On some characteristic polynomials attached to finite Drinfeld modules
Let \(K\) be a global function field over \(\mathbb{F}_q\) and \(A\) the Dedekind subring of elements regular away from a fixed place \(\infty\) of \(K\). The paper deals with questions about Drinfeld \(A\)-modules \(\varphi\) defined over a finite field \(L\) provided with a structure as an \(A\)-algebra. After recalling and discussing some results about the endomorphism ring \(\text{End} (\varphi)\) and the ``de Rham cohomology'' \(H^*_{\text{DR}} (\varphi,L)\) of \(\varphi\), the ``crystalline cohomology'' \(H^*_{\text{crys}} (\varphi,L)\) is introduced, which is a free module of \(\text{rank} r= \text{rank} (\varphi)\) over the relative Witt ring \(W(L)\) of \(L\) with respect to \(A\). It is then shown (theorem 3.2) that for each \(u\in \text{End} (\varphi)\), the characteristic polynomial of \(u\) with respect to \(K\) agrees with the characteristic polynomial of \(H^*_{\text{crys}} (u,L)\) over \(W(L)\). For purely inseparable \(u\) (e.g., a power of the Frobenius), some more detailed assertions are proved (Theorems 4.2, 4.4, 4.6).
scientific article; zbMATH DE number 1038298