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Applications of Arithmetic Algebraic Geometry to Diophantine Approximations

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Applications of Arithmetic Algebraic Geometry to Diophantine Approximations 164 Applications of Arithmetic Algebraic Geometry to Diophantine Approximations Paul Vojta∗ Department of Mathematics University of California Berkeley, CA 94720 USA Contents 1 History; integral and rational points 2 Siegel’s lemma 3 The index 4 Sketch of the proof of Roth’s theorem 5 Notation 6 Derivatives 7 Proof of Mordell, with some simplifications by Bombieri 8 Proof using Gillet-Soul´eRiemann-Roch 9 The Faltings complex 10 Overall plan 11 Lower bound on the space of sections 12 More geometry of numbers 13 Arithmetic of the Faltings complex 14 Construction of a global section 15 Some analysis 16 More derivatives 17 Lower bound for the index 18 The product theorem Let us start by recalling the statement of Mordell’s conjecture, first proved by Faltings in 1983. Theorem 0.1. Let C be a curve of genus > 1 defined over a number field k . Then C(k) is finite. In this series of lectures I will describe an application of arithmetic algebraic geom- etry to obtain a proof of this result using the methods of diophantine approximations (instead of moduli spaces of abelian varieties). I obtained this proof in 1989 [V 4]; it was followed in that same year by an adaptation due to Faltings, giving the following more general theorem, originally conjectured by Lang [L 1]: ∗Partially supported by NSF grant DMS-9001372 165 Theorem 0.2 ([F 1]). Let X be a subvariety of an abelian variety A , and let k be a number field over which both of them are defined. Suppose that there is no ¯ ¯ nontrivial translated abelian subvariety of A ×k k contained in X ×k k . Then the set X(k) of k-rational points on X is finite. In 1990, Bombieri [Bo] also found a simplification of the proof [V 4]. While it does not prove any more general finiteness statements, it does provide for a very elementary exposition, and can be more readily used to obtain explicit bounds on the number of rational points. Early in 1991, Faltings succeeded in dropping the assumption in Theorem 0.2 that ¯ X ×k k not contain any translated abelian subvarieties of A , obtaining another conjec- ture of Lang ([L 2], p. 29). Theorem 0.3 ([F 2]). Let X be a subvariety of an abelian variety A , both assumed to be defined over a number field k . Then the set X(k) is contained in a finite union S i Bi(k) , where each Bi is a translated abelian subvariety of A contained in X . The problem of extending this to the case of integral points on subvarieties of semiabelian varieties is still open. One may also rephrase this problem as showing finiteness for the intersection of X with a finitely generated subgroup Γ of A(Q) . The same sort of finiteness question can then be posed for the division group {g ∈ A(Q) | mg ∈ Γ for some m ∈ N }; this has recently been solved by M. McQuillan (unpublished); see also [Ra]. Despite the fact that arithmetic algebraic geometry is a very new set of techniques, the history of this subject goes back to a 1909 paper of A. Thue. Recall that a Thue equation is an equation f(x, y) = c, x, y ∈ Z where c ∈ Z and f ∈ Z[X, Y ] is irreducible and homogeneous, of degree at least three. Thue proved that such equations have only finitely many solutions. The lectures start, therefore, by recalling some very classical results. These include a lemma of Siegel which constructs small solutions of systems of linear equations and, later, Minkowski’s theorem on successive minima. Next follows a brief sketch of the proof of Roth’s theorem. It is this proof (or, more precisely, a slightly earlier proof due to Dyson) which motivated the new proof of Mordell’s conjecture. After that, we will consider how to apply the language of arithmetic intersection theory to this proof, and prove Mordell’s conjecture using some of the methods of Bombieri. This will be followed by the original (1989) proof using the Gillet-Soul´e Riemann-Roch theorem. These proofs will only be sketched, as they are written in detail elsewhere, and newer methods are available. Finally, we give in detail Faltings’ proof of Theorem 0.3, with a few minor simpli- fications. 