Obstructions to reciprocity laws related to division fields of Drinfeld modules

Abstract

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