RELATIVE UNITARY RZ-SPACES AND THE ARITHMETIC FUNDAMENTAL LEMMA

We prove a comparison isomorphism between certain moduli spaces of $p$ -divisible groups and strict ${\mathcal{O}}_{K}$ -modules (RZ-spaces). Both moduli problems are of PEL-type (polarization, endomorphism, level structure) and the difficulty lies in relating polarized $p$ -divisible groups and polarized strict ${\mathcal{O}}_{K}$ -modules. We use the theory of relative displays and frames, as developed by Ahsendorf, Lau and Zink, to translate this into a problem in linear algebra. As an application of these results, we verify new cases of the arithmetic fundamental lemma (AFL) of Wei Zhang: The comparison isomorphism yields an explicit description of certain cycles that play a role in the AFL. This allows, under certain conditions, to reduce the AFL identity in question to an AFL identity in lower dimension.

A MODULI STACK OF TROPICAL CURVES

We contribute to the foundations of tropical geometry with a view toward formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework, the natural forgetful morphisms between moduli spaces of curves with marked points function as universal curves. Our approach to tropical geometry permits tropical moduli problems—moduli of curves or otherwise—to be extended to logarithmic schemes. We use this to construct a smooth tropicalization morphism from the moduli space of algebraic curves to the moduli space of tropical curves, and we show that this morphism commutes with all of the tautological morphisms.

Generators in formal deformations of categories

In this paper we use the theory of formal moduli problems developed by Lurie in order to study the space of formal deformations of a$k$-linear$\infty$-category for a field$k$. Our main result states that if${\mathcal{C}}$is a$k$-linear$\infty$-category which has a compact generator whose groups of self-extensions vanish for sufficiently high positive degrees, then every formal deformation of${\mathcal{C}}$has zero curvature and moreover admits a compact generator.

-modules on spaces of rational maps

Let$X$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.