Ties in Function Field Prime Races

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 $c_1,\dots ,c_k\in ({\mathbb{F}}_q[T]\mathrm{/}m{)}^{\times }$ for which $$\pi (N;m,c_1)=\pi (N;m,c_2)=\cdots =\pi (N;m,c_k)$$ 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 Möbius inversion, where exceptional pairs of Galois-conjugate elements in the corresponding cyclotomic fields produce ties; and second, via an explicit bijection arising from the ${\mathrm{G}\mathrm{L}}_2({\mathbb{F}}_q)$-action. Our examples also include characteristic 2 cases.