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Let E be a modular elliptic curve defined over a rational function field k of odd characteristic. We construct a sequence of Heegner points on E, defined over a Z ∞ p-tower of finite extensions of k, and show that these Heegner points generate a group of infinite rank. This is a function field analogue of a result of C. Cornut and V. Vatsal.
Let X be a product of Drinfeld modular curves over a general base ring A of odd characteristic. We classify those subvarieties of X which contain a Zariski-dense subset of CM points. This is an analogue of the André-Oort conjecture. As an application, we construct non-trivial families of higher Heegner points on modular elliptic curves over global function… (More)
We prove a generalization of the Newton Identities for entire functions, which give a relation between the Taylor coefficients and sums of powers of reciprocals of the zeros of an entire function. We apply these identities to a number of special functions, yielding some interesting recursion relations.
Let Z = X 1 × · · · × X n be a product of Drinfeld modular curves. We characterize those algebraic subvarieties X ⊂ Z containing a Zariski-dense set of CM points, i.e. points corresponding to n-tuples of Drinfeld modules with complex multiplication (and suitable level structure). This is a characteristic p analogue of a special case of the André-Oort… (More)
Let ϕ be a non-isotrivial family of Drinfeld A-modules of rank r in generic characteristic with a suitable level structure over a connected smooth algebraic variety X. Suppose that the endomorphism ring of ϕ is equal to A. Then we show that the closure of the analytic fundamental group of X in SLr(A f F) is open, where A f F denotes the ring of finite… (More)
We study Ducci-sequences using basic properties of cyclotomic polynomials over F 2. We determine the period of a given Ducci-sequence in terms of the order of a polynomial, and in terms of the multiplicative orders of certain elements in finite fields. We also compute some examples and study links between Ducci-sequences, primitive polynomials and Artin's… (More)
Let k be a global field, k a separable closure of k, and G k the absolute Galois group Gal(k/k) of k over k. For every σ ∈ G k , let k σ be the fixed subfield of k under σ. Let E/k be an elliptic curve over k. We show that for each σ ∈ G k , the Mordell-Weil group E(k σ) has infinite rank in the following two cases. Firstly when k is a global function field… (More)
Let K be a number field and E/K a modular elliptic curve, with modular parametriza-tion π : X 0 (N) −→ E defined over K. The purpose of this note is to study the images in E of classes of isogenous points in X 0 (N). Theorem 1 Let S ⊂ X 0 (N)(¯ K) be an infinite set of points corresponding to elliptic curves which all lie in one isogeny class, but which are… (More)
We study modular polynomials classifying cyclic isogenies between Drinfeld modules of arbitrary rank over the ring F q [T ].
We derive asymptotically optimal upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L : K]. Our main tool is the adelic openness of the image of Galois representations attached to elliptic curves and Drinfeld modules, due to Serre and… (More)