Counting elliptic surfaces over finite fields

We count the number of isomorphism classes of elliptic curves of given height $d$ over the field of rational functions in one variable over the finite field of $q$ elements. We also estimate the number of isomorphism classes of elliptic surfaces over the projective line, which have a polarization of relative degree 3. This leads to an upper bound for the average 3-Selmer rank of the aforementionned curves. Finally, we deduce a new upper bound for the average rank of elliptic curves in the large $d$ limit, namely the average rank is asymptotically bounded by $1.5+O(1/q)$.