A Humble Example of Notions of Divisor Gohan May 16, 2013 I must apologize for my morbid procrastination. Due to my incapability of understanding advanced mathematics, I was rather tortured by my previous proposal. In the meanwhile I feel scared whenever I reflect on the pathetic falling of level by my appearance. Thank you so much for sharing wonderful time and thoughts with a moron like me. Brilliant applications of divisor and linear systems abound in literature of classical AG. Here I illustrate the significance of linear system technique through a toy problem of arithmetic geometry. Consider the most naive group action like Z/2Z on E E, where E is an elliptic curve over number field. We can show the Zariski density of rational points on the quotient surface by simple linear system account. Consider an elliptic curve over Q E : Y 2 Z = X 3 pxz 2 qz 3 and a rational point O = [0 : 1 : 0] on E. On affine open {Z 0} we take affine coordinates as x = X Z, y = Y Z Then the affine part of E on {Z 0} is E Z : y 2 = x 3 px q E Z = E {O}. On affine open {Y 0} take affine coordinates as u = X Y, v = Z Y Then the affine part of E on {Y 0} is E Y : v = u 3 puv qv 3 expwufan@gmail.com 1
2 E Y is E with three points being removed. Denote E Z,Y = E Z {Y 0}, it is E with four points of degree two being removed. There are two cases of intersection of line pencil through point O l t : u = tv t Q and E: intersect at two other Q-points intersect at a point of degree two The line l : v = 0 is tangent to E at Q-point O. Conversely, we know from Riemann-Roch theorem that all points of degree two on Ẽ should be intersections as stated above. Back to (x, y) coordinate we see that all points of degree two are of the form (t, t 3 pt q) t Q. In the viewpoint of scheme, quadratic points correspond to the maximal ideal p(t) = (x t, y 2 t 3 + pt + q) in Q[x, y]/(y 2 x 3 + px + q), where t is rational and t 3 pt q is square-free. In all, we conclude that all the residue class fields of quadratic points on E are of the form K = Q( t 3 pt q). Consider two quadratic points on E with isomorphic residue class fields, where one point corresponds to the maximal ideal p(t 1 ), while another p(t 2 ). Then we have t 3 1 p t 1 q = u 2 (t 3 2 p t 2 q) for someu Q, u 0 (0.1) Consider an affine surface S over Q defined by F (t 1, t 2, u) = (t 3 1 p t 1 q) u 2 (t 3 2 p t 2 q) = 0 S is smooth, since F t 1 = 3t 2 1 p F t 2 = u 2 (3t 2 2 p) F u = 2(t 3 2 pt 2 q)u cannot vanish simultaneously. S tells us some information on points of degree at most two on E, since there exists a 2 : 1 morphism from E Z E Z,Y to S, represented through coordinates as ((x 1, y 1 ), (x 2, y 2 )) (x 1, x 2, y 1 /y 2 )
Then a rational point (t 1, t 2, u) on S may correspond to 3 two pairs of rational points on E or a pair of quadratic points on E with isomorphic residue class fields Furthermore there is a projection from S to A 1 (Q). In coordinates it is (t 1, t 2, u) t 2 It is easy to see that every fibre of this projection is an elliptic curve with identical j-invariant. Then it is plausible to surmise the following. Theorem 0.1. For affine surface S there is a projective completion S, which is an elliptic surface with at most 16 singularities. The genus 1 fibration S P 1 is a constant family over C, but not over Q. And there exists the following commutative diagram Consequently from the structure of S we have E E 2:1 2:1 (E E)/(Z/2Z) S Theorem 0.2. The rational points S(Q) of S is Zariski dense. To facilitate the proof, we recall two simple lemmas first. Lemma 0.1. Consider two topological spaces X and Y. Let ϕ : X Y be a surjective morphism, and T be a dense subset of Y. Then ϕ 1 (T ) is a dense subset of X. Conversely, let S be a dense subset of X, then ϕ(s) is a dense subset of Y. Lemma 0.2. If T is Zariski dense in an algebraic variety X, and U is Zariski dense in an algebraic variety Y, then T U is Zariski dense in X Y. 0.1 Proof of the first theorem Proof. E E could be covered by four open affines E E = (E Z E Z ) (E Y E Y ) (E Z E Y ) (E Y E Z )
