Distribution of zeta zeroes for abelian covers of algebraic curves over a finite field

For a function field k over a finite field with Fq as the field of constants, and a finite abelian group G whose exponent divides q−1, we study the distribution of zeta zeroes for a random G-extension of k, ordered by the degree of conductors. We prove that when the degree goes to infinity, the number of zeta zeroes lying in a prescribed arc is uniformly distributed and the variance follows a Gaussian distribution.