Vojta’s_Main_Conjecture_for_blowup_surfaces

Vojta’s Main Conjecture for blowup surfaces David McKinnon November 8, 2001 Abstract In this paper, we prove Vojta’s Main Conjecture for split blowups of products of certain elliptic curves with themselves. We then deduce from the conjecture bounds on the average number of rational points lying on curves on these surfaces, and expound upon this connection for abelian surfaces and rational surfaces. MSC: 11G05, 11G35, 14G05, 14G40. Keywords: Vojta’s conjecture, heights, rational points, elliptic curves. 1 Introduction Vojta has formulated many deep conjectures about the arithmetic of algebraic varieties [Vo]. In this paper, we will be primarily concerned with the following conjecture [Vo, Conjecture 3.4.3]. Conjecture 1.1 (Vojta’s Main Conjecture) Let X be a smooth algebraic variety defined over a number field k, with canonical divisor K. Let S be a finite set of places of k. Let L be a big divisor on X, and let D be a normal crossings divisor on X. Choose height functions hK and hL for K and L, re- spectively, and define a proximity function mS(D,P ) = ∑ v∈S hD,v(P ) for D with respect to S, where hD,v is a local height function for D at v. Choose any � > 0. Then there exists a nonempty Zariski open set U = U(�) ⊂ X such that for every k-rational point P ∈ U(k), we have the following inequality: mS(D,P ) + hK(P ) ≤ �hL(P ) (1) 1 1 INTRODUCTION 2 We recall the definition of big: Definition 1.2 Let L be a divisor on a smooth algebraic variety V . The dimension of L is the integer ` = dimL satisfying `(nL) �� n` for n sufficiently divisible. If L(nL) is always empty, define dimL = −∞. A divisor L is big if and only if dimL = dimV . Big divisors are characterised by the following proposition of Kodaira ([KO, Appendix]): Proposition 1.3 Let L be a divisor on a smooth projective variety V . Then L is big if and only if nL can be written as a sum of an ample divisor and an effective divisor for some sufficiently large integer n. If X is a curve of genus g > 1, or indeed any variety of general type, then the canonical divisor K itself is big, so choosing � < 1, D = 0, and L = K in Conjecture 1.1 gives a dense open subset U on which hK(P ) ≤ 0 for all P ∈ U(k). This immediately implies that U(k) is finite, so that X(k) must be entirely contained in some Zariski closed proper subset. Thus, if X is a curve, X(k) itself must be finite, which is precisely the statement of Faltings’ Theorem. However, very little has been proven about Vojta’s conjectures in general, so the above argument is not likely to produce a new unconditional proof of Faltings’ Theorem. It is more likely that Vojta’s conjectures will be proven for other varieties first, and indeed some progress in that direction has already been made by Roth, Schmidt, Schlickewei, and others for Pn, and Faltings has indeed already proven Conjecture 1.1 for abelian varieties. (See [Vo] and [EE] for an overview of known results.) Note that the full strength of Conjecture 1.1 is not necessary above, since the term mS(D,P ) is neglected by setting D = 0. In general, setting D = 0 yields the following (possibly) weaker conjecture: Conjecture 1.4 Define X, K, k, L, hL, and hK as in Conjecture 1.1, and choose any � > 0. Then there exists a nonempty Zariski open set U = U(�) ⊂ X such that for every k-rational point P ∈ U(k), the following inequality holds: hK(P ) ≤ �hL(P ) (2) 2 ABELIAN SURFACES 3 In §2, we prove the special case of Conjecture 1.4 where X is a blowup of a product C ×C, where C is an elliptic curve whose rational points C(k) form a rank one module over the endomorphism ring Endk(C). From this and a result of Faltings [Fa], we deduce the corresponding special case of Conjecture 1.1. We further deduce that many curves on X have only finitely many rational points (Theorem 2.4) and that for certain pencils P of curves on X, the average number of rational points on curves in P is finite and can be effectively computed. In §3, we describe how specialising Conjecture 1.4 to rational surfaces implies Faltings-type theorems for curves of general type (Theorems 3.4 and 3.5). Furthermore, we can get (conjectural) quantitative bounds on the aver- age number of points on curves as they vary in plane pencils (Theorem 3.6). I would like to thank Paul Vojta for his helpful remarks and comments, and my wife Jennifer for her help with the editing of this paper. 