4 from the schematic viewpoint E Z E Z = Spec Q[x 1, y 1, x 2, y 2 ]/(y 2 1 f(x 1 ), y 2 2 f(x 2 )) E Y E Y = Spec Q[u 1, v 1, u 2, v 2 ]/(g(u 1, v 1 ), g(u 2, v 2 )) S = Spec Q[t 1, t 2, u]/(f(t 1 ) u 2 f(u 2, v 2 )) Denote A Z = Q[x 1, y 1, x 2, y 2 ]/(y 2 1 f(x 1 ), y 2 2 f(x 2 )) A Y = Q[u 1, v 1, u 2, v 2 ]/(g(u 1, v 1 ), g(u 2, v 2 )) B = Q[t 1, t 2, u]/(f(t 1 ) u 2 f(u 2, v 2 )) Then E Z E Z,Y = Spec A Z [ 1 y 2 ] The morphism E Z E Z,Y S is induced by B A Z [ 1 y 2 ], t 1 x 1, t 2 x 2, u y 1 /y 2 Consider a geometric Z/2Z group action on E E: 1 : E E E E (P, Q) ( P, Q) The group action quotient of E E exists and is a projective variety: (E E)/G = (E Z E Z )/G (E Y E Y )/G = Spec A G Z Spec AG Y The group action can be restricted onto E Z E Z,Y. Now we prove S (E Z E Z,Y )/G Since (E Z E Z,Y )/G = (Spec A Z [ 1 y 2 ])/G = Spec(A Z [ 1 y 2 ] G ), it suffices to prove B A Z [ 1 y 2 ] G The action of G on A Z [ 1 y 2 ] is given by 1(x 1 ) = x 1, 1(y 1 ) = y 1, 1(x 2 ) = x 2, 1(y 2 ) = y 2
Obviously we have B A Z [ 1 y 2 ] G A Z [ 1 y 2 ], while B A Z [ 1 y 2 ] and A Z [ 1 y 2 ] G A Z [ 1 y 2 ] are both faithfully flat embeddings of degree 2, thus B A Z [ 1 y 2 ] G is flat embedding of degree 1, thus an isomorphism. S of S. (E Z E Z,Y )/G is open affine in (E E)/G, so we could take (E E)/G as a projective completion We compute the invariant subalgebra A G Z and AG Y with Singular3-1-1. 5 Remark 0.1. For a rational point P (E E)/G, consider its preimage Q = q 1 (P ) under projection q : E E (E E)/G, we have deg Q = 2, namely Q Spec K, where [K : Q] = 2, or Q Spec(Q Q). Since Q is a G-orbit by itself, G acts on Q naturally. As Q is just one point in the former case, the action of G is represented as Q-automorphisms of K on function sheaf. On the other side, there is only one non-trivial automorphism over a quadratic field, so the action of G in this case coincides with the Galois action of K. Consequently through the geometric group action Z/2Z (E E) (E E) we could identify different Galois groups Gal(k(p) : Q) naturally, where k(p) is the residue class field of quadratic point p on E. 0.2 Proof of the second theorem Proof. Set the preimage of S(Q) under projection q be T = q 1 ( S(Q)), then T = {(P 1, P 2 ) E E κ(p 1 ) = κ(p 2 ), [κ(p 1 ) : Q] 2} From the first lemma, it suffices to prove T is dense in E E. Prove by the following three cases. Case 1 There are infinitely many rational points on E. In this case E(Q) is already Zariski dense in E. From lemma 1 we know that E(Q) E(Q) is also Zariski dense in E E. While E(Q) E(Q) T, so T is dense in E E. Case 2 There are only finitely many rational points on E but there are infinitely many points of E over some quadratic field K. In this case E(K) is already dense in E, from lemma 2 we know that E(K) E(K) is dense in E E. While E(K) E(K) T, so T is dense in E E. Case 3 For any quadratic field K, there are only finitely many K-points on E. In this case we prove by contradiction. Assume T is not Zariski dense. Consider projection pr 1 : E E E, which is proper
6 and projective, thus closed. pr 1 (T ) = Γ 2. For P Γ 2 set T P = pr 1 1 (P ) T, then T P ord P. However ord P could be arbitrarily large, thus T P could also be arbitrarily large. Since T is not dense, T is contained in some Zariski closed proper subset V of E E. dim V 1 and V has only finitely many irreducible components V 1,..., V r dim V i 1. Now the images pr 1 (V i ) of V i under pr 1 compose of two sorts. The first sort is a point in E, let A be the set of all those points of images. Then A is a finite subset of E. The second sort is E itself. For those V i, the morphism V i E is dominant. But a dominant morphism between two projective curves should be finite and flat.the morphism V i E has a finite degree d i. Set d = d i. Take a point P Γ 2 A on E, then T P pr1 1 (P ) d which contradicts with the assumption that T P could be arbitrarily large. ë z [D] Dieudonné, J, History of Algebraic Geometry, [B] Beltrametti,M.C. et al., Lectures on Curves, Surfaces and Projective Varieties, [Do] Dolgachev, I.V., Classical Algebraic Geometry, [K] Kummer, E.E., Über die Zerlegung der aus Wurzeln der Einheit gebildeten complexe Zahlen in ihre Primfaktoren,