Statistics of the Jacobians of hyperelliptic curves over finite fields

Maosheng Xiong, Alexandru Zaharescu

Research output: Contribution to journalArticlepeer-review


Let C be a smooth projective curve of genus g ≥ 1 over a finite field Fq of cardinality q. Denote by # JC the size of the Jacobian of C over Fq. We first obtain an estimate on # JC when Fq(C)/Fq(X) is a geometric Galois extension, which improves a general result of Shparlinski [19]. Then we study the behavior of the quantity #JC as C varies over a large family of hyperelliptic curves of genus g. When g is fixed and q → ∞, its limiting distribution is given by the powerful theorem of Katz and Sarnak in terms of the trace of a random matrix. When q is fixed and the genus g → ∞, we also find explicitly the limiting distribution and show that the result is consistent with that of Katz and Sarnak when both q, g → ∞.

Original languageEnglish (US)
Pages (from-to)255-272
Number of pages18
JournalMathematical Research Letters
Issue number2
StatePublished - 2012


  • Class number
  • Gaussian distribution
  • Jacobian
  • Zeta functions of curves

ASJC Scopus subject areas

  • General Mathematics


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