2 Abelian Surfaces Let C be an elliptic curve defined over a number field k, and let A = C × C. Let X be a geometrically smooth, projective k-scheme and f :X → A a birational k-morphism. Assume further that the exceptional set E of f satisfies f(E) ⊂ Z for some finite set Z of k-rational points of A. Let L be a big divisor on X, and denote by K the canonical divisor on X. We may then choose height functions hK and hL associated to K and L, respectively. In this section, we will assume either that C(k) has rank one, or that C(k) has rank two and k-rational complex multiplication. In this case, if X = A, Conjecture 1.4 is trivially true, since K = 0 and hL(P ) is uniformly bounded below. However, if X 6= A, then K 6= 0, and the conjecture becomes non-trivial. The main result of this section is the following. Theorem 2.1 (Vojta’s Main Conjecture with D = 0) Let X, L, and K be as above, and choose height functions hL and hK. For any � > 0, there is an effectively computable Zariski closed proper subset V (�) such that for all k-rational points P ∈ (X − V (�))(k), we have the following inequality: hK(P ) ≤ �hL(P ) +O(1) where the implied constant in the O(1) is independent of P . 2 ABELIAN SURFACES 4 Proof: First, we will make several reductions. Let Z = {z1, . . . , zn} be the set of points of A over which f is not an isomorphism. Then there is an effective divisor F on X and positive integers a1, . . . , an such that K+F = a1f −1(z1)+. . .+anf−1(zn). (To see this, take ai to be the maximum multiplicity of any component of the fibre f−1(zi) in K, and note that K is supported entirely on the exceptional set of f .) Conjecture 1.4 therefore becomes: ( ∑ i aihf−1(zi)(P ))− hF (P ) ≤ �hL(P ) +O(1) (3) Since F is effective, it suffices to assume that F = 0. It clearly also suffices to prove the theorem for n = 1 and a1 = 1, since if X1 and X2 are two blowups of A at a single point, then X1 is isomorphic to X2. Therefore, by adjusting �, we can reduce to the case n = 1 and a1 = 1, in which case f is the blowup of A at a single point, say, the origin. Denote the exceptional divisor of f by E. Furthermore, it suffices to prove the theorem for a single big divisor L, since if L′ is another big divisor on X, then there are constants c1 and c2 such that up to O(1), c1hL(P ) ≤ hL′(P ) ≤ c2hL(P ) for all P ∈ U(k) for some nonempty Zariski open set U ⊂ X. Thus, without loss of generality, we may assume that L = f ∗(pi∗1(O) +pi ∗ 2(O)), where pii:C×C → C is the ith projection, and O is the identity element of C. This choice of L is big but not ample. Therefore, we have reduced Conjecture 1.4 to the following inequality: hE(P ) ≤ �hL(P ) +O(1) Let v be any place of k. Then the local height hE,v can be written away from E as min{h(f◦pi1)∗Y,v, h(f◦pi2)∗Y,v} [Vo, Lemma 2.5.2], where hY,v is some choice of local height function for Y = (O), the divisor class of the identity element of C. Since the conclusion of the theorem is independent of the choice of height function, it suffices to assume that the exceptional height is given locally by this formula. Similarly, it suffices to assume that the height function hY is the canonical height function. For ease of notation, we will denote hY and hY,v by h and hv, respectively. Finally, we may assume that C is given in Weierstrass form y2 = x3 + ax+ b for some integers a, b. Fix a k-rational point Q0 on C of infinite order, and let e be the least integer such that for every element P ∈ C(k), there exists an element φ ∈ 2 ABELIAN SURFACES 5 Endk(C) = R such