An average Chebotarev density theorem for generic rank 2 Drinfeld modules with complex multiplication (Q1932380)

In this paper under review, the authors establish an average Chebotarev density theorem in a family of Galois extensions which are division fields of rank 2 Drinfeld modules with complex multiplication over the rational function field over a finite field. The main results of this paper can be viewed as in the context of generalizations of the classical Bombieri-Vinogradov theorem which gives an average of the error terms in the prime number theorem for arithmetic progressions. To state the main result, we let \(q\) be an odd prime power and let \({\mathbb F}_q\) be the finite field of \(q\) elements. Let \(A = {\mathbb F}_q[T]\) and \( k = {\mathbb F}_q(T)\) be the polynomial ring and rational function field over \({\mathbb F}_q\) respectively. Let \(\psi : A \to k\{\tau\}\) be a rank \(r\geq 1\) Drinfeld \(A\)-module of generic characteristic over \(k.\) For a non-zero \(d\in A\), we denote by \(k(\psi[d])\) the \(d\)-division field obtained by adjoining \(k\) the submodule of \(d\)-division points \(\psi[d]\) of \(\psi.\) The paper concerns with the set of primes in \(A\) that split completely in \(k(\psi[d])\) for some monic \(d \in A\). Let \(c_d\) be the degree of constant subfield in \(k(\psi[d])\) over \({\mathbb F}_q.\) Let \(x\) be a positive real number, we set \(c_d(x) = c_d \) if \(c_d \mid x\) and \(c_d(x) = 0\) otherwise. For a monic polynomial \(m\in A\), we denote by \(\ell_m(x)\) the number of primes \(p\in A\) such that \(\deg p = x\) and the ideal \(pA\) splits completely in \(k(\psi[m])\) over \(k\). By the Chebotarev density theorem, for large \(x\), \(\ell_m(x)\) is asymptotically equal to \(\left(c_m(x)/[k(\psi[m]) : k]\right) \pi_A(x)\) where as usual \(\pi_A(x)\) denotes the number of primes of \(A\) whose degrees equal \(x.\) In the case of rank 2 Drinfeld modules with complex multiplication, the authors obtain an average of the errors between \(\ell_m(x)\) and \(\left(c_m(x)/[k(\psi[m]) : k]\right) \pi_A(x)\). Let now \(\psi\) be a Drinfeld \(A\)-module of rank 2 over \(k\). Recall that \(\psi\) has complex multiplication if its endomorphism ring \(\mathrm{End}_{\bar{k}}(\psi)\) is strictly larger than \(A.\) In this case, it is isomorphic to an order of an imaginary quadratic extension of \(k.\) Assume that \(\mathrm{End}_{\bar{k}}(\psi)\) is the full ring of integers of the imaginary quadratic field (i.e. the integral closure of \(A\) in the imaginary quadratic field) in question. Then, the main result (Theorem 1) in the paper is the following: \[ \sum_{x\in A^{(1)}}\left(\ell_m(x) - \frac{c_m(x)}{[k(\psi[m]) : k]} \pi_A(x) \right) \ll_\psi q^{\frac{3}{4}x} \log x,\quad \text{ as \(x\to \infty\)} \] where \(A^{(1)}\) denotes the subset of monic polynomials in \(A\). The proof uses an effective version of the Chebotarev density theorem (for function fields). The rank 2 and CM assumptions on the Drinfeld module are key ingredients for the applicability of Chebotarev density theorem. These assumptions lead to a reinterpretation of the splitting property which enables the authors to use an elementary sieving to average the splitting primes over monic polynomials \(m\). As an application of the proof of the main result, the authors also obtain an asymptotic formula for the number of monic primes \(p\) of degree \(x\) such that the finite \(A\)-module \(\psi({\mathbb F}_{\mathfrak p})\) is \(A\)-cyclic where \({\mathbb F}_{\mathfrak p}\) is the residue field of the prime ideal \({\mathfrak p} = pA\) and \(\psi({\mathbb F}_{\mathfrak p})\) is the \(A\)-module structure on \({\mathbb F}_{\mathfrak p}\) defined by the reduction of \(\psi\) modulo \({\mathfrak p}.\)