Statistics for biquadratic covers of the projective line over finite fields

Elisa Lorenzo, Giulio Meleleo, Piermarco Milione, Alina Bucur

Research output: Contribution to journalArticleScientificpeer-review

Abstract

We study the distribution of the traces of the Frobenius endomorphisms of genus g curves which are quartic non-cyclic covers of View the MathML sourcePFq1, as the curve varies in an irreducible component of the moduli space. We show that for q fixed, the limiting distribution of the traces of Frobenius equals the sum of q+1q+1 independent random discrete variables. We also show that when both g and q go to infinity, the normalized trace has a standard complex Gaussian distribution. Finally, we extend these computations to the general case of arbitrary covers of View the MathML sourcePFq1 with Galois group isomorphic to r copies of Z/2ZZ/2Z. For r=1r=1 we recover the already known results for the family of hyperelliptic curves.
Original languageEnglish
Pages (from-to)448-477
JournalJournal of Number Theory
Volume173
DOIs
Publication statusPublished - 2017
MoE publication typeA1 Journal article-refereed

Keywords

  • Function fields
  • Biquadratic curves
  • Biquadratic covers
  • Number of points over finite fields
  • Arithmetic statistics

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