Ties in Function Field Prime Races

Abstract

The function field analogue of Chebyshev's bias was first studied by Cha. In this paper, we study *ties* in this race, namely collections of distinct congruence classes $c_1, \dots, c_k \in (\mathbb{F}_q[T] / m)^\times$ for which $$\pi(N; m, c_1) = \pi(N; m, c_2) = \dots = \pi(N; m, c_k)$$ holds for infinitely many $N$. We provide infinitely many examples of $(m, c_1, \dots, c_k)$ for which the tie holds whenever $N$ satisfies certain congruence conditions. We give two different proofs: first, via the explicit formula for prime counts in terms of $L$-functions together with a matrix analogue of M\"obius inversion, where exceptional pairs of Galois-conjugate elements in the corresponding cyclotomic fields produce ties; and second, via an explicit bijection arising from the $\mathrm{GL}_2(\mathbb{F}_q)$-action. Our examples also include characteristic 2 cases.