Local monodromy of Drinfeld modules

Abstract

Compared with algebraic varieties the local monodromy of Drinfeld modules appears to be hopelessly complex: The image of the wild inertia subgroup under Tate module representations is infinite save for the case of potential good reduction. Nonetheless we show that Tate modules of Drinfeld modules are ramified in a limited way: The image of a sufficiently deep ramification subgroup is trivial. This leads to a new invariant, the local conductor of a Drinfeld module. We establish an upper bound on the conductor in terms of the volume of the local period lattice. As an intermediate step we develop a theory of normed lattices in function field arithmetic including the notion of volume. We relate normed lattices to vector bundles on projective curves. An estimate on Castelnuovo-Mumford regularity of such bundles gives a volume bound on norms of lattice generators, and the conductor inequality follows. Last but not least we describe the image of inertia for Drinfeld modules of large residual rank. Just as in the theory of local $\ell$-adic Galois representations this image is commensurable with a commutative unipotent algebraic subgroup. However in the case of Drinfeld modules such a subgroup may be a product of several copies of $\mathbb{G}_a$.