The Tate-Voloch Conjecture in a Power of a Modular Curve

Let $p$ be a prime. Tate and Voloch proved that a point of finite order in the algebraic torus cannot be $p$-adically too close to a fixed subvariety without lying on it. The current work is motivated by the analogy between torsion points on semi-abelian varieties and special or CM points on Shimura varieties. We prove the analog of Tate and Voloch's result in a power of the modular curve Y(1) on replacing torsion points by points corresponding to a product of elliptic curves with complex multiplication and ordinary reduction. Moreover, we show that the assumption on ordinary reduction is necessary.Comment: Corrected some typos in version

Bad reduction of genus 22 curves with CM jacobian varieties

We show that a genus $2$ curve over a number field whose jacobian has complex multiplication will usually have stable bad reduction at some prime. We prove this by computing the Faltings height of the jacobian in two different ways. First, we use a formula by Colmez and Obus specific to the CM case and valid when the CM field is an abelian extension of the rationals. This formula links the height and the logarithmic derivatives of an $L$-function. The second formula involves a decomposition of the height into local terms based on a hyperelliptic model. We use results of Igusa, Liu, and Saito to show that the contribution at the finite places in our decomposition measures the stable bad reduction of the curve and subconvexity bounds by Michel and Venkatesh together with an equidistribution result of Zhang to handle the infinite places

A Note on Divisible Points of Curves

Let $C$ be an irreducible algebraic curve defined over a number field and inside an algebraic torus of dimension at least 3. We partially answer a question posed by Levin on points on $C$ for which a non-trivial power lies again on $C$. Our results have connections to Zilber's Conjecture on Intersections with Tori and yield to methods arising in transcendence theory and the theory of o-minimal structures.Comment: Published version, but with an error fixed in the formula for the function on page

O-minimality and certain atypical intersections

We show that the strategy of point counting in o-minimal structures can be applied to various problems on unlikely intersections that go beyond the conjectures of Manin-Mumford and Andr\'e-Oort. We verify the so-called Zilber-Pink Conjecture in a product of modular curves on assuming a lower bound for Galois orbits and a sufficiently strong modular Ax-Schanuel Conjecture. In the context of abelian varieties we obtain the Zilber-Pink Conjecture for curves unconditionally when everything is defined over a number field. For higher dimensional subvarieties of abelian varieties we obtain some weaker results and some conditional results

Singular Moduli that are Algebraic Units

We prove that only finitely many j-invariants of elliptic curves with complex multiplication are algebraic units. A rephrased and generalized version of this result resembles Siegel’s theorem on integral points of algebraic curves