that eP = φ(Q0). Choose � > 0, and choose a positive integer M large enough so that � > 2/M , and assume that P = f−1(P1, P2) satisfies MhE(P ) ≥ hL(P ). Lemma 2.2 Let v be a finite place of k at which C and every element of R = End(C) have good reduction. Then for any two points P1, P2 ∈ C(k) and any two elements m1,m2 ∈ R, hv(m1P1 + m2P2) ≥ hE,v(f−1(P1, P2)), provided that m1P1 +m2P2 6= 0. Proof: At a prime of good reduction, we have 2hv(P ) = ordv(x(P )). If C does not have CM, the lemma then follows immediately from Theorem III.1.1 of [La], which essentially states that the set of points P ∈ C(k) with hv(P ) ≥ x (together with the identity) forms a group for any real number x > 0. If C does have CM, then one must show in addition that hv(φ(P )) ≥ hv(P ) for any endomorphism φ ∈ R, provided that φ(P ) 6= 0. For this, note that at a good prime, we have hv(P ) = ∑ j (logNk/Qv)i(Pj, P ·O, C) where Nk/Q denotes the usual norm, i(W,X · Y, Z) denotes the intersection multiplicity of a component W of the intersection of subschemes X and Y on a scheme Z (see for example [Fu, §7]). Let C denote the smooth Weierstrass model of C over Ov (the local ring of k at v), and P and O denote the closures of P and the origin O in C. The Pj are the points of the intersection of P and O. Since φ ∈ R, it induces an isomorphism φP :P → φ(P ) over Ov. From [Fu, Example 20.2.2], we may apply the projection formula [Fu, Example 7.1.9] to get for each component Q of the intersection of φ(P ) and O: i(Q, φ(P ) ·O, C) = i(φ−1P (Q), P ·O, C) If φ−1(Q) 6∈ O, then i(φ−1P (Q), P ·O, C) = 0. ¿From this we see that∑ Q i(Q, φ(P ) ·O, C) ≥∑ j i(Pj, P ·O, C) since for each j, Pj satisfies φ(Pj) ∈ φ(P ) ∩ O. It immediately follows that hv(φ(P )) ≥ hv(P ). ♣ 2 ABELIAN SURFACES 6 Let Γ be the submodule of C(k) generated by P1 and P2, and let P0 be the element of Γ which minimises h(P0). Let S denote the set of places of k which are infinite or at which C has bad reduction. From the lemma, it follows that hE,v(P ) ≤ hv(P0) for all finite places v 6∈ S. If for any divisor D on X we define hgD = ∑ v 6∈S hD,v, then we get h g E(P ) ≤ hg(P0)[= hgY (P0)]. By Siegel’s Theorem [Si, Theorem IX.3.1], for any place v, there are only finitely many points Q in C(k) such that (2M)hv(Q) ≥ h(Q). For any positive real numberN , letW (N) denote the finite set of points f−1(P1, P2) ∈ (X − E)(k) for which N ∑v∈S hv(Pi) > h(Pi) for both i = 1, 2. Then if P 6∈ W (2M) ∪ E(k), we must have hE(P ) = ∑ v∈S hE,v(P ) + h g E(P ) ≤ 1 2M hL(P ) + h g(P0) ≤ 1 2M hL(P ) + h(P0) By assumption, we know that MhE(P ) ≥ hL(P ), so (1/2M)hL(P ) + h(P0) ≥ (1/M)hL(P ) and hence h(P0) ≥ (1/2M)hL(P ). For i = 1, 2, write ePi = miQ0 for some mi ∈ R. Then there is an element m0Q0 ∈ eΓ such that: logNR(m0)/NR(gcd(m1,m2)) ≤ logB (4) where B > 1 is a real number depending only on the ring R and NR denotes the norm map NR:R→ Z. Thus, we have (1/2M)hL(P ) ≤ h(P0) ≤ (1/e)BNR(m0)h(Q0). But: (1/2M)hL(P ) = (1/2Me)(h(m1Q0) + h(m2Q0)) = ((NR(m1) +NR(m2))/2Me)h(Q0) so it follows that NR(m1) +NR(m2) NR(m0) ≤ 2MB (5) Inequality (5) says precisely that the pair (NR(m1)/NR(m0), NR(m2)/NR(m0)) 2 ABELIAN SURFACES 7 is a rational point lying inside the triangle in the first quadrant of the xy-plane bounded by the line x + y ≤ 2MB and the coordinate axes. Inequality (4) implies that in lowest terms, the denominators are bounded above by B, so there are only finitely many choices for such pairs. Since R is either Z or an order in an imaginary quadratic field, its unit group is finite. Therefore, the norm map NR is finite-to-one. Again by inequality (4), there are therefore only finitely many choices for the pair (m1/m0,m2/m0) ∈ R2. We may choose a finite set of lines L1, . . . , Ln through the origin in R2 so that each possible pair (m1/m0,m2/m0) lies on one of the lines. For any pair α = (a, b) ∈ R2, denote by Dα the image of the map gα:C → C×C defined by gα(X,Y ) = (aX, bY ). Then P must lie on one of the