Abstract. In the relative trace formula approach to the arithmetic Gan-Gross-Prasad conjecture, we formulate a local conjecture (arithmetic transfer) in the case of an exotic smooth formal moduli space of p-divisible groups, associated to a unitary group relative to a ramified quadratic extension of a p-adic field. We prove our conjecture in the case of a unitary group in three variables.

Abstract. We define various formal moduli spaces of p-divisible groups which are regular, and morphisms between them. We formulate arithmetic transfer conjectures, which are variants of the arithmetic fundamental lemma conjecture of [41] in the presence of ramification. These conjectures include the AT conjecture of [24]. We prove these conjectures in low-dimensional cases.

We define variants of PEL type of the Shimura varieties that appear in the context of the Arithmetic Gan-Gross-Prasad conjecture. We formulate for them a version of the AGGP conjecture. We also construct (global and semi-global) integral models of these Shimura varieties and formulate for them conjectures on arithmetic intersection numbers. We prove some of these conjectures in low dimension. Contents 1. Introduction 1 2. Group-theoretic setup 7 3. The Shimura varieties 8 4. Semi-global integral models 15 5. Global integral models 28 6. The Arithmetic Gan-Gross-Prasad conjecture 31 7. L-functions and the relative trace formula 39 8. The conjectures for the arithmetic intersection pairing 44 Appendix A. Sign invariants 55 Appendix B. Local models in the case of banal signature 59 References 61

Local models are schemes, defined in terms of linear algebra, that were introduced by Rapoport and Zink to study the étale-local structure of integral models of certain PEL Shimura varieties over p-adic fields. A basic requirement for the integral models, or equivalently for the local models, is that they be flat. In the case of local models for even orthogonal groups, Genestier observed that the original definition of the local model does not yield a flat scheme. In a recent article, Pappas and Rapoport introduced a new condition to the moduli problem defining the local model, the so-called spin condition, and conjectured that the resulting "spin" local model is flat. We prove a preliminary form of their conjecture in the split, Iwahori case, namely that the spin local model is topologically flat. An essential combinatorial ingredient is the equivalence of μ-admissibility and μ-permissibility for two minuscule cocharacters μ in root systems of type D.

We define variants of PEL type of the Shimura varieties that appear in the context of the arithmetic Gan–Gross–Prasad (AGGP) conjecture. We formulate for them a version of the AGGP conjecture. We also construct (global and semi-global) integral models of these Shimura varieties and formulate for them conjectures on arithmetic intersection numbers. We prove some of these conjectures in low dimension.

We survey the theory of local models of Shimura varieties. In particular, we discuss their definition and illustrate it by examples. We give an overview of the results on their geometry and combinatorics obtained in the last 15 years. We also exhibit their connections to other classes of algebraic varieties.

Local models are certain schemes, defined in terms of linear-algebraic moduli problems, which giveétale-local neighborhoods of integral models of certain p-adic PEL Shimura varieties defined by Rapoport and Zink. When the group defining the Shimura variety ramifies at p, the local models (and hence the Shimura models) as originally defined can fail to be flat, and it becomes desirable to modify their definition so as to obtain a flat scheme. In the case of unitary similitude groups whose localizations at Qp are ramified, quasi-split GUn, Pappas and Rapoport have added new conditions, the so-called wedge and spin conditions, to the moduli problem defining the original local models and conjectured that their new local models are flat. We prove a preliminary form of their conjecture, namely that their new models are topologically flat, in the case n is odd.

Abstract. Local models are schemes which are intended to model theétale-local structure of p-adic integral models of Shimura varieties. Pappas and Zhu have recently given a general group-theoretic construction of flat local models with parahoric level structure for any tamely ramified group, but it remains an interesting problem to characterize the local models, when possible, in terms of an explicit moduli problem. In the setting of local models for ramified, quasi-split GUn, work towards an explicit moduli description was initiated in the general framework of Rapoport and Zink's book and was subsequently advanced by Pappas and Pappas-Rapoport. In this paper we propose a further refinement to their moduli problem, which we show is both necessary and sufficient to characterize the (flat) local model in a certain special maximal parahoric case with signature (n − 1, 1).