-modules on spaces of rational maps

Abstract

AbstractLet$X$be an algebraic curve. We study the problem of parametrizing geometric structures over$X$which are only generically defined. For example, parametrizing generically defined maps (rational maps) from$X$to a fixed target scheme$Y$. There are three methods for constructing functors of points for such moduli problems (all originally due to Drinfeld), and we show that the resulting functors are equivalent in the fppf Grothendieck topology. As an application, we obtain three presentations for the category of$D$-modules ‘on’$B(K)\backslash G(\mathbb{A})/G(\mathbb{O})$, and we combine results about this category coming from the different presentations.