Obstructions to reciprocity laws related to division fields of Drinfeld modules

Let q be an odd prime power, denote by \mathbb {F}_q the finite field with q elements, and set A ≔\mathbb {F}_q[T], F ≔\mathbb {F}_q(T). Let \psi : A \to F\{\tau \} be a Drinfeld A-module over F, of rank r \geq 2, with End_{\overline {F}}(\psi ) = A. For a non-zero ideal \mathfrak {n} of A, denote its unique monic generator by n, denote the degree of n as a polynomial in T by \deg n, and denote the \mathfrak {n}-division field of \psi by F(\psi [\mathfrak {n}]). A reciprocity law for \psi asserts that, if \gcd (charF, r) = 1 or if \mathfrak {n} is prime, then a non-zero prime ideal \mathfrak {p} \nmid \mathfrak {n} of A splits completely in F(\psi [\mathfrak {n}]) if and only if the Frobenius trace a_{1, \mathfrak {p}}(\psi ) of \psi at \mathfrak {p} and the first component b_{1, \mathfrak {p}}(\psi ) of the Frobenius index of \psi at \mathfrak {p} satisfy the congruences a_{1, \mathfrak {p}}(\psi ) \equiv -r \pmod n and b_{1, \mathfrak {p}}(\psi ) \equiv 0 \pmod n. We find the Dirichlet density of the set of non-zero prime ideals \mathfrak {p} for which the latter congruence never holds, that is, for which b_{1, \mathfrak {p}}(\psi ) = 1. Using similar methods, we prove an asymptotic formula for the function of x defined by the average \frac {1}{\#\{\mathfrak {p}: \ \deg p = x \}} \sum _{\mathfrak {p}: \ \deg p = x} \tau _A(b_{1, \mathfrak {p}}(\psi )), where \mathfrak {p} = A p denotes an arbitrary non-zero prime ideal of A whose monic generator p \in A has degree x and where \tau _A(b_{1, \mathfrak {p}}(\psi )) denotes the number of monic divisors of b_{1, \mathfrak {p}}(\psi ).