Endomorphism algebras of Kronecker modules regulated by quadratic function fields

Purely simple Kronecker modules M, built from an algebraically closed field K, arise from a triplet (m, h, alpha) where m is a positive integer, h: K boolean OR {infinity} -> {infinity, 0, 1, 2, 3,... }is a height function, and alpha is a K-linear functional on the. space K(X) of rational functions in one variable X. Every pair (h, a) comes with a polynomial f in K(X) [Y] called the regulator. When the module M admits non-trivial endomorphisms, f must be linear or quadratic in Y. in that case M is purely simple if and only if f is an irreducible quadratic. Then the K-algebra End M embeds in the quadratic function field K(X)[Y]/(f). For some height functions h of infinite support I, the search for a functional a for which (h, alpha) has regulator 0 comes down to having functions eta: I -> K such that no planar curve intersects the graph of eta on a cofinite subset. If K has characteristic not 2, and the triplet (m, h, alpha) gives a purely-simple Kronecker module M having non-trivial endomorphisms, then h attains the value infinity at least once on K boolean OR {infinity} and h is finite-valued at least twice on K boolean OR {infinity}. Conversely all these h form part of such triplets. The proof of this result hinges on the fact that a rational function r is a perfect square in K(X) if and only if r is a perfect square in the completions of K(X) with respect to all of its valuations.