On Characteristic Polynomials of Geometric Frobenius Associated to Drinfeld Modules

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Abstract

Let K be a function field over finite field double-struck F signq and let double-struck A be a ring consisting of elements of K regular away from a fixed place ∞ of K. Let φ be a Drinfeld double-struck A-module defined over an double-struck A-field L. In the case where L is a finite double-struck A-field, we study the characteristic polynomial Pφ(X) of the geometric Frobenius. A formula for the sign of the constant term of Pφ(X) in terms of 'leading coefficient'of φ is given. General formula to determine signs of other coefficients of Pφ(X) is also derived. In the case where L is a global double-struck A-field of generic characteristic, we apply these formulae to compute the Dirichlet density of places where the Frobenius traces have the maximal possible degree permitted by the 'Riemann hypothesis'..

Original languageEnglish
Pages (from-to)261-280
Number of pages20
JournalCompositio Mathematica
Volume122
Issue number3
DOIs
Publication statusPublished - 2000 Jan 1

Keywords

  • Characteristic polynomial
  • Dirichlet density
  • Drinfeld module
  • Endomorphism ring
  • Geometric Frobenius
  • Power residue symbol
  • Sign function
  • Tate module

ASJC Scopus subject areas

  • Algebra and Number Theory

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