# Mathematics Wiskunde

The main objective of the thesis is to prove an analogue for Drinfeld modules of a theorem due to Clark and Pollack which asserts that the cardinality of the group of $$K$$-rational torsion points of an elliptic curve $$E$$ defined over a number field $$K$$ of degree $$d$$, with complex multiplication, 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 the finite field with $$q$$ elements and $$A$$ the ring of elements of $$F$$ regular away from a fixed prime of $$F$$. Let $$r \geq 1$$ be an integer. We prove that there exists a positive constant $$C > 0$$ depending only on $$A$$ and $$r$$ such that for any field extension $$L$$ of degree $$d$$ over $$F$$ and any Drinfeld $$A$$-module of rank $$r$$ defined over $$L$$, with complex multiplication, and such that the endomorphism ring is the maximal order in its CM field, the cardinality of the $$A$$-module of $$L$$-rational torsion points is bounded by $$C d \log \log d$$. The constant depends neither on the Drinfeld module nor on $$L$$. For a given $$A$$ and $$r$$ the constant $$C$$ 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 to be the maximal order in its CM field. However, when $$A = \mathbb{F}_q [ T ] , F = \mathbb{F}_q ( T )$$ 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.