Über die Autorin bzw. den Autor

Umberto Zannier is professor of mathematics at the Scuola Normale Superiore di Pisa in Pisa, Italy. He is the author of Lecture Notes on Diophantine Analysis and the editor of Diophantine Geometry.

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Some Problems of Unlikely Intersections in Arithmetic and Geometry

PRINCETON UNIVERSITY PRESS

Contents

Preface.......................................................................................................................ixNotation and Conventions......................................................................................................xiIntroduction: An Overview of Some Problems of Unlikely Intersections..........................................................11 Unlikely Intersections in Multiplicative Groups and the Zilber Conjecture...................................................152 An Arithmetical Analogue....................................................................................................433 Unlikely Intersections in Elliptic Surfaces and Problems of Masser..........................................................624 About the André-Oort Conjecture........................................................................................96Appendix A Distribution of Rational Points on Subanalytic Surfaces by Umberto Zannier.........................................128Appendix B Uniformity in Unlikely Intersections: An Example for Lines in Three Dimensions by David Masser.....................136Appendix C Silverman's Bounded Height Theorem for Elliptic Curves: A Direct Proof by David Masser.............................138Appendix D Lower Bounds for Degrees of Torsion Points: The Transcendence Approach by David Masser.............................140Appendix E A Transcendence Measure for a Quotient of Periods by David Masser..................................................143Appendix F Counting Rational Points on Analytic Curves: A Transcendence Approach by David Masser..............................145Appendix G Mixed Problems: Another Approach by David Masser...................................................................147Bibliography..................................................................................................................149Index.........................................................................................................................159

Chapter One

As anticipated in the introduction, in this first chapter we shall describe some results of unlikely intersections in the case of multiplicative algebraic groups (also called "tori") [??]ⁿ_m, together with a sketch of some of the proofs. (The important analogue for abelian varieties shall be discussed later in Chapter 3 with other methods.)

Remark 1.0.1 Algebraic subgroups and cosets. Before going on, it shall be convenient to recall briefly the simple theory giving the structure of algebraic subgroups and cosets of [??]ⁿ_m. (Simple proofs may be found, e.g., in [BG06], Ch. 3.)

Every algebraic subgroup G of [??]ⁿ_m may be defined by equations x^a = 1 (on denoting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), where the vector a = (a₁; ...; a_n) runs through a lattice Λ = Λ_G [subset] Zⁿ; of course it suffices to choose equations corresponding to a basis of Λ. The correspondence G ↔ Λ_G is one-to-one. If Λ_G has rank r then G has dimension n - r, and is irreducible if and only if Λ_G is primitive (i.e., is a factor of Zⁿ). We say that G is proper if G [not equal to]] [??]ⁿ_m.

Any irreducible algebraic subgroup G is also called a "subtorus" and, setting d := dim G, it becomes isomorphic (as an algebraic group) to G^d_m, the isomorphism being induced by a suitable monomial change of coordinates x_i → x^ai on [??]ⁿ_m. Such a G may be also parametrized by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The torsion points in [??]ⁿ_m are those whose coordinates are roots of unity; they are Zariski-dense in any algebraic subgroup G. Further, any coset of G in [??]ⁿ_m may be defined by equations x^a = c_a, a [member of] Λ_G, for suitable nonzero constants c_a; this coset is a torsion coset (i.e., t a translate of G by a torsion point) if and only if the c_a are roots of unity (which amounts to the fact that it contains a torsion point). Any torsion coset of an irreducible G is a component of an algebraic subgroup of the same dimension.

As usual, we shall denote by [l] the multiplication-by-l map on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It is not too di_cult to prove that if X is a nonempty irreducible subvariety of [??]ⁿ_m such that [l]X [subset] X for an integer l > 1, then X is a torsion coset (and conversely). (See, for instance, [Zan09], Theorem 4.6. A proof may be also obtained by projection, from the simpler case of hypersurfaces, for instance similarly to the argument in [BZ95]; see further Remark 1.1.1 in the next section for some details.)

1.1 Torsion points on subvarieties of [??]ⁿ_m

Let us start with Lang's original problem of torsion points on plane curves in [??]²_m and its generalization to higher dimensions, i.e., describing the torsion points on a subvariety X [subset] [??]ⁿ_m. Before offering a general statement in this direction, let us recall that equations in roots of unity go back to long ago; for example, P. Gordan studied in [Gor77] the equation cos 2πx + cos 2πy + cos 2πz = -1, in rationals x; y; z, with the purpose of classifying finite subgroups of PGL₂(C). Other equations of similar shape arise in the enumeration of polytopes satisfying suitable conditions. See [CJ76] for a brief description of this, and also for a general theory of trigonometric diophantine equations (in the authors' terminology), aimed to reduce every mixed diophantine equation (in angle-variables and usual variables) to a usual one. Such paper was also partly inspired by H.B. Mann's [Man65], classifying solutions in roots of unity to linear equations with rational coe_cients. (We shall see later that these results are relevant also for Lang's issue, and represent one of the possible tools to achieve a complete solution of it.)

