On Characteristic Polynomials of Geometric Frobenius Associated to Drinfeld Modules

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'..