We construct integral models of Shimura varieties of abelian type with parahoric level structure over odd primes. These models are étale locally isomorphic to corresponding local models.

The main result of this article is the construction of a new class of weight shifting operators, similar to the theta operators of de Shalit and Goren (2019, Algebra Number Theory, 13, 1829–1877), Eischen et al. (2021, Algebra Number Theory, 15, 1469–1504), and others, which are defined on the lower Ekedahl–Oort strata of the geometric special fiber of unitary Shimura varieties of signature $(n-1, 1)$ at a good prime p, split in the reflex field E, which we assume to be quadratic imaginary. These operators act on certain graded sheaves which are obtained from the arithmetic structure of the EO strata, in particular the p-rank on each stratum. We expect these operators to have applications to the study of Hecke-eigensystems of $({\mathrm {mod}}\, p)$ modular forms and generalizations of the weight part of Serre’s conjecture.

We show that the moduli spaces of Scholze’s p-adic shtukas with framing satisfy a p-adic rigid analytic version of Borel’s extension theorem. In particular, this holds for local Shimura varieties, for all local Shimura data $(G, [b],\{\mu \})$, even for exceptional groups G, and extends work of Oswal-Shankar-Zhu-Patel who proved a p-adic Borel extension property for Rapoport-Zink spaces. As a corollary, we deduce that all these spaces satisfy a p-adic rigid analytic version of Brody hyperbolicity.

We extend the construction of the p-adic L-function interpolating unitary Friedberg–Jacquet periods in previous work of the author to include the p-adic variation of Maass–Shimura differential operators. In particular, we develop a theory of nearly overconvergent automorphic forms in higher degrees of coherent cohomology for unitary Shimura varieties generalising previous work for modular curves. The construction of this p-adic L-function can be viewed as a higher-dimensional generalisation of the work of Bertolini–Darmon–Prasanna and Castella–Hsieh, and the inclusion of this extra variable arising from the p-adic iteration of differential operators will play a key role in relating values of this p-adic L-function to p-adic regulators of special cycles on unitary Shimura varieties.

We prove an André–Oort-type result for a family of hypersurfaces in ${\mathbb{C}}^n$ that is both uniform and effective. Let $K_*$ denote the single exceptional imaginary quadratic field which occurs in the Siegel–Tatuzawa lower bound for the class number. We prove that, for $m, n \in {\mathbb{Z}}_{\gt0}$, there exists an effective constant $c(m, n)\gt0$ with the following property: if pairwise distinct singular moduli $x_1, \ldots, x_n$ with respective discriminants $\Delta_1, \ldots, \Delta_n$ are such that $a_1 x_1^m + \cdots + a_n x_n^m \in {\mathbb{Q}}$ for some $a_1, \ldots, a_n \in {\mathbb{Q}} \setminus \{0\}$ and $\# \{ \Delta_i \;:\; {\mathbb{Q}}(\sqrt{\Delta_i}) = K_*\} \leq 1$, then $\max_i \lvert \Delta_i \rvert \leq c(m, n)$. In addition, we prove an unconditional and completely explicit version of this result when $(m, n) = (1, 3)$ and thereby determine all the triples $(x_1, x_2, x_3)$ of singular moduli such that $a_1 x_1 + a_2 x_2 + a_3 x_3 \in {\mathbb{Q}}$ for some $a_1, a_2, a_3 \in {\mathbb{Q}} \setminus \{0\}$.

In this paper, we study the cohomology of the unitary unramified PEL Rapoport-Zink space of signature $(1,n-1)$ at hyperspecial level. Our method revolves around the spectral sequence associated to the open cover by the analytical tubes of the closed Bruhat-Tits strata in the special fiber, which were constructed by Vollaard and Wedhorn. The cohomology of these strata, which are isomorphic to generalized Deligne-Lusztig varieties, has been computed in an earlier work. This spectral sequence allows us to prove the semisimplicity of the Frobenius action and the non-admissibility of the cohomology in general. Via p-adic uniformization, we relate the cohomology of the Rapoport-Zink space to the cohomology of the supersingular locus of a Shimura variety with no level at p. In the case $n=3$ or $4$, we give a complete description of the cohomology of the supersingular locus in terms of automorphic representations.

