### Abstract

Let K be a function field over finite field double-struck F sign_{q} 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 P_{φ}(X) of the geometric Frobenius. A formula for the sign of the constant term of P_{φ}(X) in terms of 'leading coefficient'of φ is given. General formula to determine signs of other coefficients of P_{φ}(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 language | English |
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Pages (from-to) | 261-280 |

Number of pages | 20 |

Journal | Compositio Mathematica |

Volume | 122 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2000 Jan 1 |

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### 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

### Cite this

*Compositio Mathematica*,

*122*(3), 261-280. https://doi.org/10.1023/A:1002015330987