A Dirichlet unit theorem for Drinfeld modules.(English)Zbl 1217.11062

Let $$k$$ be a finite field of $$q$$ elements, $$K$$ a finite extension of the rational function field $$k(t)$$, $$R$$ the integral closure of $$k[t]$$ in $$K$$, $$E$$ the additive group $${\mathbf G}_{a,R}$$ equipped with an action $$\varphi$$ of $$k[t]$$ given by a $$k$$-algebra homomorphism $\varphi:k[t]\to \text{End}({\mathbf G}_{a,R}):t\mapsto t+a_1\tau+\dots+a_n\tau^n,$ $$\tau$$ the $$q$$-th power Frobenius endomorphism, $$a_i\in R$$, $$a_n\neq 0$$. This is a model over $$R$$ of a Drinfeld module of rank $$n$$ over $$K$$. Let $$\text{Lie}_E$$ be the tangent space at zero of $$E$$. Let $$K_\infty:=K\otimes_{k(t)}k((t^{-1}))$$. There is a short exact sequence $\Lambda_E\to \text{Lie}_E(K_\infty^{\text{sep}})\twoheadrightarrow E(K_\infty^{\text{sep}}),$ where the $$k[t]$$-module $$\Lambda_E$$ is discrete in $$\text{Lie}_E(K_\infty^{\text{sep}})$$ and free of rank $$n$$ times the separable degree of $$K$$ over $$k(t)$$. This sequence is $$G=\text{Gal}(k((t^{-1}))^{\text{sep}}/k((t^{-1})))$$-equivariant, and taking invariants gives an exact sequence $\Lambda_E^G\to \text{Lie}_E(K_\infty)\to E(K_\infty)\twoheadrightarrow H^1(G,\Lambda_E).$ The author proves that the cokernel of $$E(R)\to H^1(G,\Lambda_E)$$ is finite. The inverse image of $$E(R)$$ is a discrete and cocompact sub-$$k[t]$$-module of $$\text{Lie}_E(K_\infty)$$. As a corollary, the kernel of $$E(R)\to H^1(G,\Lambda_E)$$ is finitely generated. This should be seen as an analogue of Dirichlet’s unit theorem in number fields.

MSC:

 11G09 Drinfel’d modules; higher-dimensional motives, etc. 11R58 Arithmetic theory of algebraic function fields
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