We prove a general likely intersections theorem, a counterpart to the Zilber-Pink conjectures, under the assumption that the Ax-Schanuel property and some mild additional conditions are known to hold for a given category of complex quotient spaces definable in some fixed o-minimal expansion of the ordered field of real numbers.

For an instance of our general result, consider the case of subvarieties of Shimura varieties. Let S be a Shimura variety. Let $\pi :D \to \Gamma \backslash D = S$ realize S as a quotient of D, a homogeneous space for the action of a real algebraic group G, by the action of $\Gamma < G$, an arithmetic subgroup. Let $S' \subseteq S$ be a special subvariety of S realized as $\pi (D')$ for $D' \subseteq D$ a homogeneous space for an algebraic subgroup of G. Let $X \subseteq S$ be an irreducible subvariety of S not contained in any proper weakly special subvariety of S. Assume that the intersection of X with $\pi (gD')$ is persistently likely as g ranges through G with $\pi (gD')$ a special subvariety of S, meaning that whenever $\zeta :S_1 \to S$ and $\xi :S_1 \to S_2$ are maps of Shimura varieties (regular maps of varieties induced by maps of the corresponding Shimura data) with $\zeta $ finite, $\dim \xi \zeta ^{-1} X + \dim \xi \zeta ^{-1} \pi (gD') \geq \dim \xi S_1$. Then $X \cap \bigcup _{g \in G, \pi (g D') \text { is special }} \pi (g D')$ is dense in X for the Euclidean topology.

We construct a family of special cycle classes on the regular integral model of an orthogonal Shimura variety, and show that these cycle classes appear as Fourier coefficients of a Siegel modular form. Passing to the generic fiber of the Shimura variety recovers a result of Bruinier and Raum, originally conjectured by Kudla.

We show the Harris–Viehmann conjecture under some Hodge–Newton reducibility condition for a generalisation of the diamond of a non-basic Rapoport–Zink space at infinite level, which appears as a cover of the non-semi-stable locus in the Hecke stack. We show also that the cohomology of the non-semi-stable locus with coefficients coming from a cuspidal Langlands parameter vanishes. As an application, we show the Hecke eigensheaf property in Fargues’ conjecture for cuspidal Langlands parameters in the $ {\mathrm {GL}}_2$-case.

We give a construction of integral local Shimura varieties which are formal schemes that generalise the well-known integral models of the Drinfeld p-adic upper half spaces. The construction applies to all classical groups, at least for odd p. These formal schemes also generalise the formal schemes defined by Rapoport-Zink via moduli of p-divisible groups, and are characterised purely in group-theoretic terms.

More precisely, for a local p-adic Shimura datum $(G, b, \mu)$ and a quasi-parahoric group scheme ${\mathcal {G}} $ for G, Scholze has defined a functor on perfectoid spaces which parametrises p-adic shtukas. He conjectured that this functor is representable by a normal formal scheme which is locally formally of finite type and flat over $O_{\breve E}$. Scholze-Weinstein proved this conjecture when $(G, b, \mu)$ is of (P)EL type by using Rapoport-Zink formal schemes. We prove this conjecture for any $(G, \mu)$ of abelian type when $p\neq 2$, and when $p=2$ and G is of type A or C. We also relate the generic fibre of this formal scheme to the local Shimura variety, a rigid-analytic space attached by Scholze to $(G, b, \mu , {\mathcal {G}})$.

We consider the Harder–Narasimhan formalism on the category of normed isocrystals and show that the Harder–Narasimhan filtration is compatible with tensor products which generalizes a result of Cornut. As an application of this result, we are able to define a (weak) Harder–Narasimhan stratification on the $B_{\mathrm{dR}}^+$-affine Grassmannian for arbitrary $(G, b, \mu)$. When $\mu$ is minuscule, it corresponds to the Harder–Narasimhan stratification on the flag varieties defined by Dat, Orlik and Rapoport. Moreover, when b is basic, it has been studied by Nguyen and Viehmann, and Shen. We study the basic geometric properties of the Harder–Narasimhan stratification, such as non-emptiness, dimension and its relation with other stratifications.