finitely many curves f−1(Dαi), where αi = (ai, bi) is any point on Li. If we set V (�) to be the union of those curves with the set W (2M) ∪ E(k), then we have proven the theorem. One final note about effectivity. The curves f−1(Dαi) and E are obvi- ously effectively computable, but W (2M) is a little more elusive, since it is a byproduct of Siegel’s Theorem. There are effective versions of Siegel’s Theorem however (see for example [Ba]). ♣ Remark: It ought to be a straightforward matter to extend Theorem 2.1 to the case in which C is an arbitrary abelian variety whose k-rational points are a rank one module over End(C). In the proof, Siegel’s Theorem would have to be replaced by Faltings’ result quoted below (Theorem 2.6), which would destroy the effectivity, although only for the zero-dimensional part of the exceptional set V (�). Now we will show how our main result can be applied to rational points on curves. Let A/k be any abelian surface, defined over a number field k. Let P be a k-rational pencil of curves on A. We have the following results: Theorem 2.3 Assume Conjecture 1.4 holds for all blowups of an abelian surface A. Let P be any pencil of curves on A. Then the average number of k-rational points on a general curve in P is finite, if the curves in P are ordered by height. Corollary 2.4 (Faltings’ Theorem for curves on A) Assume that Con- jecture 1.4 holds for all blowups of an abelian surface A. Then for any pencil P on A, almost all curves of P have only finitely many k-rational points. 2 ABELIAN SURFACES 8 Remark: Note that for any curve C on A, we can construct a pencil P of curves on A such that C is a component of some curve in P . Proofs: We will prove both results at once. Let pi:X → A be a resolution of the rational map corresponding to P as in §2, and let f :X → P1 be the corresponding fibration. Let C be the divisor class of P , and let L = pi∗(C) be the corresponding big divisor on X. Let E be the exceptional locus of pi. By hypothesis, we know Conjecture 1.4 for X, so we get: hK(P ) ≤ �hL(P ) (6) for some dense open set U(�) and any P ∈ U(�). Since K −E is effective, it follows that hE(P ) ≤ �hL(P ). (The set U(�) may need to shrink slightly for this.) Let F = f ∗(O(1)) be the divisor class of a fibre of f . Then F − L+mE is effective for some positive integer m, so by choosing � < 1/m we get: hF (P ) ≥ αhL(P ) (7) for some α > 0. Since L is big, this implies that there is a dense open subset of V on which hL satisfies a Northcott-type finiteness theorem. Thus, there is a dense open subset U of V such that for any real number B, the set {P ∈ U(k) | hF (P ) ≤ B} is finite. Since hF (P ) depends only on the fibre of f on which P lies, it follows that every fibre of f not disjoint from U has only finitely many rational points. Moreover, the set {P ∈ U(k) | αhL(P ) ≤ B} has cardinality �� (logB)r/2, where r is the Mordell-Weil rank of A over k, and so the same will be true for {P ∈ U(k) | αhL(P ) ≤ B}. The set {P ∈ P1(k) | hO(1)(P ) ≤ B} has cardinality �� B2 [Sch]. Therefore, there must be a density one set of points P ∈ P1(k) whose preimages under f contain no k-rational points in U(k). Since only finitely many fibres of f are not disjoint from U(k), the average number of points on curves in P must be finite. ♣ It would be nice to be able to apply Theorem 2.1 directly to Theorems 2.4 and 2.3 to deduce unconditional results about rational points on curves. Unfortunately, these theorems only apply when the basepoints of the pencil are k-rational, and if we extend k to include the basepoints of the pencil, the hypothesis on the rank of C(k) may become false. However, there are some cases in which we do have a basepoint locus which splits completely over k. For instance, we have the following result: 2 ABELIAN SURFACES 9 Corollary 2.5 Let C be an elliptic curve of rank one over a number field k, or of rank two over k with k-rational complex multiplication. Let A = C×C, and let F1 and F2 be