Heights and multiplicative relations on algebraic varieties

Points on a subvariety X of a semi-abelian variety A that are contained in a subgroup, let the subgroup be of finite rank or algebraic, are subject to severe restrictions arithmetical nature. Finiteness results for intersections of X with subgroups of finite rank have been studied by Faltings, Hindry, Laurent, McQuillan, Raynaud, Vojta and others. More recently several authors ([CZ00], [BMZ99], [BMZ03], [BMZ06a], [BMZ06b], [BMZ04], [Via03], [RV03], [R´em05b], [R´em07], [Pin05b], [Zan00], [Zil02], [Mau06]) have considered the intersection of X with A[r], the set of complex points in A contained in an algebraic subgroup of codimension greater or equal to r. If H is a fixed algebraic subgroup of A with codimension strictly less than dimX, then a dimension counting argument shows that X\H is either empty or contains a curve. As we are allowing H to vary with fixed codimension, the intersection X \ A[r] may be quite large if r < dimX. In this thesis we are only interested in the case r � dimX. If not stated otherwise we will also assume throughout the introduction that all varieties are defined over Q, the field of algebraic numbers. One can define a height function on the set of algebraic points of A. Throughout this thesis we work only in the algebraic torus Gn m or an abelian variety. So we can take the Weil height or the N´eron-Tate height associated to an ample line bundle. We will pursue two types of questions. First, for which r does the set X0(Q) \ A[r] have bounded height and how do these bounds depend on X? Second, for which r is the set X00(Q) \ A[r] finite? Here X0 and X00 are obtained from removing from X certain subvarieties in order to to eliminate trivial counterexamples. For example if X is a proper algebraic subgroup of Gn m with positive dimension, then there is no hope for a boundedness of height or finiteness result for U(Q) \ (Gn m)[r] if r � dimX and if U is Zariski open and dense in X. In this case X0 and X00 are both empty. The simplest non-trivial example seems to be the curve defined by x + y = 1 in G2 m. Here we can take X0 and X00 to equal our curve. Algebraic subgroups of G2 m can be described by at most two monomial relations x�y� = 1 with integer exponents � and �. For subgroups of dimension 1, one non-trivial relation suffices. If (x, y) is contained in such a subgroup then x and y are called multiplicatively dependent. Hence the intersection of our curve with the union of all proper algebraic subgroups of G2 m can be described by the solutions of (0.0.1) x�(1 − x)� = 1. This is an equation in three unknowns x, �, and �, so one should not expect finitely many solutions. Indeed, taking x 6= 1 a root of unity gives infinitely many solutions. In [CZ00] Cohen and Zannier showed that if H denotes the absolute non-logarithmic Weil height then (0.0.1) implies the sharp inequality max{H(x),H(1−x)} � 2. In chapter 2 we start off by giving an alternative proof of Cohen and Zannier’s Theorem. We even show that the possibly larger height H(x, 1−x) is at most 2. In their paper, Cohen and Zannier also proved that 2 is an isolated point in the range of max{H(x),H(1−x)}. We make this result explicit in Theorem 2.2, working instead with H(x, 1 − x). The proof applies Smyth’s Theorem on lower bounds for heights of non-reciprocal algebraic numbers and a Theorem of Mignotte. As was already noticed in [CZ00], solutions of (0.0.1) are closely linked to roots of certain trinomials whose coefficients are roots of unity. In chapter 3 Theorem 3.2 we follow this avenue by factoring such trinomials over cyclotomic fields. Having essentially a minimal polynomial in our hands, we obtain a new proof for the boundedness of H(x, 1 − x) with x as in (0.0.1). More importantly, in Theorem 3.1 we show that not only is 2 isolated in the range of the height function, but also that H(x, 1−x) converges to an absolute constant if [Q(x) : Q] goes to infinity. The proof determines the value of this limit: it is the Mahler measure of the two-variable polynomial X + Y − 1. In a certain sense this Mahler measure is the height of the curve in our problem. In Theorem 3.3 we prove a conjecture of Masser stated in [Mas07]: the number of solutions of (0.0.1) with [Q(x) : Q] � D is asymptotically equal to c0D3 with c0 = 2.06126 . . . as D ! 