Coming back to Lang's formulation, as mentioned in the introduction, work of M. Laurent [Lau84] and independently of Sarnak-Adams [SA94] led to the following general result:

Theorem 1.1. Torsion points theorem. Let Σ be any set of torsion points in [??]ⁿ_m([bar.Q]). The Zariski closure of Σ is a finite union of torsion cosets.

This may be reformulated by saying that

The torsion points in a subvariety X [subset] [??]ⁿ_m all lie and are Zariski-dense in a finite number of torsion cosets contained in X. In particular, if X is irreducible, the torsion points in X are Zariski-dense if and only if X is a torsion coset.

This result may be proved in several different ways. Since the arguments are elementary and the essentials may be explained in short, we recall some of the possible approaches.

Let first X be an irreducible curve in [??]²_m, as in the case of the original problem of Lang, and let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be a torsion point on X, of exact order N (i.e., (r, s, N) = 1). We have to prove that either N is bounded (in terms of X) or X is a torsion coset. If X is defined by an irreducible polynomial equation f(x, y) = 0, to be a torsion coset amounts to f being, up to a monomial factor, of the shape x^ay^b - ρ, for a, b [member of] Z and ρ a root of unity.

- Intersecting X and [l]X. One approach consists in observing that a conjugation σ [member of] Gal([bar.Q]/Q) sends the torsion point ζ to a power ζ^l, where l = l_σ may be chosen as any integer coprime to N. Now, if ζ [member of] X, then ζ^σ = ζ^l lies in X_. If we choose σ fixing a number field k of definition for X we then have [l] ζ := ζ^l 2 X (where [l] denotes the algebraic map of multiplication by l in [??]ⁿ_m); and of course [l] ζ [member of] [l]X.

This fact also holds for the conjugates of ζ over k, hence either [l]X and X have a component in common (which amounts to being equal, since they are irreducible) or by Bezout's theorem we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As recalled in the opening Remark 1.0.1, the first alternative, which amounts to [l]X [subset] X, may be shown to entail that X is a torsion coset, as in the following:

Remark 1.1.1 [l]X [subset] X implies that X is a torsion coset. For the present case of curves this implication may easily be proved directly: for instance, note that if [l]X [subset] X then f(θx; ηy) divides f(x^l; y^l) for all l-th roots of unity θ, η. Comparison of degrees then shows that X is invariant by at least l distinct multiplicative translations (x, y) [??] (θx, ηy), where (θ, η) has order dividing l (and varies in a set of ≥ l elements). Replacing l by powers l^m, m = 1; 2; ..., one deduces that X is invariant under multiplicative translations X → τ · X by an infinite set of τ [member of] [??]²_m; and certainly it is invariant by multiplicative translation by any element in the Zariski-closure of that set. This Zariski-closure must contain a curve Z, and Z · X [subset] X, so X must be a translate of Z and now one easily deduces Z · Z [subset] τ Z and the sought conclusion follows. Such a strategy applies in any dimensions; see also [Zan09], Theorem 4.6, for a different general argument, and see further [Hin88], Lemme 10, for another method relying on degrees and valid for any extension of a complex abelian varieties by a torus. (Still another, p-adic analytic, argument related to the Skolem-Mahler-Lech theorem is given in the appendix to [Lan65].)

In the other cases, the estimate deg([l]X) ≤ l · deg X⁵ yields [k(ζ) : k] ≤ l · (deg X)².

On the other hand, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so on choosing l "small" and such that σ fixes k we deduce that N is bounded, concluding the argument.

This approach is the one adopted in the first solutions to Lang's question by J-P. Serre and by J. Tate. (Lang's paper [Lan65] reproduces this, together with another argument by Y. Ihara; see also [Lan83], see Sec. 8.6.)

Finally, note that actually for this argument we do not need that X^σ = X: in fact, arguing as above would lead either to the sought bound or to [l]X [subset] X^σ; but an iteration of this yields [l^m]X [subset] X for some m ≥ 1, so again we deduce that X is a torsion coset. This observation is relevant if we want to forget about the field of definition of X; in fact, a detailed analysis on these lines, with careful choice of σ, was carried out by F. Beukers and C. Smyth [BS02]; in particular, they use that any root of unity ζ has a conjugate of the shape -ζ or ± ζ², and obtain upper bounds on the number of torsion points remarkably dependent only on the degree of X (in fact, bounded by 22 times the area of the Newton polygon of f(x, y)).

