### 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 |
---|---|

Pages (from-to) | 261-280 |

Number of pages | 20 |

Journal | Compositio Mathematica |

Volume | 122 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2000 Jan 1 |

### Fingerprint

### 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

**On Characteristic Polynomials of Geometric Frobenius Associated to Drinfeld Modules.** / Hsia, Liang-Chung; Yu, Jing.

Research output: Contribution to journal › Article

*Compositio Mathematica*, vol. 122, no. 3, pp. 261-280. https://doi.org/10.1023/A:1002015330987

}

TY - JOUR

T1 - On Characteristic Polynomials of Geometric Frobenius Associated to Drinfeld Modules

AU - Hsia, Liang-Chung

AU - Yu, Jing

PY - 2000/1/1

Y1 - 2000/1/1

N2 - 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'..

AB - 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'..

KW - Characteristic polynomial

KW - Dirichlet density

KW - Drinfeld module

KW - Endomorphism ring

KW - Geometric Frobenius

KW - Power residue symbol

KW - Sign function

KW - Tate module

UR - http://www.scopus.com/inward/record.url?scp=0041727258&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0041727258&partnerID=8YFLogxK

U2 - 10.1023/A:1002015330987

DO - 10.1023/A:1002015330987

M3 - Article

AN - SCOPUS:0041727258

VL - 122

SP - 261

EP - 280

JO - Compositio Mathematica

JF - Compositio Mathematica

SN - 0010-437X

IS - 3

ER -