We construct universal G-zips on good reductions of the Pappas-Rapoport splitting models for PEL-type Shimura varieties. We study the induced Ekedahl-Oort stratification, which sheds new light on the mod p geometry of splitting models. Building on the work of Lan on arithmetic compactifications of splitting models, we further extend these constructions to smooth toroidal compactifications. Combined with the work of Goldring-Koskivirta on group theoretical Hasse invariants, we get an application to Galois representations associated to torsion classes in coherent cohomology in the ramified setting.

We formulate Guo–Jacquet type fundamental lemma conjectures and arithmetic transfer conjectures for inner forms of $GL_{2n}$. Our main results confirm these conjectures for division algebras of invariant $1/4$ and $3/4$.

We consider Shimura varieties associated to a unitary group of signature $(n-s,s)$ where n is even. For these varieties, we construct smooth p-adic integral models for $s=1$ and regular p-adic integral models for $s=2$ and $s=3$ over odd primes p which ramify in the imaginary quadratic field with level subgroup at p given by the stabilizer of a $\pi $-modular lattice in the hermitian space. Our construction, which has an explicit moduli-theoretic description, is given by an explicit resolution of a corresponding local model.

We construct explicit generating series of arithmetic extensions of Kudla’s special divisors on integral models of unitary Shimura varieties over CM fields with arbitrary split levels and prove that they are modular forms valued in the arithmetic Chow groups. This provides a partial solution to Kudla’s modularity problem. The main ingredient in our construction is S. Zhang’s theory of admissible arithmetic divisors. The main ingredient in the proof is an arithmetic mixed Siegel-Weil formula.

We prove a conjecture of Pappas and Rapoport for all Shimura varieties of abelian type with parahoric level structure when $p>2$ by showing that the Kisin–Pappas–Zhou integral models of Shimura varieties of abelian type are canonical. In particular, this shows that these models are independent of the choices made during their construction, and that they satisfy functoriality with respect to morphisms of Shimura data.

For a smooth affine group scheme G over the ring of p-adic integers and a cocharacter $\mu $ of G, we develop the deformation theory for G-$\mu $-displays over the prismatic site of Bhatt–Scholze, and discuss how our deformation theory can be interpreted in terms of prismatic F-gauges introduced by Drinfeld and Bhatt–Lurie. As an application, we prove the local representability and the formal smoothness of integral local Shimura varieties with hyperspecial level structure. We also revisit and extend some classification results of p-divisible groups.

We prove new fundamental lemma and arithmetic fundamental lemma identities for general linear groups over quaternion division algebras. In particular, we verify the transfer conjecture and the arithmetic transfer conjecture from Li and Mihatsch (2023, Preprint, arXiv:2307.11716) in cases of Hasse invariant $1/2$.

We consider integral models of Hilbert modular varieties with Iwahori level structure at primes over p, first proving a Kodaira–Spencer isomorphism that gives a concise description of their dualizing sheaves. We then analyze fibres of the degeneracy maps to Hilbert modular varieties of level prime to p and deduce the vanishing of higher direct images of structure and dualizing sheaves, generalizing prior work with Kassaei and Sasaki (for p unramified in the totally real field F). We apply the vanishing results to prove flatness of the finite morphisms in the resulting Stein factorizations, and combine them with the Kodaira–Spencer isomorphism to simplify and generalize the construction of Hecke operators at primes over p on Hilbert modular forms (integrally and mod p).

The linear arithmetic fundamental lemma (AFL) is a conjectural identity of intersection numbers on Lubin–Tate deformation spaces and derivatives of orbital integrals. It was introduced for elliptic orbits by Li, and Howard and Li. For elliptic orbits, the relevant intersection problem is formulated for the basic isogeny class. In the present article, we extend the conjecture to all orbits and all isogeny classes. Our main result is a reduction of the non-basic cases of the AFL to the basic ones, which relies on an analysis of the connected-étale sequence. Our results will be relevant in the global setting, where also locally non-elliptic orbits may contribute in a non-trivial way.