Related papers: Manin problems for Shimura varieties of Hodge type

We construct a (smooth, projective) surface over the field of rational numbers, which is a counterexample to the Hasse principle not accounted for by the Manin obstruction. The construction relies on the classical 4-descent on elliptic…

We generalise Birch's seminal work on forms in many variables to handle a system of forms in which the degrees need not all be the same. This allows us to prove the Hasse principle, weak approximation, and the Manin-Peyre conjecture for a…

We study Nygaard-, conjugate-, and Hodge filtrations on the many variants of Breuil--Kisin modules associated to integral semi-stable Galois representations. This leads to an integral Sen operator satisfying certain ``$1$-degree shrinking"…

Given an integral p-adic variety, we observe that if the integral Hodge--de Rham spectral sequence behaves nicely, then the special fiber knows the Hodge numbers of the generic fiber. Applying recent advancements of integral p-adic Hodge…

We study the formal neighbourhood of a point in $\mu$-ordinary locus of an integral model of a Hodge type Shimura variety. We show that this formal neighbourhood has a structure of a shifted cascade. Moreover we show that the CM points on…

In this paper, we give a geometric construction of the Jacquet-Langlands transfer for automorphic forms of higher weights by studying the geometry of the mod $p$ fibres of different Hodge type Shimura varieties which satisfy a mild…

We prove the congruence relation for the mod-p reduction of Shimura varieties associated to a unitary similitude group GU(n-1,1), when p is inert and n odd. When n is even, this result was obtained by T. Wedhorn and O. B\"ultel using the…

We discuss and extend some of the results obtained in "Arakelov inequalities and the uniformization of certain rigid Shimura varieties" (math.AG/0503339), restricting ourselves to the two dimensional case, i.e. to surfaces Y mapping…

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $X$ a smooth proper $K$-variety with good reduction. Under a mild assumption on the behaviour of Hodge numbers under reduction modulo $p$, we prove that the existence of a non-zero global…

We investigate homological properties of perfect algebras of prime characteristic. The principle is as follows: perfect algebras resolve the singularities. For example, we show any module over the ring of absolute integral closure has…

Let $k$ be an algebraically closed field of positive characteristic $p$. We first classify the $D$-truncations mod $p$ of Shimura $F$-crystals over $k$ and then we study stratifications defined by inner isomorphism classes of these…

In this paper we prove sharp multipolar Hardy-type inequalities in the Riemannian $L^p-$setting for $p\geq 2$ using the method of super-solutions and fundamental results from comparison theory on manifolds, thus generalizing previous…

We give a new definition -- and in some cases, a new construction -- of integral canonical models of Shimura varieties that uses the notion of an aperture appearing in work of Gardner--Madapusi on some conjectures of Drinfeld. This applies…

Let $K$ be a complete discrete valuation field of characteristic $0$ with not necessarily perfect residue field of characteristic $p>0$. We define a Faltings extension of $\mathcal{O}_K$ over $\mathbb{Z}_p$, and we construct a Hodge-Tate…

We construct exotic Hecke correspondences between the special fibers of different PEL type Shimura varieties, when the local groups are restrictions of scalars of unramified groups. In particular, the local groups themselves are not…

We obtain existence results for a class of fully nonlinear Yamabe-type problems on non-compact manifolds, addressing both the so-called positive and negative cases. We also give explicit examples of manifolds with warped product ends and…

We prove a conjecture of Pappas and Rapoport for all Shimura varieties of abelian type with parahoric level structure when $p>3$ by showing that the Kisin-Pappas-Zhou integral models of Shimura varieties of abelian type are canonical. In…

Let $F$ be a totally real field in which a prime number $p>2$ is inert. We continue the study of the (generalized) Goren--Oort strata on quaternionic Shimura varieties over finite extensions of $\mathbb F_p$. We prove that, when the…

After R.~Schoen completed the solution to the Yamabe problem, compact manifolds could be categorized into three classes, depending on whether they admit a metric with positive, non-negative, or only negative scalar curvature. Here we follow…

We study the rigid cohomology of the ordinary locus in some compact PEL Shimura varieties of type C with values in automorphic local systems and use it to prove a small slope criterion for classicality of overconvergent Hecke eigenforms.…