Explicit Description Of Isogeny And Isomorphism Classes Of Drinfeld Modules Of Higher Rank Over Finite Fields

When jumping from the number fields theory to the function fields theory, one cannot miss the deep analogy between rank 1 Drinfeld modules and the group of root of unity and the analogy between rank 2 Drinfeld modules and elliptic curves. But so far, there is no known structure in number fields theory that is analogous to the Drinfeld modules of higher rank r ≥ 3. In this thesis we investigate the classes of those Drinfeld modules of higher rank r ≥ 3 defined over a finite field L. We describe explicitly the Weil polynomials defining the isogeny classes of rank r Drinfeld modules defined over a finite field L for any rank r ≥ 3, which generalizes what Yu already did for r = 2. We also provide a necessary and sufficient condition for an order O in the endomorphism algebra corresponding to some isogeny classes, to be the endomorphism ring of a Drinfeld module. To complete the classification, we define the notion of fine isomorphy invariants for any rank r Drinfeld module defined over a finite field L and we prove that the fine isomorphy invariants together with the J-invariants describe the L-isomorphism classes of rank r Drinfeld modules defined over the finite field L.