Ordinary cyclotomic function fields (Q1930113)

Let \({\mathbb F}_q\) be the field with \(q\) elements and of characteristic \(p\). Let \(k={\mathbb F}_q(T)\) be a rational function field over \({\mathbb F}_q\) and let \(R_T={\mathbb F}_q[T]\) be the polynomial ring. For a monic polynomial \(M\in R_T\), let \(k(\Lambda_M)\) and \(k(\Lambda_M)^+\) be the \(M\)-th cyclotomic function and, its maximal real subfield, respectively. For any global function field \(K\), let \(J_K\) be the Jacobian of \(K\bar{\mathbb F}_q\). If \(l\) is a prime number, the \(l\)-primary subgroup \(J_K(l)\) of \(J_K\) is isomorphic to \(R_l^{2 g_K}\) if \(l \neq p\) and to \(R_p^{\lambda_K}\) if \(l=p\) where \(R_l={\mathbb Q}_l/ {\mathbb Z}_l\), \(R_p={\mathbb Q}_p/{\mathbb Z}_p\) and \(g_K\), \(\lambda_K\) denote the genus and the Hasse-Witt invariant of \(K\) respectively. We have \(0\leq \lambda_K\leq g_K\). The field \(K\) is called supersingular if \(\lambda_K=0\), and ordinary if \(\lambda_K=g_K\). The objective of this paper is the study of \(J_{k(\Lambda_M)}(p)\) and \(J_{k(\Lambda_M)^+}(p)\). The main result is to find a necessary and sufficient condition on \(M\) such that \(k(\Lambda_M)\) (resp. \(k(\Lambda_M)^+\)) is ordinary. As consequences, the author obtains that when \(q\neq p\) and \(M\) is a monic irreducible polynomial, \(k(\Lambda_M)\) is ordinary if and only if \(\deg M\leq 1\) and \(k(\Lambda_M)^+\) is ordinary if and only if \(\deg M\leq 2\). The case \(q=p\) is harder. It is proved that if \(M\) is a monic irreducible polynomial of degree two then \(k(\Lambda_M)^+\) and \(k(\Lambda_M)\) are ordinary. Finally if \(M\) is a monic irreducible polynomial of degree three, then \(k(\Lambda_M)^+\) is ordinary but \(k(\Lambda_M)\) not necessarily is ordinary. For instance, for \(p=3\) and \(M=T^3 +2T+1\), then \(g_{k(\Lambda_M)}=19\) and \(\lambda_{ k(\Lambda_M)}=18\).