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

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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 will contain asymptotically 2g|I| angles as the genus grows. We show that for the variance of number of angles in is asymptotically (2/π2)log (2g) 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.

Original languageEnglish
Pages (from-to)81-101
Number of pages21
JournalCompositio Mathematica
Volume146
Issue number1
DOIs
StatePublished - Jan 2010

Keywords

  • central limit theorem
  • hyperelliptic curve
  • random matrix theory
  • zeros of L-functions

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