Joint Extension Field Point Counts for Superelliptic Curves

Abstract

Fix a finite field $\mathbf{F}_q$, an integer $m\ge2$ prime to $q$, and $2\leκ\le m$. For a uniformly chosen $κ$-power-free polynomial $F\in\mathbf{F}_q[x]$ of degree $D$, we determine the limiting joint distribution of the affine point counts of $y^m=F(x)$ over $\mathbf{F}_q,\ldots,\mathbf{F}_{q^n}$ as $D\to\infty$, with $q$ and $n$ fixed. The probability generating function is a finite Euler product with explicit local factors. Equivalently, the limiting vector is a sum of independent closed-point contributions, each of which may affect several extension degrees. We obtain the formula by specializing the equidistribution and motivic Euler product formalism of Bertucci-Howe. We then compute means, covariances, and moment expansions as $q$ varies in a fixed reduced residue class modulo $m$, after the degree limit has been taken. The one-field specialization recovers the power-free affine point-count distribution of Cheong-Wood-Zaman.