More regular formal moduli spaces and arithmetic transfer conjectures: the ramified quadratic case

TL;DR

This paper develops new regular formal moduli spaces for unitary groups over ramified quadratic extensions, formulates related arithmetic transfer conjectures, and proves these in low-dimensional cases, advancing understanding of arithmetic geometry in this setting.

Contribution

It introduces novel regular formal moduli spaces, characterizes special divisors, and formulates and proves arithmetic transfer conjectures for ramified quadratic cases.

Findings

Defined regular formal moduli spaces with parahoric levels

Characterized exceptional special divisors

Proved conjectures in lowest dimensional cases

Abstract

For unitary groups associated to a ramified quadratic extension of a $p$-adic field, we define various regular formal moduli spaces of $p$-divisible groups with parahoric levels, characterize exceptional special divisors on them, and construct correspondences between them. We formulate arithmetic transfer conjectures, which are variants of the arithmetic fundamental lemma conjecture in this context. We prove the conjectures in the lowest dimensional cases.