On Characteristic Polynomials of Geometric Frobenius Associated to Drinfeld Modules

AbstractLet K be a function field over finite field $$\mathbb{F}_q $$ and let $$\mathbb{A}$$ be a ring consisting of elements of K regular away from a fixed place ∞ of K. Let ϕ be a Drinfeld $$\mathbb{A}$$ -module defined over an $$\mathbb{A}$$ -field L. In the case where L is a finite $$\mathbb{A}$$ -field, we study the characteristic polynomial $$P_\phi (X)$$ of the geometric Frobenius. A formula for the sign of the constant term of $$P_\phi (X)$$ in terms of ‘leading coefficient’ of ϕ is given. General formula to determine signs of other coefficients of $$P_\phi (X)$$ is also derived. In the case where L is a global $$\mathbb{A}$$ -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’.