1. The constant c0 is defined properly in chapter 3 as a converging series. This counting result is a further application of Theorem 3.2. In chapter 4 we generalize the method from chapter 2 to bound the height of multiplicatively dependent solutions of (0.0.2) x + y = �. Here � is now any non-zero algebraic number. In [BMZ99] Bombieri, Masser, and Zannier prove a more general result which also implies boundedness of height in this case. Their Proposition A leads to an explicit upper bound for the height; the bound is polynomial in H(�). We are mainly interested in upper bounds for H(x, y) which have good dependency in H(�). The value H(�) can be regarded as the height of the defining equation (0.0.2). In Theorems 4.1 and 4.2 we get the bound H(x, y) � 2H(�) min{H(�), 7 log(3H(�))}. By Theorem 4.3 the exponent of the logarithm cannot be less than 1. But in some special cases, e.g. if � is a rational integer, we improve the upper bound to 2H(�), see Theorem 4.4. In this theorem we also show that if � is a rational integer then 2H(�) is attained as a height if and only if � is a power of two. Thus if � is a power of two, then our bound is sharp. For such � and if also � � 2 we prove in Theorem 4.5 that 2H(�) is isolated in the range of the height. Starting from chapter 6 we work in an algebraic torus of arbitrary dimension. Algebraic subgroups can still be described by a finite set of monomial equations. For example (x1, . . . , xn) 2 Gn m(C) is contained in a proper algebraic subgroup if and only if the xi satisfy a non-trivial multiplicative relation. In [BMZ99] Bombieri, Masser, and Zannier proved that if X is an irreducible curve which is not contained in the translate of a proper algebraic subgroup, then points on X that lie in a proper algebraic subgroup have bounded height. Moreover, they showed that this statement is false if X is contained in the translate of a proper algebraic subgroup. The authors also showed that there are only finitely many points on X that lie in an algebraic subgroup of codimension at least 2. This finiteness result was generalized by the same authors in [BMZ03] to algebraic curves defined over the field of complex numbers. Hence for curves it makes sense to take X0 = X if X is not contained in the translate of a proper algebraic subgroup and X0 = ; else wise. But X00 is more subtle: we take X00 = X if X is not contained in a proper algebraic subgroup and X00 = ; else wise. The point in making this distinction is that in [BMZ06a] the authors conjectured that X00 contains only finitely many points in an algebraic subgroup of codimension at least 2. They proved this conjecture for n � 5. Recently, in [Mau06] Maurin gave a proof for all n. Let X � Gn m be an irreducible subvariety, not necessarily a curve. In the higher dimensional case we finally need a definition of X0: we get X0 by removing from X all positive dimension subvarieties that show up in an improper component of the intersection of X with the translate of an algebraic subgroup. The definition of X00 is similar but we require the translates of algebraic subgroups to be algebraic subgroups. In [BMZ06b] Bombieri, Masser, and Zannier showed that X0 is Zariski open in X. Let h be the absolute logarithmic Weil height. Our contribution in chapter 6 is Theorem 6.1 where we give an explicit bound for the height of algebraic points p in X0 that lie “uniformly close” to an algebraic subgroup of codimension strictly greater than n − n/ dimX. By uniformly close we mean that there exist an � > 0, independent of p, and an a in an algebraic subgroup of said codimension with h(pa−1) � �. Actually, in Theorem 6.1 we will use a weaker notion of uniformly close. The terminology comes from the fact that the map (p, a) 7! h(pa−1) has similar properties as a distance function. For example it satisfies the triangle inequality. This notion of distance was considered by several authors ([Eve02], [Poo99], [R´em03]) in connection with subgroups of finite rank. Theorem 6.1 generalizes the Bounded Height Theorem for curves by Bombieri, Masser, and Zannier. We state our theorem such that it also gives an explicit version of a Theorem of Bombieri and Zannier in [Zan00] on the intersection of varieties with one dimensional subgroups. To do this we will need a slightly more general definition of X0 which is provided in chapter 6. The height upper bound in Theorem 6.1 involves, along with n, the degree and height of the variety X. We define these two notions in chapter 5. In simple terms, the height of X controls the heights of the coefficients of a certain set of defining equations for X whereas the degree of X controls their degrees. Just as in the second proof for height bounds on curves given in [BMZ99], our proof of Theorem 6.1 uses ideas from the geometry of numbers. Given p 2 X(Q) uniformly close to an algebraic subgroup we construct a new algebraic subgroup H of codimension dimX and controlled degree, such that pH has normalized height small compared to the height of p. We then intersect pH with X. The Arithmetic B´ezout Theorem bounds the height of isolated points in this intersection leading to an explicit height bound for p. Lehmer-type lower bounds for heights