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On Characteristic Polynomials of Geometric Frobenius Associated to Drinfeld Modules

Published online by Cambridge University Press:  04 December 2007

Liang-Chung Hsia
Affiliation:
Department of Mathematics, National Central University, Chung-Li, Taiwan, Republic of China. E-mail: [email protected]
Jing Yu
Affiliation:
Institute of Mathematics, Academia Sinica and NSC Center for Theoretical Sciences, Taipei, Taiwan, Republic of China. E-mail: [email protected]
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Abstract

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Let K be a function field over finite field ${\Bbb F}_q$ and let ${\Bbb A}$ be a ring consisting of elements of K regular away from a fixed place ∞ of K. Let φ be a Drinfeld ${\Bbb A}$-module defined over an ${\Bbb A}$-field L. In the case where L is a finite ${\Bbb A}$-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 ${\Bbb A}$-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’.

Type
Research Article
Copyright
© 2000 Kluwer Academic Publishers