Semi-simplicity of the Galois representations attached to Drinfeld modules over fields of ``infinite characteristics''
Let \(K\) be a function field of transcendence degree one over a finite field, let \(\infty\) be a place of \(K\), and let \(A\) be the ring of elements of \(K\) regular outside \(\infty\). Let \(\varphi: A\to \text{End}_ F(\mathbb{G}_ a)\) be a rank \(r\) Drinfel'd module over a finite extension \(F\) of \(K\). If \(\wp\) is a prime of \(A\), the \(\wp^ n\) division points of \(\varphi\) give rise to a \(\wp\)-adic Tate module \(T_ \wp(\varphi)\) which carries a continuous action of the Galois group \(\text{Gal}(F^{\text{sep}}/F)\). This Galois representation is analogous to the \(\ell\)-adic representations obtained from division points on abelian varieties. This paper proves an analogue of Falting's semi-simplicity theorem: Theorem: \(T_ \wp(\varphi)\otimes_{A_ \wp} K_ \wp\) is a semi- simple \(K_ \wp[\text{Gal}(F^{\text{sep}}/F)]\) module. The author obtains this result by a process modelled on Faltings' results for abelian varieties. In particular, he deduces the theorem above from a Shafarevich-type finiteness result. Theorem: Let \(W\) be a \(\text{Gal}(F^{\text{sep}}/F)\)-invariant \(A_ \wp\)-direct summand of \(T_ \wp(\varphi)\) corresponding to a chain \(\varphi\to\varphi_ 1\to \varphi_ 2\cdots\) of isogenies of Drinfel'd modules over \(F\). Then the set of isomorphism classes of the \(\varphi_ i\) is finite. In the first part of the paper, the author proves a number of basic results on the bad reduction of Drinfel'd modules and the action of the inertia group on \(T_ \wp(\varphi)\). In the second part of the paper, the main results are obtained by comparing two heights on Drinfel'd modules. The first height is analogous to Faltings' height coming from Arakelov theory and behaves well under isogeny; the second height, which is essentially the naive logarithmic height on projective space restricted to the Drinfel'd symmetric space, enjoys good finiteness properties. The Shafarevich result is obtained by comparing the two heights. The author refers to \textit{J. Tate's} fundamental paper [``Endomorphisms of abelian varieties over finite fields'', Invent. Math. 2, 134-144 (1966; Zbl 0147.203)] for the passage from the Shafarevich result to semi-simplicity. The paper in question in a continuation of the author's work [Duke Math. J. 62, 593-599 (1991; Zbl 0739.14029)].
analogue of Falting's semi-simplicity theorem
scientific article; zbMATH DE number 431330