Remarks on rank one Drinfeld modules and their torsion elements (Q2080513)

The torsion elements of rank one Drinfeld modules of a global function field \(k\) give an explicit description of abelian class fields, produce units, Euler systems and annihilators of the ideal class group of the abelian extensions of \(k\). In this paper, the authors consider these torsion elements. Let \(\rho\) be a sgn-normalized Drinfeld \(A\)-module of generic characteristic, where \(A\) is the Dedekind ring of elements of \(k\) regular outside a fixed place \(\infty\) of \(k\). Let \(\Omega\) be the completion of a fixed algebraic closure of \(k_{\infty}\), the completion of \(k\) at \(\infty\). Let \(H_A^*\) be the subfield of \(\Omega\) generated by the coefficients of the polynomials \(\rho_x\), for \(x\in A\). David Hayes called \(H_A^*\) the {\em normalizing field with respect to} sgn. The first main result of this paper is that if \(E_{\rho}\subseteq H_A^* [\tau]\) is the vector space generated by all the polynomials \(\rho_x\), \(x\in A\), then \(R_{\rho}=H_A^*[\tau]/E_{\rho}\) is a finitely generated vector space of dimension less than or equal to the genus \(g\) of \(k\). Further, if \(\infty\) is of degree \(1\), then \(\dim_{H_A^*} ( R_{\rho})=g\). The second main result is the following. There exists a suitable ideal \({\mathfrak m}_{\rho}\) such that if \({\mathfrak m}\) is an ideal of \(A\) prime to \({\mathfrak m}_{\rho}\), then the \(B\)-module \(B\Lambda_{ \mathfrak m}\) is free of rank \(\deg({\mathfrak m})-s+1\) if \(q=2\) and \(s\geq 1\) and of rank \(\deg({\mathfrak m})\) if \(q>2\) or (\(q=2\) and \(s=0\)), where \(s\) is the number of prime ideals \({\mathfrak p}\) dividing \({\mathfrak m}\) and such that \(\deg({\mathfrak p})=1\), \(\Lambda_{\mathfrak m}\) is the \(A\)-module of the \({\mathfrak m}\)-torsion elements of \(\Omega\) and \(B\) is the integral closure of \(A\) in \(H_A^*\).