linearly equivalent effective divisors whose components are fibres of the two projection maps A → C. Assume F1 and F2 intersect properly, and let P be the pencil of curves through F1 and F2. Then every component of a curve in P either has genus 1, or has finitely many k-rational points. Moreover, there exists an effectively computable Zariski dense open set U such that the average number of k-rational points in Z(k) ∩ U(k) is zero, where Z varies amongst all curves in P, ordered by height. Remarks: Note that divisors described in the theorem are easy to construct. Let D1 be any very ample divisor on C which is a sum of k-rational points. Then we can certainly find a disjoint divisor D2 which is linearly equivalent to D1 and is also a sum of k-rational points. We may then set Fi = pi ∗ 1Di+pi ∗ 2Di. Note also that every curve in P has several rational points, corresponding to the basepoints of the pencil. These points are not in U(k), which consists of the intersection of the set U(1/2) from Theorem 2.1 with its translates by the basepoint locus of P . This result is related to the results of Manin and Demjanenko (see for example [Se, §5.2]). They prove, for example, that any curve of genus greater than one which admits more than r different k-rational morphisms to an elliptic curve of k-rank r has only finitely many rational points. The first part of Corollary 2.5 is contained in this result, but the second part is not. Proof: By blowing up the basepoint locus of P to get a surface p:X → A, we obtain a map f :X → P1 whose fibres are the curves in P . Let E be the exceptional locus of p, and let F = f ∗(O(1)). Then the canonical divisor on X is just E, since the basepoint locus of P is reduced, and F + E is big. (Indeed, it’s linearly equivalent to p∗F1 and p∗F2.) By Theorem 2.1, there exists an effectively computable dense open subset U of X such that for every P ∈ U(k), we have the following inequality: hE(P ) ≤ (1/2)hF+E(P ) This implies imediately that: hF (P ) ≥ (1/2)hF+E(P ) Since F + E is big, we may shrink U slightly so that the set {P ∈ U(k) | hF+E(P ) ≤ B} is finite for any real number B. Thus, the set {P ∈ U(k) | 2 ABELIAN SURFACES 10 hF (P ) ≤ B} must also be finite. But hF (P ) depends only on f(P ), so it follows that every curve in P not disjoint from U must contain only finitely many k-rational points. Since the complement of U is a union of points and curves of genus 1, the first part of the corollary follows. For the second part, we will compare the counting functions for U relative to hF and hF+E. We have: #{P ∈ P1(k) | hO(1)(P ) ≤ B} �� B2 #{P ∈ U(k) | hF+E(P ) ≤ B} �� logB � B2 so since hF (P ) = hO(1)(f(P )), it follows that almost all fibres of f cannot have any k-rational points at all in U , and hence the average number of such rational points must be 0. ♣ Theorem 2.1 also implies the full strength of Conjecture 1.1 for X by way of a result of Faltings [Fa, Theorem 2]: Theorem 2.6 (Faltings, 1991) Let A be an abelian variety, hL a height associated to an ample divisor L on A, and � ≥ 0 any positive real number. Let D be any divisor on A. Then for all but finitely many k-rational points P on A−D, the following inequality holds: mS(D,P ) ≤ �hL(P ) Remark: Faltings proves this result only in the case that S contains a single place and D is irreducible, but this is obviously equivalent to the more general result. He also proves considerably more than this, but the above version will suffice for our purposes. ¿From this we can deduce the following corollary of Theorem 2.1: Corollary 2.7 (Vojta’s Main Conjecture) Let X, L, and K be as in Theorem 2.1, and choose height functions hL and hK. Let D be any divi- sor on X, and fix a finite set S of places of k. For any � > 0, there is a Zariski closed set V (�) such that for all k-rational points P ∈ (X−V (�))(k), we have the following inequality: mS(D,P ) + hK(P ) ≤ �hL(P ) +O(1) where the implied constant in the O(1) is independent of P . 3 RATI