- Constructing auxiliary monomials. Another, partially similar, approach, due to Liardet, consists in constructing a nontrivial monomial µ = x^ay^b, a, b [member of] Z, of \small" degree, such that µ(ζ) = 1; we want ar + bs ≡ 0 (mod N) and by an easy application of the pigeon-hole principle this may be done with 0 < deg(µ) := max(|a|; |b|) ≤ [square root of N]: consider, for instance, the ([[square root of N] + 1)² pairs (u, v) [member of] Z² with 0 ≤ u; v ≤ [square root of N], and take the difference of two pairs producing a same value ur + vs (mod N).

Then ζ lies in the intersection of X with the curve H defined by µ = 1 (which is an algebraic subgroup of [??]²_m), and the same holds for the conjugates of ζ over k. By Bezout's theorem again, one may conclude that either φ(N) [much less than]_k [square root of N] deg X or H contains X. In the first case, N is bounded. In the second case, i.e., X [subset] H, we again easily find (on factoring x^ay^b - 1) that X must be of the predicted special shape. Note that in this argument the bound takes the shape N [much less than]_{k,[member of]} (deg X)^{2+[member of]} for every [member of] > 0, where the exponent `2' is best-possible. (See [Zan09], Prop. 4.1 and Ex. 4.4, for more details.)

Remark 1.1.2 Taking conjugates. We remark at once that the idea of taking conjugates, present in both proofs that we have seen, shall appear again in different shapes in many of the problems considered in our future discussions, and has proved to be fundamental in the context. Often, Bezout's theorem may then be used as above, to compare estimates; eventually one concludes that either the degree (over Q) of the torsion point is bounded (and then the same holds for its order N) or we have a geometrical consequence for which we fall in the special varieties. (In the present case they are the torsion cosets.) For instance, an advanced version of this kind of idea, relying on deep results by Serre on homotheties in the image of Galois on torsion points, led M. Hindry [Hin88] to a quantitative proof of the Manin-Mumford statement (to be discussed later in Chapter 3 with different methods).

- Considering heights. Still another approach, working for arbitrary X and ambient dimension n, comes from considerations of heights; we explain the essentials. Let f(ζ) = 0, where f [member of] Z[x] and ζ [member of] [??]²_m is a torsion point. Then we have by Fermat's Little Theorem, f(ζ^p) ≡ f(ζ)^p = 0 (mod p) (in the ring Z[ζ]). Now, ζ^p has zero Weil height, so if p is large enough with respect to f, this congruence yields f(ζ^p) = 0. Hence we gain a new equation; actually, applying this argument to a set of de_ning equations for X shows that if the torsion points are Zariski-dense in X we must have X [subset] [p]^-1 X; but this implies [p]X [subset] X for large p, which as already noted in turn implies that X is special.

This procedure may be carried out for arbitrary varieties, as done in [BZ95], and leads more generally to a positive lower bound for the height of algebraic points in X, valid outside the translates of torsion cosets (i.e., a solution of the toric case of Bogomolov problems, first obtained by S. Zhang, recalled below in Remark 1.1.7); see [BZ95] or [Zan09] for the details of this approach.

Remark 1.1.3 Weil height. We have referred here to the logarithmic Weil absolute height on [bar.Q]. We refer, e.g., to [BG06] for this, but we recall here that this height may be defined as follows: if α [member of] [bar.Q] has minimal polynomial a₀(x - α₁) ··· (x - α_d) over [??], we have dh(α) = log |a₀| + [summation]^d_i=1 max(0; log |α_i). One usually writes H(α) := exp h(α). Another, equivalent, definition is in terms of absolute values: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where k is any number field containing α and v runs through the places of k, suitably normalized in terms of k (which is important to make the definition independent of k).

For instance, the Height of a rational number p=q in lowest terms is H(p/q) = max(|p|; |q|).

This height satisfies h(x^-1) = h(x), h(xy) ≤ h(x) + h(y), h(x + y) ≤ h(x) + h(y) + log 2. It vanishes precisely at 0 and at the roots of unity. (See [BG06].)

There is also a related notion of Weil height h(x₀ : ... : x_n) for algebraic points in projective spaces P_n: see (2.1.1); we have h(x₀ : x₁) = h(x₁/x₀) if x₀ [not equal to] 0.

(Continues...)

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