in spirit of Dobrowolski’s Theorem and its generalization to higher dimension provide a method for deducing finiteness results from height bounds as given in chapter 6. This method was used together with algebraic number theory in Bombieri, Masser, and Zannier’s article [BMZ99] to prove the finiteness of the set of points on X0 in an algebraic subgroup of codimension at least 2 if X is a curve. Meanwhile, their intricate argument has been simplified in [BMZ04] by applying a more advanced height lower bound due to Amoroso and David [AD04]. In this lower bound the degree over Q of a point is essentially replaced by its degree over the maximal abelian extension of Q. Using this approach we show in Corollary 6.2 that if X is a surface in G5 m, then there are only finitely many points on X0 contained in an algebraic subgroup of codimension at least 3. Thus we have finiteness for the correct subgroup size at least in an isolated case. Even in presence of a uniform height bound as in Theorem 6.1, the approaches in [BMZ99] and [BMZ04] cannot be used to prove the finiteness of the set of p 2 X0(Q) with h(pa−1) small and a contained in an algebraic subgroups of appropriate dimension: although pa−1 has small height, its degree cannot be controlled. In chapter 7 we pursue a new approach using a Bogomolov-type height lower bound. This bound was proved by Amoroso and David in [AD03]; it bounds from below the height of a generic point on a variety not equal to the translate of an algebraic subgroup. The main result of chapter 7 is Theorem 7.1: we show that for B 2 R there exists an � = �(X,B) > 0 with the following property: there are only finitely many p 2 X0(Q) with h(pa−1) � � where a is contained in an algebraic subgroup of dimension strictly less than m(dimX, n). In other words, there are only finitely many algebraic points on X0 of bounded height which are uniformly close to an algebraic subgroup of dimension less than m(dimX, n). Just as was the case in Theorem 6.1 we actually use a relaxed version of uniformly close in Theorem 7.1. The somewhat unnatural function m(·, ·) is defined in (7.1.1). At least in the case of curves we have n − 2 < m(1, n) and so we can take the subgroups to have the best possible dimension n−2. Unfortunately this is the only interesting case where m(r, n) > n − r − 1. With the height upper bound from chapter 6 we can deduce a corollary to Theorem 7.1 which proves finiteness independently of B and where the subgroup dimension is strictly less than min{n/ dimX,m(dimX, n)}. Let X be a curve, then this result is optimal with respect to the subgroup dimension. Let us assume that X is not contained in the translate of a proper algebraic subgroup, hence X0 = X. Then our corollary says that there are only finitely many algebraic points on X that are close to an algebraic subgroup of codimension at least 2. Moreover, in Corollary 7.2 we use Dobrowolski’s Theorem to show that if � in the definition of uniformly close is small enough, then all points on X close to an algebraic subgroup of codimension at least 2 are actually contained in such a subgroup. We now shift our focus from the algebraic torus to abelian varieties: we want to study the intersection X0(Q)\A[r] where A is an abelian variety and X is an irreducible closed subvariety of A. The definitions of X0 and X00 make sense in the abelian setting and are completely analog to the multiplicative case. Let X be a curve, then in [Via03] Viada proved that X0(Q)\A[1] has bounded height if A is a power of an elliptic curve. If the elliptic curve has complex multiplication she also proved that X0(Q)\A[2] is finite. R´emond in [R´em05b] generalized Viada’s height bound to any abelian variety. In [R´em07] R´emond applied a generalization of Vojta’s inequality which he proved in [R´em05a] and in Theorem 1.2 showed boundedness of height of (X(Q)\Z(r) X )\A[r]. Here X\Z(r) X � X is a new deprived subset which depends on r. In fact his result holds for a set larger than A[r] involving also the division closure of finitely generated group. If A is isogenous to a product of elliptic curves and if X is a sufficiently general surface which is not contained in the translate of a proper algebraic subgroup then X\Z(r) X is non-empty and Zariski open in X for r � (dimA + 3)/2. In [RV03], R´emond and Viada proved that if X is a curve then X00(Q) \ A[2] is finite if A is a power of an elliptic curve E with complex multiplication. In a recent preprint, Viada [Via07] announced the finiteness of X00(Q) \ A[3] for unrestricted E, the optimal subgroup codimension 2 is thus just missed. We announce the following result called the Bounded Height Theorem: if A = Eg is a power of an elliptic curve E and X is an irreducible closed subvariety of arbitrary dimension, then