Analytic Homomorphisms into Drinfeld Modules

Linear independence of Gamma values in positive characteristic

“…In that case, we can apply Yu's Kolchin-type result [27], Thm. 1.3, to conclude that, for some fixed i, there are non-zero endomorphisms Θ ℓ ∈ End(H i ), ℓ = 1, .…”

“…Seen on a large enough scale, the proofs here and in [24] also run somewhat parallel, based as they are on J. Yu's Theorem of the Sub-t-module [27], reproduced below as Theorem 5.1.1, and G. Wüstholz's Theorem of the Subgroup [25], respectively. Luckily, as stated above, the theory of bracket relations even provides the analogue of the Deligne-Koblitz-Ogus characterization for the algebraicity of the product of values of the (normalized) classical Gamma function at rational points predicted by the classical relations.…”

“…Now if f ∈ End(E), then the kernel of f is a sub-algebraic group invariant under the action of t. Its connected component will be a sub-t-module of E, and so the kernel of f must be finite. Thus Lemma 1.1 of [27] gives a ∈ A and g ∈ End(E) so that, (as elements of End(E)), f g = Φ(a).…”

“…A morphism Θ will be called an isogeny if the dimensions of E 1 and E 2 are the same and if Θ has a finite kernel as a map of algebraic groups. We will write E 1 ∼ E 2 to denote that E 1 is isogenous to E 2 ; isogeny is an equivalence relation (see [27], Lem. 1.1).…”

Some examples toward a Manin-Mumford conjecture for abelian uniformizable T-modules

“…Indeed, any tensor power C ⊗m , for any power m ∈ N \ {0}, of the Carlitz module C is always a simple, abelian (see [13], Corollary 5.9.38) and uniformizable T −module (see [31], Proposition 1.2), but one can prove that it possesses sometimes (as in the present case) nontrivial sub-T j −modules for some j depending on m and q. By choosing 0 × G a as an algebraic subvariety of C ⊗2 , we now see that it contains infinitely many torsion points, which correspond, by Proposition 1.14 and discussion subsequent to Remark 1.17, to the F 2 (T 2 )−rational points of the torsion part of:…”