Endomorphism Algebras of Kronecker Modules Regulated by Quadratic Function Fields

Okoh, F. ; Zorzitto, F.

Canadian journal of mathematics, Tome 59 (2007) no. 1, pp. 186-210

Résumé

Purely simple Kronecker modules $M$ , built from an algebraically closed field $K$ , arise from a triplet $\left( m,\,h,\,\alpha \right)$ where $m$ is a positive integer, $h:\,K\,\cup \,\{\infty \}\,\to \,\{\infty ,\,0,\,1,\,2,\,3,\,.\,.\,.\,\}$ is a height function, and $\alpha$ is a $K$ -linear functional on the space $K\left( X \right)$ of rational functions in one variable $X$ . Every pair $\left( h,\,\alpha \right)$ comes with a polynomial $f$ in $K\left( X \right)\left[ Y\, \right]$ 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 ${K\left( X \right)\left[ Y\, \right]}/{(f\,)}\;$ . For some height functions $h$ of infinite support $I$ , the search for a functional $\alpha$ for which $\left( h,\,\alpha \right)$ has regulator $0$ comes down to having functions $\eta \,:\,I\,\to \,K$ such that no planar curve intersects the graph of $\eta$ on a cofinite subset. If $K$ has characterictic not $2$ , and the triplet $\left( m,\,h,\,\alpha \right)$ gives a purely-simple Kronecker module $M$ having non-trivial endomorphisms, then $h$ attains the value $\infty$ at least once on $\text{K}\,\cup \,\text{ }\!\!\{\!\!\text{ }\infty \text{ }\!\!\}\!\!\text{ }$ and $h$ is finite-valued at least twice on $\text{K}\,\cup \,\text{ }\!\!\{\!\!\text{ }\infty \text{ }\!\!\}\!\!\text{ }$ . 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 $K\left( X \right)$ if and only if $r$ is a perfect square in the completions of $K\left( X \right)$ with respect to all of its valuations.

Détail

DOI : 10.4153/CJM-2007-008-7

Mots-clés : 16S50, 15A27, Purely simple Kronecker module, regulating polynomial, Laurent expansions, endomorphism algebra

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