 1. Then Cg /Z2g = C/Z2 has dimension 1 and is therefore a curve. Fix x0 ∈ X. The A-J map gives a map X → Pic◦ (X), where we send x to x − x0 . Then this is an isomorphism. Corollary 1. Every normal curve of genus 1 has a group structure (they are called the elliptic curves). As an example, consider X ⊆ P2 is the projective closure y 2 = P (x) = x3 + ax + b (char k 6= 2, 3) (and assume no multiple roots). We’ll check today that X is a smooth curve by showing it’s normal and irreducible. Assume k = C, we claim that g = 1, i.e. the topological Euler character is 0. Consider the map (x, y) 7→ x, which extends to a morphism X → P1 . This is of degree 2 and has four ramification points: the roots of P (x) as well as the infinity. Thinking in classical topology and choose your favorite argument, we know that Eul(X) = 2Eul(CP1 ) − 4 = 0. Now let’s consider how to write down the composition (group) law. To do so, we first fix the initial point x0 = (0 : 1 : 0), where we see that {x0 } = X ∩ P1∞ . The complex story suggests that we have a group law on X, such that for every x, y ∈ X, we have the divisor equivalence (x +E y) − x0 ∼ (x − x0 ) + (y − x0 ) (where +E denotes the addition using the group law), in other words, (x +E y) − x − y ∼ −x0 . We know that for every two lines l, l0 = P1 ⊆ P2 we have (l ∩ X) ∼ (l0 ∩ X) (we discussed this before). Now take l0 = P1∞ , then (l0 ∩ X) = 3x0 . Write l ∩ X = x1 + x2 + x3 , then (x1 − x0 ) + (x2 − x0 ) + (x3 − x0 ) ∼ 0 in Pic(X). So we should expect x1 +E x2 +E x3 = 0. Now we construct the group law. For x = (a, b) ⊆ X, x0 = (a, −b), we have x + x0 + x0 ∼ 3x0 ∈ Pic, so we define x +E x0 = 0. Now in general, define x +E y to be the 3rd point in l ∩ X, where l passes through x0 and y 0 . One can directly check that this is a group law that makes X an abelian algebraic group. Remark 1. Over C, X = C/Z2 makes it clear that for all N > 0 we have {x ∈ X | N x = 0} ∼ = (Z/N Z)2 . This can be checked algebraically to hold for k of characteristic p - N . If N = p, then this group is Z/p, or trivial if X is respectively ordinary or supersingular. Consider X0 ⊆ A2 given by {(x, y) | y 2 = P (x)}. If X0 − {z} is affine, then it corresponds to k[X0 ](f ) where f is a function in k[X0 ] such that f (x) = 0 ⇔ x = z, which is iff (f ) = N z − N x0 for some N (where x0 is the group law identity, which is the infinite point). For a given N there are N 2 − 1 such z. Last time we proved that if X is normal irreducible complete curve, f ∈ K(X), then it defines some f : X → P1 , then the divisor (f ) is (f0 ) − (f∞ ) where deg(f0 ) = deg(f∞ ) = deg(f ). We proved this modulo the following proposition, which we shall prove today: Proposition 2. A non-constant map between irreducible compact curves is finite. 1 Lecture 17 18.725 Algebraic Geometry I Normalization Let X be an irreducible variety, F = K(X) be the field of rational functions on X. Let E/F be a (finite) field extension. Then build a new variety as follows: Proposition 3. There exists a variety Y along with a finite map f : Y → X such that for every affine open U ⊆ X, k[f −1 (U )] = k[U ]E (the integral closure). If E = F , then Y is called a renormalization of X. In fact, Y is the unique normal variety with a finite onto map to X with the fractional field being E. k[U ]E is finitely generated as a k[U ]-module, or equivalently, as a ring. In other words k[U ] is a Nagata ring. Sketch of proof to this: using Noether normalization reduce to X = An . Consider separately the case of purely inseparable and the separable extensions. For separable extension case, the bilinear form (x, y) 7→ Tr(xy) on E as an F -vector space is not degenerate, so if we pick a basis (yi ) for E/F which lies in k[x1 , . . . , xn ]E , then k[x1 , . . . , xn ]E ⊆ {e ∈ E | Tr(exi ) ∈ A} is a finitely generated algebra for A = k[x1 , . . . , xn ]. Now the assignment U 7→ k[U ]E extends to a coherent sheaf A of rings on X, and let Y = SpecX (A). Corollary 2. Given f : X → Y where X, Y are irreducible, if X is normal, f is finite, onto, then X can be reconstructed from Y and f −1 (U ) for some open U = 6 ∅⊆Y. Example 1. Let X = V (x3 − y 2 ), then the normalization of X is A1 , and the map is t 7→ (t2 , t3 ). Lemma 1. If f : X → Y is a map of irreducible curves, suppose f is onto, birational, Y is normal, then f is an isomorphism. Proof. Let ϕ ∈ K(Y ), ϕ on f −1 (U ) ⇔ ϕ is regular on U . If ϕ is not, ϕ−1 is regular and is 0 at some x ∈ U . Suppose y 7→ x, then ϕ is not regular at y. Lemma 2. Suppose X → Y is birational map, X is complete, Y is normal, then X ∼ = Y iso. Proof. Since f (X) is closed and not finite, we know f must be onto. Proof of Proposition 2. X → Y is a map of complete curves. We can assume X is normal. Then it factors through normalization X → Nor(Y ) → Y . The first is isomorphism by assumption, and the second map is finite by construction. Tangent Space Now let X be an algebraic variety, x ∈ X. Let us define the Zariski tangent space Tx X. We first we note the tangent space to a smooth manifold is the fiber of the bundle of vector fields Vect(M ) = Der(C ∞ (M )). Each vector field v gives a linear map δv : Fun(M ) → C that maps f to v · f |x , so we see that δv (f g) = f (x)δv (g) + g(x)δv (f ). This suggests the definition Tx X ⊆ Homk (OX,x , k) given by {ξ | ξ(f g) = f (x)ξ(g) + g(x)ξ(f )}. The cotangent space Tx∗ X is the dual (Tx X)∗ , and we can describe it as mx /mx2 . In particular, for X = Spec(A), Vect(X) = Der(A) = {δ : A → A k-linear | δ(f g) = δ(f )(g)+f δ(g)}. 2