On Characteristic Polynomials of Geometric Frobenius Associated to Drinfeld Modules

Abstract

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’.