166 In this paper, places v of a number field k will be taken in the classical sense, so that places corresponding to complex conjugate embeddings into C will be identified. Also, absolute values k·kv will be normalized so that kxkv = |σ(x)| if v corresponds to 2 a real embedding σ : k ,→ R ; kxkv = |σ(x)| if v corresponds to a complex embedding, −ef and kpkv = p if v is p-adic, where p is ramified to order e over a rational prime p and f is the degree of the residue field extension. With these normalizations, the product formula reads Y (0.4) kxkv = 1, x ∈ k, x 6= 0. v A line sheaf on a scheme X means a sheaf which is locally isomorphic to OX ; i.e., an invertible sheaf. Similarly a vector sheaf is a locally free sheaf. More notations appear in Definition 2.3 and in Section 5. §1. History; integral and rational points In its earliest form, the study of diophantine approximations concerns trying to prove that, given an algebraic number α , there are only finitely many p/q ∈ Q (written in lowest terms) satisfying an inequality of the form p c − α < q |q|κ for some value of κ and some constant c > 0 . It took many decades to obtain the best value of κ : letting d = [Q(α): Q] , the progress is as follows: κ = d, c computable Liouville, 1844 d+1 κ = 2 + Thue, 1909 κ = min{ d + s − 1 | s = 2, . , d} + Siegel, 1921 √ s κ = 2d + Dyson, Gel’fond (independently), 1947 κ = 2 + Roth, 1955 Of course, stronger approximations may be conjectured; e.g., p −2 −1− − α < c|q| (log q) . q See ([L 3], p. 71). Beginning with Thue’s work, these approximation results can be used to prove finiteness results for certain diophantine equations, as the following example illustrates. Example 1.1. The (Thue) equation 3 3 (1.2) x − 2y = 1, x, y ∈ Z has only finitely many solutions. Indeed, this equation may be rewritten x √ 1 − 3 2 = √ √ . y y(x2 + 3 2xy + 3 4y2) 167 But for |y| large the absolute value of the right-hand side is dominated by some multiple of 1/|y|3 ; if (1.2) had infinitely many solutions, then the inequalities of Thue, et al. would be contradicted. For a second example, consider a particular case of Mordell’s conjecture (Theorem 0.1). Example 1.3. The equation 4 4 4 (1.4) x + y = z , x, y, z ∈ Q in projective coordinates (or x4 + y4 = 1 in affine coordinates) has only finitely many solutions. The intent of these lectures is to show that Theorem 0.1 can be proved by the methods of diophantine approximations. At first glance this does not seem likely, since it is no longer true that solutions must go off toward infinity. But let us start by considering how, in the language of schemes, these two problems are very similar. In the first example, let W = Spec Z[X, Y ]/(X3 − 2Y 3 − 1) and B = Spec Z be schemes, and let π : W → B be the morphism corresponding to the injection 3 3 Z ,→ Z[X, Y ]/(X − 2Y − 1). Then solutions (x, y) to the equation (1.2) correspond bijectively to sections s: B → W of π since they correspond to homomorphisms 3 3 Z[X, Y ]/(X − 2Y − 1) → Z,X 7→ x, Y 7→ y and the composition of these two ring maps gives the identity map on Z . In the second example, let W = Proj Z[X, Y, Z]/(X4 + Y 4 − Z4) and B = Spec Z . Then sections s: B → W of π correspond bijectively to closed points on the generic fiber of π with residue field Q . In one direction this is the valuative criterion of properness, and in the other direction the bijection is given by taking the closure in W . These closed points correspond bijectively to rational solutions of (1.4). Thus, in both cases, solutions correspond bijectively to sections of π : W → B . The difference between integral and rational points is accounted for by the fact that in the first case π is an affine map, and in the second it is projective. Note that, in the second example, any ring with fraction field Q can be used in place of Z as the affine ring of B (by the valuative criterion of properness). But, in the 1 case of integral points, localizations of Z make a difference: using B = Spec Z[ 2 ] , for example, allows solutions in which x and y may have powers of 2 in the denominator. §2. Siegel’s lemma Siegel’s lemma is a corollary of the “pigeonhole principle.” Actually, the idea dates back to Thue, but he did not state it explicitly as a separate lemma. 168 Lemma 2.1 (Siegel’s lemma). Let A be an M × N matrix with M < N and having entries in Z of absolute value at most Q . Then there exists a nonzero vector N x = (x1, . , xn) ∈ Z with Ax = 0 , such that h M/(N−M)i |xi| ≤ (NQ) =: Z, i = 1, . , N. Proof. The number of integer points in the box (2.2) 0 ≤ xi ≤ Z, i = 1,...,N is (Z + 1)N .
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