Heights and multiplicative relations on algebraic varieties
Philipp Habegger · edoc (University of Basel) · 2007
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 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 controll