Torsion bounds for Drinfeld modules with complex multiplication.

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.