Browsing by Author "Rabenantoandro, Andry Nirina"

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    Torsion bounds for Drinfeld modules with complex multiplication
    (Stellenbosch : Stellenbosch University., 2020-04) Rabenantoandro, Andry Nirina; Breuer, Florian; Wagner, Stephan; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: The main objective of the present thesis is to prove an analogue for Drinfeld modules of a theorem due to Clark and Pollack. The cardinality of the group of K-rational torsion points of an elliptic curve EjK with complex multiplication defined over a number field K of degree d is uniformly bounded by Cd log log d for some absolute and effective constant C > 0, i.e. the constant C > 0 depends neither on E nor on K. Let F be a global function field over Fq and A the ring of elements of F regular away from a fixed prime ¥. Let r 1 be an integer. We prove that there exists a positive constant CA,r > 0 depending only on A and r such that for any field extension L of degree d over F and any Drinfeld A-module jjL of rank r with complex multiplication defined over L and such that the endomorphism ring of j is the maximal order in its CM field, the cardinality of the A-module of L-rational torsion points of j is bounded by CA,rd log log d. The constant depends neither on j nor on L. For a given A and r the constant CA,r is effective and we get an explicit formula for it. The above result is not the full analogue of Clark and Pollack’s theorem but rather a weaker version since it requires the endomorphism ring of j to be the maximal order in its CM field. However, when A = Fq[T], F = Fq(T) and r = 2 we obtain the full analogue of Clark and Pollack’s result by proving the analogue of what they called the Isogeny Torsion Theorem in [CP15].