Quartic K3 surfaces without nontrivial automorphisms

We will deal with a gap in a result of Bjorn Poonen. He found explicit examples of hypersurfaces of degree d³3 and dimension n³1 over any field, such that the group of automorphisms over the algebraic closure is trivial, except for the pairs (n,d)=(1,3) or (2,4). Examples of the former pair, cubic curves, do not exist. We deal with the remaining case, quartic surfaces. For any field k of characteristic at most 19 we exhibit an explicit smooth quartic surface in projective threespace over k with trivial automorphism group over the algebraic closure of k. We also show how this can be extended to higher characteristics. Over the rationals we also construct an example on which the set of rational points is Zariski dense.