Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field

Dmitry Faifman, Zeév Rudnick
2009 Compositio Mathematica  
AbstractWe study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genusga fixed interval ℐ will contain asymptotically 2g∣ℐ∣ angles as the genus grows. We show that for the variance of number of angles in ℐ is asymptotically (2/π2)log (2g∣ℐ∣) and
more » ... π2)log (2g∣ℐ∣) and prove a central limit theorem: the normalized fluctuations are Gaussian. These results continue to hold for shrinking intervals as long as the expected number of angles 2g∣ℐ∣ tends to infinity.
doi:10.1112/s0010437x09004308 fatcat:2ko7limoezeajhpodui54ucoi4