Abstract. Let Fq (q = p ) be a finite field. In this note the number of irreducible polynomials of degree m in Fq[x] with prescribed trace and norm coefficients is calculated in certain special cases and general bounds for that number are obtained. As a corollary, sharp bounds are obtained for the number of elements in Fq3 with prescribed trace and norm over Fq improving the estimates by Katz. ...

Introduction. In [Ge1] Gekeler classified all elliptic curves over F2r (T ) with one rational place of multiplicative reduction (without loss of generality located at ∞), one further rational place of bad reduction (without loss of generality located at 0) and good reduction elsewhere. So these curves have conductor ∞ · T where n is a natural number (which actually can be arbitrarily large). In...