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

@article{Faifman2009StatisticsOT,
  title={Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field},
  author={Dmitry Faifman and Ze'ev Rudnick},
  journal={Compositio Mathematica},
  year={2009},
  volume={146},
  pages={81 - 101}
}
Abstract We 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 genus g a 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… Expand
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