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.


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