Endomorphism Algebras of Kronecker Modules Regulated by Quadratic Function Fields

Abstract

Abstract. Purely simple Kronecker modules M, built from an algebraically closed field K, arise from a triplet (m, h, α) where m is a positive integer, h : K ∪ {∞} → {∞, 0, 1, 2, 3, . . . } is a height function, and α is a K-linear functional on the space K(X) of rational functions in one variable X. Every pair (h, α) comes with a polynomial f in K(X) [Y ] called the regulator. When the module M admits nontrivial 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 functionFor some height functions h of infinite support I, the search for a functional α for which (h, α) has regulator 0 comes down to having functions η : I → K such that no planar curve intersects the graph of η on a cofinite subset. If K has characterictic not 2, and the triplet (m, h, α) gives a purely-simple Kronecker module M having non-trivial endomorphisms, then h attains the value ∞ at least once on K ∪ {∞} and h is finite-valued at least twice on K ∪ {∞}. 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.