Abstract :
We show that if φ is a Drinfeld A-module over a field L, then any polynomial g(x) set membership, variant L[x] which maps the a-torsion into itself for all a set membership, variant A must be an endomorphism of φ. In generic characteristic, we prove a stronger result: if there is an infinite A-submodule S of the torsion submodule (over image) such that g maps S into S, or if g maps the a-torsion into itself for infinitely many a, then g is an endomorphism followed by a translation. The first of these results generalizes easily to maps between different Drinfeld modules. The second can be generalized as well, assuming a conjecture which would follow from an analogue of Serre′s theorem on the image of Galois. Analogous rigidity results are known to hold for abelian varieties. As one application, we show that the ring of semi-invariants of a Drinfeld module (defined by D. Goss) is nothing more than the ring of endomorphisms.
