Drinfeld modules and subfields of division fields (Q2828637)

The objective of this article is to generalize some results of \textit{A. C. Cojocaru} and \textit{W. Duke} [Math. Ann. 329, No. 3, 513--534 (2004; Zbl 1062.11039)] where they obtained some asymptotic formulas for the splitting of the rational primes \(p\) in some division fields associated to an elliptic curve \(E\) defined over \({\mathbb Q}\). The results of Cojocaru and Duke are conditional on the generalized Riemann hypothesis. The authors study the Tate-Shafarevich groups \(\mathrm{TS}_p\) of the reduction modulo rational primes \(p\) of an elliptic curve \(E/{\mathbb Q}\).NEWLINENEWLINEIn the paper under review the results are unconditional, as the generalized Riemann hypothesis for the function field case is known.NEWLINENEWLINELet \(F\) be a finite extension of \(k={\mathbb F}_q(T)\), and let \({\mathbb F}_F\) be the constant field of \(F\). For \({\mathfrak p}\) a prime of \(F\), let \({\mathbb F}_{{\mathfrak p}}\) be the residue field at \({\mathfrak p}\). Let \(\phi\) be a Drinfeld \(A\)-module over \(F\) of rank \(r\), where \(A={\mathbb F}_q[T]\). For \(m\in A\), consider the field \(F(\phi[m])\) where \(\phi[m]\) is the \(m\)-torsion of \(\phi\) and let \(F(\phi[m])'\) be the subfield of \(F(\phi[m])\) fixed by the scalar elements of \(\mathrm{Gal}(F(\phi[m])/F) \subseteq \mathrm{GL}_r(A/mA)\). For \(x\in{\mathbb N}\), let \(f_{\phi,F}(x)\) be the number of primes \({\mathfrak p}\) with good reduction for \(\phi\) such that \(\deg_F{\mathfrak p}=x\) and \({\mathfrak p}\) does not split completely in \(F(\phi[m])'\) for any \(m\in A\) with \(\gcd(N_{F/k}{\mathfrak p},m)=1\).NEWLINENEWLINEThe main result is that if \(\phi\) is a Drinfeld \(A\)-module over \(F\) of rank \(r\geq 3\), with some mild condition, then for \(x\in{\mathbb N}\), \(f_{\phi,F}(x)=c_{\phi,F}(x)\pi_F(x) +O((q^{d_F x})^{\Delta})\), where \(\Delta\) is given explicitly in terms of \(r\), \(c_{\phi,F}(x)=\sum\limits_{m\in A \atop m\text{ is monic}} \frac{\mu_q(m)r_m(x)}{[F(\phi[m])' :F]}\), \(\mu_q\) is the Mobius function of \(A\), \(r_m(x)= r_m=[F(\phi[m])'\cap \bar{\mathbb F}_F:{\mathbb F}_F]\) if \(r_m\mid x\) and \(r_m(x)=0\) otherwise, \(d_F=[{\mathbb F}_F:{\mathbb F}_q]\), and \(\pi_F(x)\) is the number of primes of degree \(x\).NEWLINENEWLINEOne of the main tools in the proof is Chebotarev density theorem.