X0(Q) \ A[dimX] has bounded N´eron-Tate height. Also, using a result from Kirby’s Thesis [Kir06] and ideas from Bombieri, Masser, and Zannier’s [BMZ06b] one can show that X0 is Zariski open and give a criterion on X to decide when X0 is non-empty. Using height lower bounds on abelian varieties with complex multiplication due to Ratazzi in [Rat07] we can use the Bounded Height Theorem to show that X0(Q)\A[dimX+1] is finite if E has complex multiplication. For an elliptic curve without complex multiplication, finiteness of X0(Q)\A[r] can also be obtained, using for example R´emond’s Theorem 2.1 from [R´em05b]. But r is in general sub-optimal for such elliptic curves. The essential difference between the Bounded Height Theorem and Theorem 6.1 is that the subgroups are now allowed to have the best-possible codimension dimX for all X. In the future we plan to publish these results. Pink has stated a general conjecture on mixed Shimura varieties, see [Pin05a] and [Pin05b]. One special implication is his Conjecture 5.1 from [Pin05b]: if A is a semiabelian variety defined over C and if X � A is a subvariety also defined over C which is not contained in a proper algebraic subgroup of A, then X(C)\A[dimX+1] is not Zariski dense in X. Zilber’s stronger Conjecture 2 in [Zil02] implies the same conclusion. With the Bounded Height Theorem we can prove this assertion under the following stronger hypothesis on A and X: A is a power of an elliptic curve E with complex multiplication and if ' : Eg ! EdimX is a surjective homomorphism of algebraic groups, then the restriction '|X : X ! EdimX is dominant. The proof of the Bounded Height Theorem uses the completeness of A (and X) in an essential way as it relies on intersection theory. Nevertheless, a proof for the boundedness of height of X0(Q)\A[dimX] for the non-complete X � A = Gn m along the lines of the proof of the Bounded Height Theorem must not be ruled out. For instance one could compactify Gn m ,! Pn and work in Pn. Still, there seems to be no suitable Theorem of the Cube for Gn m. Future research could consist in finding a proof of the Bounded Height Theorem in the multiplicative case or in abelian varieties other than a power of an elliptic curve. In the two appendices we leave the main path of the thesis. Let P be an irreducible polynomial in two variables with algebraic coefficients. Say x and y are algebraic with P(x, y) = 0. In appendix A, motivated by Proposition B of [BMZ99], we consider the problem of bounding | degX(P)h(x)−degY (P)h(y)| explicitly and with good dependency in h(x), h(y), and P. For simple examples such as P = Xp − Y q with p and q coprime integers, the absolute value is zero. But for general and fixed P it may even be unbounded as (x, y) runs over all algebraic solutions of P. In Theorem A.1 we prove an upper bound which is of the form c max{1, hp(P)}1/2 max{1, h(x), h(y)}1/2 where the constant c is completely explicit and depends only on the partial degrees of P. Here hp(P) is the projective logarithmic Weil height of the coefficient vector of P. This type of height inequality is often referred to as quasi-equivalence of heights. In appendix B we demonstrate four known results using the Quasi-equivalence Theorem from appendix A. The first application is the Theorem of Bombieri, Masser, and Zannier, already discussed above, in the case of curves in G2 m. We then prove a version of Runge’s Theorem on the finiteness of the number of solutions of certain diophantine equations. Next we show a result of Skolem from 1929: we first generalize the greatest common divisor of pairs of integers to pairs of algebraic numbers. We then show that if x and y are coprime algebraic numbers and P(x, y) = 0 where P is an irreducible polynomial in Q[X, Y ] without constant term, then x and y have uniformly bounded height. This result has been proved independently by Abouzaid in [Abo06] who used it to prove a variant of the Quasi-equivalence Theorem. The fourth and final application is an explicit version of Sprindzhuk’s Theorem: let P have rational coefficients, again without constant term and such that not both partial derivatives of P vanish at (0, 0). Then for a sufficiently large prime l, the polynomial P(l, Y ) 2 Q[Y ] is irreducible. Since the Quasi-equivalence Theorem gives explicit bounds, so do its four applications. Chapters 1 and 5 contain no new results but serve as reference for certain theorems which we apply in the rest of the thesis. Chapter 1 introduces the Weil height and related subjects. It is used throughout the thesis. Chapter 5 contains some results from algebraic geometry and gives a definition for the height of a positive dimensional variety. These definitions and results will be used in the second part of the thesis, chapters 6 and 7