Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field
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 2gI angles as the genus grows. We show that for the variance of number of angles in I is asymptotically a constant multiple of log(2gI) 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 2gI tends to infinity.
 Publication:

arXiv eprints
 Pub Date:
 March 2008
 arXiv:
 arXiv:0803.3534
 Bibcode:
 2008arXiv0803.3534F
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Algebraic Geometry;
 11G20;
 14G10;
 15A52
 EPrint:
 Added references to the CLT in RMT