Class numbers of definite binary quadratic lattices over algebraic function fields

Let k be an algebraic function field of one variable X having a finite field GF(q) of constants with q elements, q odd. Confined to imaginary quadratic extensions $\text{K}\text{k}$, class number formulas are developed for both the maximal and nonmaximal binary quadratic lattices L on (K, N), where N denotes the norm from K to k. The class numbers of L grow either with the genus g(k) of k (assuming the fields under consideration have bounded degree) or with the relative genus $\text{g(}\text{K}\text{k}\text{)}$ (assuming the lattices under consideration have bounded scale). In contrast to analogous theorems concerning positive definite binary quadratic lattices over totally real number fields, k is not necessarily totally real.